Quasiharmonic Thermodynamic Properties of Minerals

Renata M. M. Wentzcovitch

Department of Chemical Engineering and Materials Science Minnesota Supercomputer Institute U. of Minnesota • Motivation

• First Principles Thermodynamic Method How reliable is it?

• Examples MgSiO3- Ilmenite to perovskite phase transition Thermoelasticity of perovskite Crystal structures at high (P,T)

• Summary

•

The Contribution from Seismology

VP K

4

3G

VS G

Longitudinal (P) waves

Transverse (S) wave

from free oscillations

“660 km” topographyJ. M. Kendall, 2000

Seismic Discontinuities and Phase Transitions

PREM Dziewonski and Anderson, 1981

Methods

• Local Density Approximation

• Soft norm-conserving pseudopotentials

• Born-Oppenheimer variable cell shape molecular dynamics

• Density functional perturbation theory for phonons

Thermodynamic Method

qj B

qjB

qj

qj

Tk

VTk

VVUTVF

)(exp1ln

2

)()(),(

• VDoS and F(T,V) within the QHA

PVTSFG TV

FP

VT

FS

N-th (N=3,4,5…) order isothermal (eulerian or logarithm) finite strain EoS

IMPORTANT: structural parameters and phonon frequencies depend on volume alone!!….

equilibrium structure

i

re-optimize

(Thermo) Elastic constant tensor

ji

Tij

GPTc

2

),(

V

jiTij

Sij C

VTPTcPTc

),(),(

Tii

S

Zero Point Motion Effect

Volume (Å3)

F (

Ry)

MgO

Static 300K Exp (Fei 1999)V (Å3) 18.5 18.8 18.7K (GPa) 169 159 160K´ 4.18 4.30 4.15K´´(GPa-1) -0.025 -0.030

-

-

Elasticity of MgO

(Karki et al., Science 1999)

MgSiO3-Akimotoite to perovskite transition

From Fukao et al., Rev. Geophys. (2001)

dS

dV

dP

dT

Clapeyron equation:

00 dP

dTdS

P

T

Tc

Pc

Ak

Pv

T<Tc

P>Pc

Akimotoite bearing slab

Transformation inhibited in cold regions!!

23 GPa1980 K

MgSiO3-ilmenite (Akimotoite)

Si2O3 layer

Mg2O3 layer1.77 A < Si-O < 1.83 A

1.99 A < Mg-O < 2.16 A

oo

o o

corundum

ilmenite

LiNbO3

Mg Si

Al

SiMgR3

MgSiO3-perovskite (Pbnm)

SiO3 octahedra

1.78 A < Si-O < 1.80 A

2.01 A < Mg-O < 3.12 Ao o

o o

Phonon dispersion of MgSiO3-ilmenite and perovskite

Calc Exp

Calc Exp

Pv: Raman [Durben and Wolf 1992] Infrared [Lu et al. 1994]

0 GPa

Calc Exp

Aaaaa

Aaaaaaa

Ak: Raman [Reynard and Rubie, 1996] Infrared [Madon and Price, 1989]

Octahedraldeformation

Octahedraldeformation

Mg displacement

Mg displacement

Octahedral rotation

NEW!

30

25

20

15

10

5

0

Pre

ssur

e (G

Pa)

2000150010005000

Temperature (K)

akimotoite

perovskite

MgSiO3

Theory

Experiment

Static

Pre

ssur

e (G

Pa)

Temperature (K)

MgSiO3

akimotoite

perovskiteStatic

Experiment

Theory

Thermodynamic phase boundary

Issue I: Change in PT after inclusion of zero point motion energy (Ezp)

Issue II: discrepancy between theory and experiments

Exp:Ito & Takahashi (1996)

Gil(P,T) X Gpv(P,T)

“…Useful rule…”

•Issue I

ix

iB

xB i

i

e

xkekTVS

11ln),(

00 zpEdP

dT

dV

dSPc decreases

F(V

,T)

V

pv

ak

Ezp shiftsPc

Tkx

B

ii

30

25

20

15

10

5

0

Pre

ssur

e (G

Pa)

2000150010005000

Temperature (K)

akimotoite

perovskite

MgSiO3

Theory

Experiment

Static

Pre

ssur

e (G

Pa)

Temperature (K)

MgSiO3

akimotoite

perovskiteStatic

Experiment

Theory

Thermodynamic phase boundary

Issue I: Change in PT after inclusion of zero point motion energy (Ezp)

Issue II: discrepancy between theory and experiments

Exp:Ito & Takahashi (1996)

Gil(P,T) X Gpv(P,T)

…a posteriori criterion for the validity of the QHA

(

10-5 K

-1)

MgSiO3

Karki et al, GRL (2001)

Issue II…

30

25

20

15

10

5

0

Pre

ssur

e (G

Pa)

2000150010005000

Temperature (K)

akimotoite

perovskite

MgSiO3

Theory

Experiment

Static

Pre

ssur

e (G

Pa)

Temperature (K)

MgSiO3

akimotoite

Static

Experiment

Theory

Exp:Ito & Takahashi (1996)

perovskite

Not OK!!

QHA OK

Properties of MgSiO3-perovskite and -ilmenite

(gr/cm-3)

V (A3)

KT

(GPa) d KT/dP d KT

2/dP2

(GPa-1) d KT/dT (Gpa K-1)

10-5 K-1

3.580 18.80 159 4.30 -0.030 -0.014 3.12 Calc. MW

3.601 18.69 160 4.15 ~ -0.0145 3.13 Exp. MW

4.210 164.1 247 4.0 -0.016 -0.031 2.1 Calc. Pv

4.247 162.3 246 | 266

3.7 | 4.0

~ -0.02 | -0.07

1.7 | 2.2

Exp. Pv

Exp.: [Ross & Hazen, 1989; Mao et al., 1991; Wang et al., 1994; Funamori et al., 1996; Chopelas, 1996; Gillet et al., 2000; Fiquet et al., 2000; Weidner & Ito, 1985; Reynard& Rubie, 1996; Hofmeister and Ito, 1992; Chopelas, 1999]

Ak

Ak

1.88

1.67 |2.44

-0.025

~

-0.042

4.8

4.7201

212

176.8

175.2

3.908

3.943

Pv

Pv

(256)

Ad hoc correction to DFT results…

GPaGPaLDA VV 0

exp0 %1.1

0exp

GPaV

V GPaGPaLDA VV 0

exp5.2

(perovskite)

Ad hoc correction to DFT results…

GPaGPaLDA VV 0

exp0

GPaGPaLDA KK 0

exp0

%1.10

exp

GPaV

V GPaGPaLDA VV 0

exp5.2

GPaGPaLDA KGPaK 0

exp5.2 257 !!!...

(perovskite)

but…

Ad hoc correction to DFT results…

GPaGPaLDA VV 0

exp0

GPaGPaLDA KK 0

exp0

%1.10

exp

GPaV

V GPaGPaLDA VV 0

exp5.2

GPaGPaLDA KGPaK 0

exp5.2 257 !!!...

)()( exp VKVKLDA ?!

(perovskite)

but…

Ad hoc correction to DFT results…

GPaGPaLDA VV 0

exp0

GPaGPaLDA KK 0

exp0

%1.10

exp

GPaV

V GPaGPaLDA VV 0

exp5.2

GPaGPaLDA KGPaK 0

exp5.2 257 !!!...

)()( exp VKVKLDA ?!

CVPVPLDA )()( exp

(perovskite)

but…

EoS for Perovskite

C = 2.5 GPa

EoS for Ilmenite

C = 1.9 GPa Exp.:Reynard et al., 1996

Calc.:Karki & Wentzcovitch, 2002.

Ad hoc correction to Pc…

'5.2)()( CVVFVF pvLDA

pvideal

(ilmenite to perovskite)

''9.1)()( CVVFVF ilLDA

ilideal

Pc at 300K should increase

(not really conclusive…!!)

cij

(Wentzcovitch, Karki, Cococciono, de Gioroncoli, 2003)

300 K1000K2000K3000 K4000 K

(Oganov et al,2001)

Cij(P,T)

…IMPORTANT: structural parameters and phonon frequencies depend on volume alone!!

• Structures at high P are determined at T= 0

P(V,0)

• P’(V,T’) within the QHA

• At T 0… V(P’,T’)=V(P,0) structure(P’,T’) = structure(P,0)

Corresponding States

Comparison with Experiments(Ross & Hazen, 1989)

77 K < T < 400K

0 GPa < P < 12 GPa

o

o

o

Calc.

Comparison with Experiments(Ross & Hazen, 1989)

77 K < T < 400K

0 GPa < P < 12 GPa

o

o

o

Calc.

LDALDA+ZPExp.

(Funamori et al., 1996)

300 K < T < 2000 K

21 GPa < P < 29 GPa

(Fiquet et al., 1998)

300 K < T < 2000 K

26 GPa < P < 58 GPa

Predictions a,b,c(P,T)

4000 K3000 K2000 K1000 K 300 K

Summary

LDA + QHA is a good and useful FP method for high P,T thermodynamics (..lots of insights)

The validity criterion based on suggests avoidance of phase boundaries

Prediction of high P,T crystal structures through corresponding states

Acknowledgements

Bijaya B. Karki (LSU)

Stefano de Gironcoli, Stefano Baroni, Matteo Coccocioni (SISSA, Italy)

NSF-EAR and NSF-COMPRES, SISSA and INFM (Italy)