First Principles Thermoelasticity of Minerals:Insights into the Earth’s LM

• Problems related with seismic observations T and composition in the lower mantle Origin of lateral heterogeneities Origin of anisotropies

• How and what we calculate MgSiO3-perovskite MgO

• Geophysical inferences

• Future directions

Renata M. Wentzcovitch U. of Minnesota (USA) and SISSA (Italy)

The Contribution from Seismology

VP K

4

3G

VS G

Longitudinal (P) waves

Transverse (S) wave

from free oscillations

PREM (Preliminary Reference Earth Model)

(Dziewonski & Anderson, 1981)

0 24 135 329 364P(GPa)

Mantle Mineralogy

SiO2 45.0MgO 37.8FeO 8.1Al2O3 4.5CaO 3.6Cr2O3 0.4Na2O 0.4NiO 0.2TiO2 0.2MnO 0.1

(McDonough and Sun, 1995)

Pyrolite model (% weight)

De

pth

(km

)

P (

Kba

r)

V %

8

4

12

16

20

6020 40 80 1000

100

300

500

700

Olivine

perovskite

-phase

spinel

MW

garnets

opx

cpx(Mg1--x,Fex)2SiO4

(‘’)

(‘’)

MgSiO3

(Mg,Al,Si)O3

(Mg,Fe) (Si,Al)O3

(Mg1--x,Fex) O

(Mg,Ca)SiO3

CaSiO3

Mantle convection

Temperature and Composition of LM

Lateral Heterogeneities

3D Maps of Vs and Vp

Vs V Vp

(Masters et al, 2000)

RS / P lnVs

lnVP P

(MLDB-Masters et al., 2000)(KWH-Kennett et al., 1998)(SD-Su & Dziewonski, 1997)(RW-Robertson & Woodhouse,1996)

Lateral variations in VS and VP

(Karato & Karki, JGR 2001)

R / S lnV

lnVS P

(MLDB-Masters et al., 2000)(SD-Su & Dziewonski, 1997)

Lateral variations in V and VP

(Karato & Karki, 2001)

Relations

RS / P 1

(1 A)R / S AP

A 4VS

2

3VP2

0.42 ≤ A ≤ 0.37with

R / S (S 1)

( 1)P

S lnKS

lnP

lnG

lnP

(Karato & Karki, 2001)

Anisotropy

isotropic

transverse

azimuthal

VP

VS1= VS2

VP () VS1 () VS2 ()

VP (,) VS1 (,) VS2 (,)

(VSH VSV)%

VS

Anisotropy in the Earth

(Karato, 1998)

Mantle AnisotropySH>SVSV>SH

Slip system

Zinc wire

F

Slip systems and LPO

Lattice Preferred Orientation (LPO) Shape Preferred Orientation

(SPO)

Mantle flow geometry

LPO Seismic anisotropy

slipsystem Cij

Anisotropic Structures

+

Mineral sequence II

Lower Mantle

410 kmTransition Zone

(520 km (?))

670 km

(Mgx,Fe(1-x))O

(Mgx,Fe(1-x))SiO3

(Mgx,Fe(1-x))2SiO4 (Olivine)

Upper Mantle

(Spinel)

Method

• Structural optimizations

• First principles variable cell shape MD for structural optimizations xxxxxxxxxxxxxxxxxx(Wentzcovitch, Martins,& Price, 1993)

• Self-consistent calculation of forces and stresses (LDA-CA)

• Phonon thermodynamics

• Density Functional Perturbation Theoryxxxxxxxxxxxxxxxxxx(Gianozzi, Baroni, and de Gironcoli, 1991)

(http://www.pwscf.com)

• Soft & separable pseudopotentials (Troullier-Martins)

Typical Computational Experiment

Damped dynamics (Wentzcovitch, 1991)

)(~ PI),(~ int rffr

P = 150 GPa

abcxP

K Vo

dP

dV

Kth = 259 GPa K’th=3.9

Kexp = 261 GPa K’exp=4.0

(a,b,c)th < (a,b,c)exp ~ 1%

Tilt angles th - exp < 1deg

( Wentzcovitch, Martins, & Price, 1993)

( Ross and Hazen, 1989)

c11 c12 c13 * * *

c21 c22 c23 * * *

c31 c32 c33 * * *

* * * c44 * *

* * * * c55 *

* * * * * c66

(m P (m;1,2,3 )) cmnn

• Crystal (Pbnm)

equilibrium structure

kl

re-optimize

kl

ijcijkl

(i,j) m

Elastic constant tensor

Yegani-Haeri, 1994Wentzcovitch et al, 1995Karki et al, 1997

within 5%

S-waves (shear)

P-wave (longitudinal)

n propagation direction

Elastic Waves

VI2 iI ikkICristoffel’s eq.: with ik cijkln jnl

n is the propagation direction

Wave velocities in perovskite (Pbnm)

Anisotropy

P-azimuthal:

S-azimuthal:

S-polarization:

avP

PPP

V

VVV

minmax

avS

SSS

V

VVV

minmax

avS

SS

SV

VVV

max

21

C11 C12 C12 * * *

C12 C11 C12 * * *

C12 C12 C11 * * *

* * * C44 * *

* * * * C44 *

* * * * * C44

C44G

C12 C11 2C44

C11 2C12

3K•Voigt: uniform strain

MN

MN ij

ij

MN CMNRSRS

MN SMNRSRS

CMNRS niMnj

NnkRnlagg S cijklf(n)

SMNRS niMnj

NnkRnlagg S sijklf(n)

•Reuss: uniform stress

•Voigt-Reuss averages:

CMNRS CMNRS SMNRS

1

2

• Poly-Crystalline aggregate

Polarization anisotropy in transversely isotropic medium

High P, slip systems

MgO: {100} ? (c44 < c11-c12)

MgSiO3 pv: {010} ? (soft c55)

D

Seismic anisotropy

Isotropic in bulk LM2% VSH > VSV in

SH

/SV

Ani

sotr

opy

(%)

(Karki et al. 1997; Wentzcovitch et al1998)

-

-

-

Theory x PREM

Acoustic Velocities of Potential LM Phases

(Karki, Stixrude, Wentzcovitch,2001)

Effect of Fe alloying

(Kiefer, Stixrude,Wentzcovitch,2002)

(Mg0.75Fe0.25)SiO3

K (P=0 GPa) = + 2%K (P=135 GPa) = + 1%G (P = 0 GPa) = - 6%G (P = 135 GPa) = - 8%

TM of mantle phases

Core T

Mantle adiabat

solidusHA

Mw

(Mg,Fe)SiO3

CaSiO3

peridotite

P(GPa)0 4020 60 80 100 120

2000

3000

4000

5000

T (

K)

(Zerr, Diegler, Boehler, 1998)

High Temperature calculations

• MgO and MgSiO3 perovskite

• Phonon dispersions from density functional perturbation theory (DFPT).

• Quasiharmonic approximation (QHA) and thermal properties (e.g., , CP, S, KS,T, Cij’s).

Phonon dispersions in MgO

Exp: Sangster et al. 1970

(Karki, Wentzcovitch, de Gironcoli and Baroni, PRB 61, 8793, 2000)

-

Phonon dispersion of MgSiO3 perovskite

Calc Exp

Calc Exp

Calc: Karki, Wentzcovitch, Gironcoli, Baroni PRB 62, 14750, 2000

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

0 GPa

100 GPa

--

Quasiharmonic approximation

qj B

qjB

qj

qj

Tk

VTk

VVUTVF

)(exp1ln

2

)()(),(

Volume (Å3)

F (

Ry)

4th order finite strain equation of state

static zero-point

thermal

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

-

-

-

-

Thermal expansivity of MgO and MgSiO3

(Karki, Wentzcovitch, de Gironcoli and Baroni, Science 286, 1705, 1999)

(

10-5 K

-1)

Elastic moduli of MgO

EoS:K = (c11 + 2c12 )/3

Tetragonal strain: cs = c11 - c12

Shear strain: c44

(Karki, Wentzcovitch, de Gironcoli and Baroni, Science 286, 1705, 1999)

ij

jiijcPTGPTG 2

1),,0(),,(

�

Elastic moduli of MgO at high P and T(Karki et al., Science 1999)

Elastic anisotropy of MgO

12

11

1244

c

ccA

A P 2A

3

A S1 A

2

A S2 3A

8

(Karki et al., 1997, 1999)

Velocity anisotropy

-

Adiabatic bulk modulus at LM P-T(Karki, Wentzcovitch, de Gironcoli and Baroni, GRL, 2001)

LM geothermsLM Geotherms

1000

2000

3000

4000

5000

6000

500 1000 1500 2000 2500 3000

T (

K)

Depth (km)

Pv

Solidus

Isentropes

Pyrolite

CMB|

Tc

Elasticity of MgSiO3 at LM Conditions

Adiabatic Moduli

Vji

Tij

Sij C

TVTPcTPc ),(),(

where

Tii

S

Seismic Velocities

Summary• Building a consistent body of knowledge obout LM phases

• QHA is suitable for studying thermal properties of minerals at

LM conditions

• A homogeneous and adiabatic LM model appears to be incompatible with PREM.

• LPO in aggregates of MgO and MgSiO3 can exhibit strong anisotropy at LM conditions.

• We have all ingredients now to re-examine what has been said about lateral variations.

Future directions

• Properties of solid solutions, e.g., Fe, Al, bearing perovskites and oxides

• Rheology (deformations, slip systems, diffusion, anelasticity) of materials

• Computationally intensive, e.g, large-scale MD simulations!!

Acknowledgements

Bijaya B. Karki (U. Of MN)

Lars Stixrude (Ann Arbor)Shun-ichiro Karato (U. of MN)

Stefano de Gironcoli (SISSA)Stefano Baroni (SISSA)

Funding: NSF/EAR