First Principles Thermoelasticity of Minerals: Insights into the Earth

1
First Principles Thermoelasticity of
MineralsInsights into the Earths LM
Renata M. Wentzcovitch U.
of Minnesota (USA) and SISSA (Italy)
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
2
The Contribution from Seismology
Longitudinal (P) waves
Transverse (S) wave
from free oscillations
3
PREM (Preliminary Reference Earth
Model)(Dziewonski Anderson, 1981)
P(GPa)
0
24
135
329
364
4
Mantle Mineralogy
MgSiO3
Pyrolite model ( weight)
opx
100
4
Olivine
SiO2 45.0 MgO 37.8 FeO
8.1 Al2O3 4.5 CaO 3.6 Cr2O3
0.4 Na2O 0.4 NiO
0.2 TiO2 0.2 MnO
0.1 (McDonough and Sun, 1995)
8
cpx
(Mg1--x,Fex)2SiO4
300
(Mg,Ca)SiO3
12
P (Kbar)
Depth (km)
garnets
16
500
?-phase
()
(Mg,Al,Si)O3
20
spinel
()
700
perovskite
MW
(Mg,Fe) (Si,Al)O3
CaSiO3
60
20
40
80
100
0
(Mg1--x,Fex) O
V
5
Mantle convection
6
Temperature and Composition of LM
7
Lateral Heterogeneities
8
3D Maps of Vs and Vp
(Masters et al, 2000)
Vs
V?
Vp
9
Lateral variations in VS and VP
(Karato Karki, JGR 2001)
(MLDB-Masters et al., 2000) (KWH-Kennett et al.,
1998) (SD-Su Dziewonski, 1997) (RW-Robertson
Woodhouse,1996)
10
Lateral variations in V? and VP
(MLDB-Masters et al., 2000) (SD-Su Dziewonski,
1997)
(Karato Karki, 2001)
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Relations
0.42 A 0.37
with
(Karato Karki, 2001)
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Anisotropy
?

?
isotropic
azimuthal
VP VS1 VS2
VP (?,?) VS1 (?,?) ? VS2 (?,?)
transverse
VP (?) VS1 (?) ? VS2 (?)
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Anisotropy in the Earth
(Karato, 1998)
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Mantle Anisotropy
SHgtSV
SVgtSH
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Slip systems and LPO
Zinc wire
Slip system
F
16
Anisotropic Structures
(SPO)
(LPO)
Shape Preferred Orientation
Lattice Preferred Orientation
Mantle flow geometry
LPO
Seismic anisotropy
slip system
Cij
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Mineral sequence II
Upper Mantle
Transition Zone
410 km
(Spinel)
(520 km (?))
(Mgx,Fe(1-x))2SiO4 (Olivine)
Lower Mantle
670 km

(Mgx,Fe(1-x))O
(Mgx,Fe(1-x))SiO3
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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 Theory
• xxxxxxxxxxxxxxxxxx(Gianozzi, Baroni, and de
  Gironcoli, 1991)
• 
  (http//www.pwscf.com)
• Soft separable pseudopotentials
  (Troullier-Martins)

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Typical Computational Experiment
Damped dynamics (Wentzcovitch, 1991)
P 150 GPa
20
abcxP
(a,b,c)th lt (a,b,c)exp 1
Tilt angles ?th - ?exp lt 1deg
Kth 259 GPa Kth3.9
Kexp 261 GPa Kexp4.0
( Wentzcovitch, Martins, Price, 1993)
( Ross and Hazen, 1989)
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Elastic constant tensor ?
?ij
cijkl
?kl
?kl
equilibrium structure
(i,j) m
re-optimize
Crystal (Pbnm)
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Elastic Waves
P-wave (longitudinal)
S-waves (shear)
n propagation direction
Yegani-Haeri, 1994 Wentzcovitch et al, 1995 Karki
et al, 1997
within 5
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Wave velocities in perovskite (Pbnm)
Cristoffels eq.
with
is the propagation direction
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Anisotropy
P-azimuthal S-azimuthal
S-polarization
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Poly-Crystalline aggregate
Voigt-Reuss averages

Voigt uniform strain
Reuss uniform stress
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Polarization anisotropy in transversely isotropic
medium
(Karki et al. 1997 Wentzcovitch et al1998)
Seismic anisotropy Isotropic in bulk LM 2 VSH gt
VSV in
-
-
SH/SV Anisotropy ()
High P, slip systems MgO 100 ?
(c44 lt c11-c12) MgSiO3 pv 010 ? (soft
c55)
-
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Theory x PREM
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Acoustic Velocities of Potential LM Phases
(Karki, Stixrude, Wentzcovitch,2001)
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Effect of Fe alloying

• (Kiefer, Stixrude,Wentzcovitch,2002)
• (Mg0.75Fe0.25)SiO3
• ?K (P0 GPa) 2
• ?K (P135 GPa) 1
• ?G (P 0 GPa) - 6
• ?G (P 135 GPa) - 8

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TM of mantle phases
CaSiO3
(Mg,Fe)SiO3
5000
Mw
Core T
4000
HA
solidus
T (K)
3000
Mantle adiabat
2000
peridotite
0
40
20
60
80
100
120
P(GPa)
(Zerr, Diegler, Boehler, 1998)
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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, Cijs).

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Phonon dispersions in MgO
(Karki, Wentzcovitch, de Gironcoli and Baroni,
PRB 61, 8793, 2000)
-
Exp Sangster et al. 1970
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Phonon dispersion of MgSiO3 perovskite
Calc Exp
-
Calc Exp
0 GPa
-
Calc Karki, Wentzcovitch, Gironcoli, Baroni
PRB 62, 14750, 2000 Exp Raman Durben and
Wolf 1992 Infrared Lu et al. 1994
100 GPa
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Quasiharmonic approximation
MgO
-
static
zero-point
-
F (Ry)
-
thermal
-
4th order finite strain equation of state
Static 300K Exp (Fei
1999) V (Å3) 18.5 18.8
18.7 K (GPa) 169 159 160 K
4.18 4.30
4.15 K(GPa-1) -0.025 -0.030
Volume (Å3)
35
Thermal expansivity of MgO and MgSiO3
(Karki, Wentzcovitch, de Gironcoli and Baroni,
Science 286, 1705, 1999)
? (10-5 K-1)
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Elastic moduli of MgO
(Karki, Wentzcovitch, de Gironcoli and Baroni,
Science 286, 1705, 1999)
EoS K (c11 2c12 )/3 Tetragonal strain cs
c11 - c12 Shear strain c44
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Elastic moduli of MgO at high P and T
(Karki et al., Science 1999)
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Elastic anisotropy of MgO
(Karki et al., 1997, 1999)
Velocity anisotropy
-
39
Adiabatic bulk modulus at LM P-T
(Karki, Wentzcovitch, de Gironcoli and Baroni,
GRL, 2001)
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LM geotherms
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Elasticity of MgSiO3 at LM Conditions
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Adiabatic Moduli
where
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Seismic Velocities
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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.

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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!!

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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