Transcript First Principles Thermoelasticity of Mantle Minerals

Thermoelasticity of (Mg,Fe)SiO3perovskite at lower mantle conditions
Renata M. M. Wentzcovitch
Department of Chemical Engineering and Materials Science
U. of Minnesota, Minneapolis
• Research in the early 90’s (first principles MD)
• Current research (NSF/EAR funded)
Geophysical motivation (thermo-chemical state of the LM)
Thermoelasticity of (Mg,Fe)SiO3 and MgO
Comparisons with PREM
• Summary
Research in the early nineties
• Development of a variable cell shape (VCS) molecular
dynamics (MD) method (Wentzcovitch, PRB,1991)
• Development of first principles MD
I. Self-consistent method with iterative diagonalization
used in MD simulations (Wentzcovitch and Martins, SSC,1991)
II. Implementation of finite temperature DFT
(Wentzcovitch, Martins, and Allen, PRB ,1992)
• Some original applications of combined methodologies
Collaborators: J. L. Martins (INESC, Lisbon) and
P. B. Allen (SUNY-Stony Brook, CHiPR)
Methods
• Density Functional Theory ( E[n(r )], H
 E )
• Local Density Approximation
• First Principles Pseudopotentials
•


( ikr )
e
u
Plane-wave expansion ( (r )  
)
k

k
• Self-consistent Forces and Stresses (molecular dynamics)
First Principles VCS-MD
(Wentzcovitch, Martins, Price, PRL 1993)
Damped dynamics
 ~ (  PI )
r ~ f int  f (r, )
MgSiO3
P = 150 GPa
Lattice
(a,b,c)th < (a,b,c)exp ~ 1%
Tilt angles
th - exp < 1deg
dP
K  Vo
dV
Kth = 259 GPa K’th=3.9
Kexp = 261 GPa K’exp=4.0
(• Wentzcovitch, Martins, & Price, 1993)
( Ross and Hazen, 1989)
Acknowledgements
• David Price (UCL-London)
• Lars Stixrude (U. of Michigan, Ann Arbor)
• Shun-ichiro Karato (U. of Minnesota/Yale)
• Bijaya B. Karki (Louisiana S. U.)
• Boris Kiefer (Princeton U.)
PREM
(Preliminary Reference Earth Model)
(Dziewonski & Anderson, 1981)
P(GPa)
0
24
135
329
364
Mantle Mineralogy
MgSiO3
Pyrolite model (% weight)
Olivine
cpx
(Mg1--x,Fex)2SiO4
300
8
(Mg,Ca)SiO3
12
garnets
500
-phase
spinel
700
(Mg1--x,Fex) O
(McDonough and Sun, 1995)
4
0
(‘’)
(Mg,Al,Si)O3
20
(‘’)
perovskite
MW(Mg,Fe) (Si,Al)O3
20
40
16
60
V%
CaSiO3
80
100
P (Kbar)
45.0
37.8
8.1
4.5
3.6
0.4
0.4
0.2
0.2
0.1
Depth (km)
SiO2
MgO
FeO
Al2O3
CaO
Cr2O3
Na2O
NiO
TiO2
MnO
opx
100
Mantle convection
Mineral sequence II
Lower Mantle
+
+
(Mgx,Fe(1-x))SiO3
(Mgx,Fe(1-x))O
CaSiO3
Mineral sequence II
Lower Mantle
+
+
(Mg(1-x-z),Fex Alz)(Si(1-y),Aly)O3
(Mgx,Fe(1-x))O
CaSiO3
Elastic constant tensor 
kl
ij
cijkl
kl
(i,j)
equilibrium
structure
• Crystal (Pbnm)
(m P (m;1,2,3 ) )  cmn n
re-optimize
c11
c21
c31
*

*

*
m
c12
c22
c13
c23
*
*
*
*
c32
*
c33
*
*
c44
*
*
*
*
*
c55
*
*
*
*
* 
* 
* 
* 

*

c66 
Elastic Waves
P-wave
(longitudinal)
S-waves
(shear)
n propagation
direction
Yegani-Haeri, 1994
Wentzcovitch et al, 1995
Karki et al, 1997
within 5%
Wave velocities in perovskite (Pbnm)
Cristoffel’s eq.:
VI  iI   ik kI
2
n
with
 ik  cij kln j nl
is the propagation direction
(Wentzcovitch, Karki, Karato, EPSL 1998)
Theory x PREM
(Voigt-Reuss-Hill averages)
TM of mantle phases
CaSiO3
(Mg,Fe)SiO3
5000
T (K)
Mw
4000
HA
Core T
solidus
3000
Mantle adiabat
2000
peridotite
0
20
40
60
P(GPa)
80
100
120
(Zerr, Diegler, Boehler, Science1998)
Thermodynamic Method
• VDoS and F(T,V) within the QHA
F (V , T )  U (V )  
qj
 qj (V )
2

  qj (V )  


 k BT  ln1  exp

k BT  
qj


N-th order isothermal finite strain EoS (N=3,4,5…)
 F 
P   
 V T
 F 
S   
 T V
G  F  TS  PV
• Density Functional Perturbation Theory for phonons
xxxxxxxxxxxxxxxxxx(Gianozzi, Baroni, and de Gironcoli, 1991)
(Thermo) Elastic constant tensor 
2


G 
T
cij (T , P )  

  i  j 
kl
cij (T , P)  cij (T , P) 
S
equilibrium
structure
re-optimize
T
S
i 
 i
T
i  jVT
CV
Phonon dispersion of MgSiO3 perovskite
Calc Exp
-
Calc Exp
0 GPa
-
Calc: Karki, Wentzcovitch, de Gironcoli, Baroni
PRB 62, 14750, 2000
Exp: Raman [Durben and Wolf 1992]
Infrared [Lu et al. 1994]
100 GPa
MgSiO3-perovskite and MgO

(gr/cm-3)
V
(A3)
KT
(GPa)
d KT/dP
d KT2/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
3.7
|
4.0
~
162.3
246
|
266
-0.02
|
-0.07
1.7
|
2.2
Exp.
Pv
4.247
~

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]
Elasticity of MgO
(Karki et al., Science 1999)
Thermal expansivity of MgSiO3
 (10-5 K-1)
(Karki, Wentzcovitch, de Gironcoli and Baroni, GRL 2001)
The QHA
Criterion: inflection point of (T)
B&S geotherm
Adiabatic bulk modulus at LM P-T
(Karki, Wentzcovitch, de Gironcoli and Baroni, GRL, 2001)
LM Geotherms
6000
LM geotherms
5000
T (K)
Pv
Tc
4000
Solidus
Pyrolite
3000
2000
1000
500
Isentropes
1000
1500
2000
De pth (k m )
2500
CMB
|
3000
Elastic constant tensor
300 K
1000K
2000K
3000 K
4000 K
1500 K
2500 K
3500 K
Oganov et al
2001
(Wentzcovitch, Karki, & Coccociono, 2002)
V (km/sec) &  (gr/cm3)
Velocities
Effect of Fe alloying
(Kiefer, Stixrude,Wentzcovitch, GRL 2002)
(Mg0.75Fe0.25)SiO3
||
+
+
4
+
Comparison with PREM
Pyrolite
Perovskite
Brown & Shankland T(r)
38 GPa
88 GPa
Moduli
Pyrolite
Perovskite
Brown & Shankland T(r)
38 GPa
88 GPa
Moduli
38 GPa
88 GPa
Pyrolite
Perovskite
Brown & Shankland T(r)
38 GPa
88 GPa
Moduli
38 GPa
100 GPa
Pyrolite
Perovskite
Brown & Shankland T(r)
Me
“…At depths greater than 1400 km, the rate of rise of the bulk and shear moduli are too small and
too large respectively for the lower mantle to consist of a homogeneous isotropic layer of pure
perovskite or pyrolite composition. It seems that changes in chemical composition, or subtle phase
changes, or anisotropy, or a combination of all, are required to account for the elastic moduli of the
deeper part of the LM ,….” (2002)
Intermediate Model of Mantle Convection
(Kellogg, Hager, van der Hilst, Science, 1999)
Summary
• Building a consistent body of knowledge obout LM phases
• We have adequate methods (DFT, QHA) to examine elasticity
of major mantle phases
• The objective is to interpret seismic observations (1D, 3D,
anisotropy) in terms of composition, temperature, ``flow’’…
Acknowledgements
Bijaya B. Karki (LSU)
Stefano de Gironcoli and Matteo Coccocioni
(SISSA, Italy)