Transcript First Principles Thermoelasticity of Mantle Minerals

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
Longitudinal (P) waves
VP 
4
K G
3

Transverse (S) wave
VS 
G

 from free oscillations
Seismic Discontinuities and Phase Transitions
PREM
Dziewonski and Anderson, 1981
“660 km” topography
J. M. Kendall, 2000
Methods
• Local Density Approximation
• Soft norm-conserving pseudopotentials
• Born-Oppenheimer variable cell shape molecular dynamics
• Density functional perturbation theory for phonons
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 (N=3,4,5…) order isothermal (eulerian or logarithm) finite strain EoS
 F 
P   
 V T
 F 
S   
 T V
G  F  TS  PV
IMPORTANT: structural parameters and phonon frequencies
depend on volume alone!!….
(Thermo) Elastic constant tensor 
2


G 
T
cij (T , P )  

  i  j 
i
cij (T , P)  cij (T , P) 
S
equilibrium
structure
re-optimize
T
S
i 
 i
T
i  jVT
CV
Zero Point Motion Effect
F (Ry)
MgO
Volume (Å3)
V (Å3)
K (GPa)
K´
K´´(GPa-1)
Static 300K
18.5 18.8
169
159
4.18 4.30
-0.025 -0.030
Exp (Fei 1999)
18.7
160
4.15
Elasticity of MgO
(Karki et al., Science 1999)
MgSiO3-Akimotoite to perovskite transition
Akimotoite bearing slab
Clapeyron equation:
dT dV

dP dS
23 GPa
1980 K
dS  0 
dT
0
dP
Tc
Pv
T
Ak
From Fukao et al., Rev. Geophys. (2001)
T<Tc
Pc
P
P>Pc
Transformation inhibited in cold regions!!
MgSiO3-ilmenite (Akimotoite)
corundum
Si2O3 layer
ilmenite
Al
o
o
1.77 A < Si-O < 1.83 A
LiNbO3
Mg2O3 layer
Mg
R3
o
o
1.99 A < Mg-O < 2.16 A
Si
Mg
Si
MgSiO3-perovskite (Pbnm)
SiO3 octahedra
o
o
2.01 A < Mg-O < 3.12 A
o
o
1.78 A < Si-O < 1.80 A
Phonon dispersion of MgSiO3-ilmenite and perovskite
Pv: Raman [Durben and Wolf 1992]
Infrared [Lu et al. 1994]
Ak: Raman [Reynard and Rubie, 1996]
Infrared [Madon and Price, 1989]
Calc Exp
Calc Exp
Calc Exp

Octahedral
deformation

Mg
displacement

Octahedral
rotation
NEW!

0 GPa
Aaaaaaa
Aaaaa

Octahedral
deformation
Mg
displacement
Thermodynamic phase boundary
Exp:Ito & Takahashi (1996)
Issue I: Change in PT after inclusion
of zero point motion energy (Ezp)
Issue II: discrepancy between
theory and experiments
30
perovskite
perovskite
Static
Static
Pressure (GPa)
Gil(P,T)
X
Gpv(P,T)
Pressure (GPa)
25


Experiment
Experiment
20
Theory
Theory
15
akimotoite
akimotoite
10
5
MgSiO3
MgSiO
3
0
0
500
1000
1500
Temperature
(K)
Temperature (K)
2000
• I
s
s
“…Useful rule…”

xi 
 xi
S (V , T )  kB   ln 1  e  kB xi
e  1 
i 





 i
 xi 
k BT

F(V,T)
pv
Pc
Ezp shifts
ak
dS dT

 0  E zp  0
dV dP
V

Pc decreases



Thermodynamic phase boundary
Exp:Ito & Takahashi (1996)
Issue I: Change in PT after inclusion
of zero point motion energy (Ezp)
Issue II: discrepancy between
theory and experiments
30
perovskite
perovskite
Static
Static
Pressure (GPa)
Gil(P,T)
X
Gpv(P,T)
Pressure (GPa)
25


Experiment
Experiment
20
Theory
Theory
15
akimotoite
akimotoite
10
5
MgSiO3
MgSiO
3
0
0
500
1000
1500
Temperature
(K)
Temperature (K)
2000
Issue II…
 (10-5 K-1)
…a posteriori criterion for the validity of the QHA



MgSiO3
Karki et al, GRL (2001)
30
perovskite
perovskite
Static
Static
25
Pressure (GPa)
(GPa)
Pressure

Experiment
Experiment
20
Not OK!!
Theory
Theory
15
QHA OK
10
akimotoite
akimotoite
5
MgSiO
3
MgSiO
3
0
0
500
1000
1500
2000
Temperature
(K)
Temperature (K)
Exp:Ito & Takahashi (1996)
Properties of MgSiO3-perovskite and -ilmenite

(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
3.908
18.80
176.8
159
201
4.30
4.7
-0.030
-0.042
-0.014
-0.025
1.88
3.12
Calc.
MW
Ak
-0.0145
~
1.67
|
3.13
2.44
Exp.
MW
Ak
3.601
3.943
18.69
175.2
160
212
4.15
4.210
164.1
247
4.8
4.0
-0.016
-0.031
2.1
Calc.
Pv
Pv
246
|
266
(256)
3.7
|
4.0
~
-0.02
|
-0.07
1.7
|
2.2
Exp.
Pv
Pv
4.247
162.3
~

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]
Ad hoc correction to DFT results…
(perovskite)
0GPa
LDA
V
V
0GPa
exp
2.5GPa
0GPa
V
V

V


1
.
1
%
LDA
exp
0GPa
Vexp
Ad hoc correction to DFT results…
(perovskite)
0GPa
LDA
V
K
V
0GPa
LDA
0GPa
exp
K
0GPa
exp
2.5GPa
0GPa
V
V

V


1
.
1
%
LDA
exp
0GPa
Vexp
but…
K
2.5GPa
LDA
 257GPa  K
0GPa
exp
!!!...
Ad hoc correction to DFT results…
(perovskite)
0GPa
LDA
V
K
V
0GPa
LDA
0GPa
exp
K
0GPa
exp
2.5GPa
0GPa
V
V

V


1
.
1
%
LDA
exp
0GPa
Vexp
but…
K
2.5GPa
LDA
 257GPa  K
KLDA (V )  Kexp (V )
0GPa
exp
?!
!!!...
Ad hoc correction to DFT results…
(perovskite)
0GPa
LDA
V
K
2.5GPa
0GPa
V
V

V


1
.
1
%
LDA
exp
0GPa
Vexp
V
0GPa
LDA
0GPa
exp
K
0GPa
exp
but…
K
2.5GPa
LDA
 257GPa  K
KLDA (V )  Kexp (V )
0GPa
exp
?!
PLDA (V )  Pexp (V )  C
!!!...
EoS for Perovskite
C = 2.5 GPa
EoS for Ilmenite
C = 1.9 GPa
Calc.:
Karki & Wentzcovitch, 2002.
Exp.:
Reynard et al., 1996
Ad hoc correction to Pc…
(ilmenite to perovskite)
pv
ideal
F
(V )  F
pv
LDA
(V )  2.5V  C'
il
il
Fideal
(V )  FLDA
(V )  1.9V  C' '

Pc at 300K should increase
(not really conclusive…!!)
c
i
j
300 K
1000K
2000K
3000 K
4000 K
Cij(P,T)
(Oganov et al,2001)
(Wentzcovitch, Karki, Cococciono, de Gioroncoli, 2003)
…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)
o
Calc.
o
o
77 K < T < 400K
0 GPa < P < 12 GPa
Comparison with Experiments
(Ross & Hazen, 1989)
o
Calc.
77 K < T < 400K
0 GPa < P < 12 GPa
o
o
LDA
Exp.
LDA
+ZP
(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 K
3000 K
2000 K
1000 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)