Quasiharmonic Thermodynamic Properties of Minerals Renata M. M. Wentzcovitch Department of Chemical Engineering and Materials Science Minnesota Supercomputer.

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Presentation transcript:

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 MgSiO 3 - Ilmenite to perovskite phase transition Thermoelasticity of perovskite Crystal structures at high (P,T) Summary

The Contribution from Seismology Longitudinal (P) waves Transverse (S) wave from free oscillations

“660 km” topography J. 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 VDoS and F(T,V) within the QHA 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i re-optimize (Thermo) Elastic constant tensor 

Zero Point Motion Effect Volume (Å 3 ) F (Ry) MgO Static 300K Exp (Fei 1999) V (Å 3 ) K (GPa) K´ K´´(GPa -1 )

Elasticity of MgO (Karki et al., Science 1999)

MgSiO 3 -Akimotoite to perovskite transition From Fukao et al., Rev. Geophys. (2001) Clapeyron equation: P T TcTc PcPc Ak Pv T<T c P>P c Akimotoite bearing slab Transformation inhibited in cold regions!! 23 GPa 1980 K

MgSiO 3 -ilmenite (Akimotoite) Si 2 O 3 layer Mg 2 O 3 layer 1.77 A < Si-O < 1.83 A 1.99 A < Mg-O < 2.16 A oo o o corundum ilmenite LiNbO 3 MgSi Al SiMg R3

MgSiO 3 -perovskite (Pbnm) SiO 3 octahedra 1.78 A < Si-O < 1.80 A 2.01 A < Mg-O < 3.12 A oo oo

Phonon dispersion of MgSiO 3 -ilmenite and perovskite 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] Octahedral deformation Octahedral deformation Mg displacement Mg displacement Octahedral rotation NEW !

Pressure (GPa) Temperature (K) MgSiO 3 akimotoite perovskite Static Experiment Theory Thermodynamic phase boundary Issue I: Change in P T after inclusion of zero point motion energy (E zp ) Issue II: discrepancy between theory and experiments Exp:Ito & Takahashi (1996) G il (P,T) X G pv (P,T)

“…Useful rule…” I s s u e I P c decreases F(V,T) V pv ak  E zp shifts PcPc

Pressure (GPa) Temperature (K) MgSiO 3 akimotoite perovskite Static Experiment Theory Thermodynamic phase boundary Issue I: Change in P T after inclusion of zero point motion energy (E zp ) Issue II: discrepancy between theory and experiments Exp:Ito & Takahashi (1996) G il (P,T) X G pv (P,T)

…a posteriori criterion for the validity of the QHA  (10 -5 K -1 ) MgSiO 3    Karki et al, GRL (2001) Issue II…

Pressure (GPa) Temperature (K) MgSiO 3 akimotoite Static Experiment Theory Exp:Ito & Takahashi (1996) perovskite Not OK!! QHA OK

Properties of MgSiO 3 -perovskite and -ilmenite 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 | ~ Pv (256)

Ad hoc correction to DFT results… (perovskite)

Ad hoc correction to DFT results… !!!... (perovskite) but…

Ad hoc correction to DFT results… !!!... ?! (perovskite) but…

Ad hoc correction to DFT results… !!!... ?! (perovskite) but…

EoS for Perovskite C = 2.5 GPa

EoS for Ilmenite C = 1.9 GPaExp.: Reynard et al., 1996 Calc.: Karki & Wentzcovitch, 2002.

Ad hoc correction to P c … (ilmenite to perovskite) P c at 300K should increase (not really conclusive…!!)

cijcij (Wentzcovitch, Karki, Cococciono, de Gioroncoli, 2003) 300 K 1000K 2000K 3000 K 4000 K ( Oganov et al,2001) C ij (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. LDA +ZP Exp.

(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)