Mauro Prencipe Dip. Scienze della Terra, Università di Torino Accademia Nazionale dei Lincei Roma – 13, 14 giugno 2016

Ab initio modelling of the Earth's structure and dynamics Ab Initio Modelling: What is it? One of the continuing scandals in the physical sciences is that it remains in general impossible to predict the structure of even the simplest crystalline solids from a knowledge of their chemical composition John Maddox, Nature, 1988 That’s no longer true! Artem Oganov

Ab Initio Modelling: What is it? Ab initio modelling of the Earth's structure and dynamics In practice, often the knowledge of the stoichiometry, the symmetry (space group) together with an approximate geometric model (cell parameters and atomic fractional coordinates), at standard P/T conditions, is a very useful starting point  Structure at different P/T conditions (usually high pressure and/or high temperature)  Vibrational properties (frequencies, normal modes…)  P(V,T) Equation of state  Specific heat Cp (P,T) enthalpy, entropy, free energy Ab Initio computing: no reference to other experimental quantities Well, indeed nothing is required. Just state what elements you want…

Ab initio modelling of the Earth's structure and dynamics Ab Initio Modelling: Why? You truly understand a given phenomenon only when you are able to calculate it [from first principles] Leonard Susskind

 To provide insights and interpretations of experimentally observed facts  To provide reliable and accurate data when experimental results are lacking or not accurate enough (i.e. in the case of simultaneus high pressure/high temperature measurements)  To cross-check and validate experiments  To predict new facts and phenomena and driving new experiments Ab Initio Modelling: Why? Ab initio modelling of the Earth's structure and dynamics

Ab initio modelling of the Earth's structure and dynamics The General Theory (in a small nutshell…) Partition functions: sum over all the possible energy levels of the system Energies εn can be quantum-mechanically calculated by solving the Schrödinger equation To solve any problem, just minimize G I formulated the equation but its solution is upon you…

Ab initio modelling of the Earth's structure and dynamics An Example: the Case of Diamond Ab Initio calculated Gibbs free energy (G) curves for diamond (at the hybrid HF/DFT level of the theory) as function of T (P constant). Properties can be directly calculated from the derivatives of G(P, T) with respect to P and/or T. -1 Experimental value

Ab initio modelling of the Earth's structure and dynamics ol ring wad X P ΔG P X Experiments Ab Initio computations Stability fields and equilibrium data Global inversion Thermodynamics database Computed phase diagrams & properties Predictions 0 Cross-check & interpretation ρ P The Process: A General Overview

The Process: The implementation Ab initio modelling of the Earth's structure and dynamics Lars Stixrude I did the database that Mauro is talking about, for minerals at the P/T conditions of the Earth’s mantle James Connolly A database without a program to play with it, is of little use… Use my «Perplex» program!

Ab initio modelling of the Earth's structure and dynamics Gasparik (2003): high pressure experiments Pv Gt(maj) Aki 10 12 14 28 18 16 22 24 20 26 1200 1800 2400 3000 P (GPa) T (K) Stixrude & Lithgow-Bertelloni (2011) Holland & Powell (2011) Belmonte (2013) Gasparik (2003) Yu et al. (2011) LDA Yu et al. (2011) GGA Belmonte (2013), PhD thesis: B3LYP calculation + disorder (empirical) Stixrude & Lithgow-Bertelloni (2011): PBE calculation + mixing Yu et al. (2011); LDA, GGA calculations (fully ordered) Holland & Powell (2011): new database Test Case: Garnet-Perovskite-Akimotoite equilibria Original elaboration by Donato Belmonte The Stixrude’s database is OK, however my results are the best!

Mg2 SiO4 fo wad ring pv+per aki + per per+st maj+per 305 573 813 Depth (Km) fo: forsterite wad: wadsleyite ring: ringwoodite maj: majorite pv: perovskite per: periclase aki: akimotoite st: stishovite Wadsleyite: Mg2 SiO4 Space group Imma Ringwoodite: spinel type structure Ab initio modelling of the Earth's structure and dynamics Mg2 SiO4 at High Pressures (Stixrude’s database)

Ab initio modelling of the Earth's structure and dynamics Bulk composition (wt%) MgO 39.8 CaO 3.3 FeO 8.0 Al2 O3 3.7 SiO2 46.2 Fe/(Fe+Mg) = 0.1 10 25 13 16 19 22 P (GPa) 305 388 470 548 624 697 Depth (km) 0.06 0.15 0.105 Mineral Phases in a Pyrolitic Mantle P-X section, at variable iron content, along the geotherm T(K) = 1600 + 1.6·10-3 P(bar) Stixrude’s database

10 25 13 16 19 22 Ol Opx Cpx Gt Ol Cen Cpx Gt Wad Cen Cpx Gt(maj) Wad Cpx Gt(maj) Wad Gt(maj) Ring Gt(maj) Ca-Pv Ring Gt(maj) Ca-Pv St + Ca-Pv + Aki Pv Fe-Per Gt Ca-Pv Pv Fe-Per Ca-Pv 1.04 1.46 2.31 Al2 O3 wt% P (GPa) Depth (km) 9 16 36 Gt (Wt%) 0.30 0.43 0.83 XMaj 305 388 470 548 624 697 Ab initio modelling of the Earth's structure and dynamics Bulk composition (wt%) MgO 41.4 CaO 3.1 FeO 8.3 Al2 O3 1.5 SiO2 45.7 Fe/(Fe+Mg) = 0.1 Low Al-content and the Akimotoite phase

olivine wadsleyite ringwoodite bridgmanite (Mg-pv) garnet (majorite) o/c-enstatite clino-pyroxene Fe-per Ca-pv akimotoite Vol (%) 20 40 60 80 100 10 15 20 25 30 P (GPa)  550 km  410 km  660 km 442 km 576 km Ab initio modelling of the Earth's structure and dynamics Minerals Abundances in the Pyrolitic Mantle Bulk composition (wt%) MgO 41.4 CaO 3.1 FeO 8.3 Al2 O3 1.5 SiO2 45.7 Fe/(Fe+Mg) = 0.1 Typical abundances of minerals at increasing depth (P/T trajectory along the estimated geotherm) for the pyrolitic bulk composition. At shallow depths, olivine: 60%; garnet: 20%; pyroxenes: 20%. At higher depths, pyroxenes dissolve into garnet (majorite). Both ringwoodite and garnet transform into perovskite (plus Fe-periclase). Calcium enters in the Ca-perovskite structure.

Ab initio modelling of the Earth's structure and dynamics A Cross-Check of the Preliminary Reference Model (PREM) Pressure as a function of depth, within the [300, 670 km] range of depth, derived from calculated densities of mineral assemblages, by assuming a linear variation of the temperature from 1650 K (300 km) to 1850 K (670 km) Green points represent data from the Preliminary REference Model (PREM), derived from the inversion of seismological observations Estimations for pyrolite and MORB bulk compositions are plotted The pyrolite model well agrees with the PREM data P (GPa) Depth (km) Seismic discontinuity at 400 km: olivine wadsleyite transition P = 13.4 GPa, T = 1690K

Ab initio modelling of the Earth's structure and dynamics Temperature at the  650 km Discontinuity 23.1 23.3 23.5 1750 1850 1950 2050 660 650 655 1920 K  30 K P (GPa) Depth (km) T (K) ring (+ gt, pv, Ca-pv) pv + Fe-per (+ gt, Ca-pv) Density (kg/m3) 3970 4300 4200 The ring pv + Fe-per transition (negative Clapeyron slope) defines a rather sharp change of the density By setting at 655 km the depth of the density change (seismic observations) and, thus, at a corresponding pressure of 23.3 GPa (PREM), a temperature of about 1920 K can be estimated Bulk composition (wt%) MgO 39.8 CaO 3.3 FeO 8.0 Al2 O3 3.7 SiO2 46.2 Fe/(Fe+Mg) = 0.1

8.94 10.70 Vp (km/sec) 10 25 13 16 19 22 P (GPa) Depth (km)  10 km  410 km 20 km  660 km Ol Wad Ring Pv + Fe-Per Ol: 60% vol (fo 0.90) Gt: 17% (py 0.55) Cpx 11% (di 0.76) Opx 12% (en 0.84) 305 388 470 548 624 Ab initio modelling of the Earth's structure and dynamics Seismic Velocities (P-waves) in a Pyrolitic Mantle Bulk composition (wt%) MgO 39.8 CaO 3.3 FeO 8.0 Al2 O3 3.7 SiO2 46.2 Fe/(Fe+Mg) = 0.1

Ol: 64% vol (fo 0.90) Gt: 7% (py 0.54) Cpx 11% (di 0.76) Opx 18% (en 0.84) Ab initio modelling of the Earth's structure and dynamics Bulk composition (wt%) MgO 41.4 CaO 3.1 FeO 8.3 Al2 O3 1.5 SiO2 45.7 Fe/(Fe+Mg) = 0.1 Seismic Velocities (P-waves) in a (Al-poor) Pyrolitic Mantle P (GPa) 10 25 13 16 19 22 300 380 470 535 615  400 km  8 km  600 km  645 km  8 km 8.9 10.9 Vp (km/sec) Depth * (km) AK135 * Pressure/depth relation from the AK135 densities

10 14 22 18 26 30 P (GPa) 3.57 3.76 3.95 4.14 4.33 r (gr/cm3) T=2000 K T=1700 K Ol (fo = 0.90) Wad Ring Pv + Fe-Per 10 14 22 18 26 P (GPa) 1700 2000 T (K) Negative Clapeyron slope Ring Pv + Fe-Per Wad Ol 305 Depth (km) 416 522 624 721 Ab initio modelling of the Earth's structure and dynamics Profile Density in the Mg1.8 Fe0.2 SiO4 System The ringwoodite perovskite + Fe-periclase reaction has a negative Clapeyron slope: as the temperature decreases, the transition pressure shifts to higher values

10 14 22 18 26 P (GPa) r (gr/cm3) 4.444 3.485 Ol + Gt Wad + Gt(maj) Ring + Gt(maj) Pv + Fe-Per T=2000 K T=1700 K Depth (km) 305 416 522 624 721 Ab initio modelling of the Earth's structure and dynamics Profile Density in the Pyrolitic Mantle The garnet (maj) perovskite + Fe-periclase has a positive Clapeyron slope: the transition pressure increases at higher temperature By taking into account the amounts of ringwoodite and majorite, for the specified bulk composition, at any depth, overall the density of the colder slab is higher than that of the hotter one Bulk composition (wt%) MgO 39.8 CaO 3.3 FeO 8.0 Al2 O3 3.7 SiO2 46.2 Fe/(Fe+Mg) = 0.1

Ab initio modelling of the Earth's structure and dynamics 5 10 15 20 25 30 P (GPa) 1080 1240 1400 1320 1160 T (K) cpx gt coe cpx gt st Adiabatic P/T path ring cpx gt st ring gt st Ca-pv Fe-per gt st Ca-pv Fe-per Ca-pv st aki cf pv + cf Fe-per st Ca-pv cf aki 305 575 697 444 813 Depth (km) The Subduction of Mid Ocean Ridge Basalts (MORB) Bulk composition (wt%) MgO 10.2 CaO 13.2 FeO 8.0 Na2 O 2.0 Al2 O3 16.5 SiO2 50.1 Fe/(Fe+Mg) = 0.3 Model of the subduction of a MORB along the adiabat through the point at T = 1000 K and P = 5 GPa (160 km depth). No olivine or wadsleyite present

Ab initio modelling of the Earth's structure and dynamics r (gr/cm3) 5 10 15 20 25 30 305 575 697 444 813 157 pyrolite MORB (adiabat) MORB (P/T Akaogi) P (GPa) Depth (km) 3.67 3.91 4.14 4.38 23.6 GPa 663 km ring Mg-pv + per ol wad gt Mg-pv MORB (BAB, adiabat) The Subduction of Oceanic Basalts Densities of pyrolite (along a typical geotherm) and of subducting slabs (MORB: Mid Ocean Ridge Basalts bulk composition) following different P/T trajectories The density of a cold slab (blue and green curves) is higher than that of the surrounding (hot) pyrolitic mantle (black curve), up to about 660 km: the slab cannot sink beyond the transition zone. On the contrary, a cold Back Arc Basin basalt (BAB; red curve) has an higher density than the surrounding hot mantle, at any depth and it could sink. Wt% MORB BAB SiO2 49.7 53.2 Al2 O3 16.4 15.6 MgO 10.1 10.2 FeO 8.7 6.9 CaO 13.1 11.2 Na2 O 2.0 2.9

Ab initio modelling of the Earth's structure and dynamics From: Bass and Parise (2008) Elements, 4, 157-163 The Subduction of Oceanic Basalts 3-D seismic tomographic structures of the mantle near several subduction zones in the western Pacific and central America

Ab initio modelling of the Earth's structure and dynamics gph diam wal lrn + cstn Ca-pv ol wad ring Mg-pv 5 25 21 17 13 9 P (GPa) 1300 1600 1900 2200 2500 T (K) geotherm 697 600 497 389 276 Depth (km) Not consistent pseudosection (Mg0.9 , Fe0.1 )2 SiO4 calculated from Stixrude & Bertelloni (2011); Ca-silicates from Holland & Powell (2011) Mix of databases… Tomorrow you will hear about some of these minerals we found as inclusions in diamonds Stability fields of Ca-pv, larnite (lrn) + titanite (cstn; CaSi2 O5 ), and walstromite (wal)

Thank you for your Attention!