Computational Mineral Physics and the Physical Properties of Mantle Minerals John Brodholt UCL

Outline of the talk Different computational techniques: Molecular dynamics Ab initio (quantum mechanics) calculations Ab initio molecular dynamics. Ab initio calculations on elastic constants of perovskite at lower mantle P and T and using this to make an estimate of temperature variations on the lower mantle. The effect of aluminium on the physical properties of perovskite. Future directions?

P and T at centre of Earth ~ 3.5 Mbar and ~ 6000 K P and T at core/mantle ~ 1.3 Mbar and ~ 3000 to 4000 K Piston cylinder ~ 40 kbar Multi-anvil ~ 400 kbar Diamond-anvil ~ 2 Mbar (temperatures uncertain, gradients high) Shock guns ~ 2 Mbar (temperatures extremely uncertain) High P and T experiments are hard and can have large uncertainties. Why use computational mineral physics? Use computational minerals physics to predict the properties of Earth materials at mantle and core conditions. Also to understand properties at an atomistic level.

Predict: Volumes, bulk modulus Elastic constants (seismic velocities) Heat capacities Free energies of reactions (phase diagrams) Defects Diffusion Viscosities Element partitioning etc

Molecular Dynamics Simulate the properties of minerals, fluids, melts, etc at an atomistic (or sub-atomic) level. If you know the forces (or potential energy) between the constituent atoms, you can find the total force on the atoms (or ions) and solve the equations of motions and move them through time Obtain averages over time that give you the properties of the material of interest. i.e. the kinetic energy gives the temperature.

In classical simulations, the potentials are parameterised empirically; fit to some experimental data.

Simulation of Na diffusing in Quartz

Matsui, P.E.P.I., 2000: (Uses breathing shell for the oxygen)

Pros and Cons of Empirical Potentials Pros: Large systems (millions of atoms) Long runs (millions of time steps ~ ns) Cons Very difficult to assess how transferable the potentials are: will work in one phase but not another. So, solution is to use ab initio quantum mechanical techniques to get the forces.

“I think I can safely say nobody understands quantum mechanics” Richard Feynman H  = E  Alternatively,

Ab Initio Techniques Numerically solving Schrodinger’s equation One major approximation; the effect on any one electron from all the other electrons (a very serious many body problem) is wrapped up into a term called the exchange- correlation. The exchange correlation functional can be either the Local Density Approximation (LDA) or the Generalised Gradient Approximation (GGA). Also use pseudopotentials for the interactions between valence electrons and the tightly bound core electrons (not to be confused with empirical pair potentials). Run on national supercomputers – T3E. Until recently has been used only at 0K.

LDA vs GGA LDA tends to underestimate volume (1-3%) and overestimate the bulk modulus (by a few %). GGA tends to do the opposite – overestimate the volume and underestimate the bulk modulus. GGA is better for reaction energies. GGA is better for hydrogen bonded systems. This is just a fact of life.

Masters et al., 2000

Forte and Mitrovica, Nature, 26 April km depth ~ D’’

Oganov, Brodholt and Price, EPSL, 2001

ATHERMAL (0 K) ELASTIC PROPERTIES OF PEROVSKITE Volume is constrained to experimental zero pressure V. Calc. Exp. K(GPa) G C11(GPa) C C C C C C C C According to Artem Oganov, “ These are simply brilliant ”

Ab Initio Molecular Dynamics VASP (Vienna Ab-Initio Simulation Package) Generalised Gradient Approximation (GGA) Ultrasoft pseudopotential for O Norm-conserving pseudopotential with partial core corrections for Mg and Si Plane-wave cutoff of 500 eV 1 kpt (gamma point) 80 atom cell NVT Run lengths of about 2 ps with a 1 fs timestep Run on CSAR T3E (which NERC has ¼).

Elastic constants of MgSiO 3 Perovskite Oganov, Brodholt & Price, Nature 2001 – QMD on perovskite at P and T. Remains orthorhombic in mantle P-T space. Elastic constants obtained from finite deformation of box and stress/strain relations:

38 GPa 1500K 38 GPa 2500K 88 GPa 1500K 88 GPa 3500K C C C C C C C C C K G At 38 GPa dlnVs/dlnVp = 1.48 At 88 GPa dlnVs/dlnVp = 1.91

High Temperature Vp and Vs from Ab Initio Molecular Dynamics R = 1.48 to 1.91 If we extrapolate, R ~ 2.3 at core-mantle boundary This is in accordance with other minerals. Observed R is about 1.7 to perhaps greater that 3. So, this supports the suggestion that the deeper lower mantle is either chemically heterogeneous or it is caused by anelasticity. Anelasticity could perhaps take R up to about 2.7 (Karato, 2001). Need to know the dependence of elastic constants on chemistry.

High Temperature Vp and Vs from Ab Inito Molecular Dynamics V  = Sqrt(K/  ) is relatively independent of composition and anelasticity. Therefore, use V  to make an estimate of temperature variations in the lower mantle. Masters et al, (2000) give a maximum δV  /V  of about 1.4% at 1225 and 2195 km. Our results give: dlnV  /dT = -1.7×10 -5 K -1 at 1000 km dlnV  /dT = -0.9×10 -5 K -1 at 2000 km Maximum temperature contrast is ~800 K at 1000 km and about ~1500 K at 2000 km. Possibly > 2000 K at core-mantle boundary Using Kennett et al, (1998) we get smaller temperature variations: 500 and 1000K.

Forte and Mitrovica, Nature, 26 April km depth ~ top of D’’

Summary 1.Experiments on MgSiO 3 perovskites with only 5% Al 2 O 3 have unexpectedly high compressibilities (mantle has ~ 4 wt%). 2.This is only consistent with a mechanism where Al 3+ substitutes for Si 4+, charge balanced by oxygen vacancies. 3.But, this mechanism is only favoured in the shallow part of the lower mantle. In the deeper part Al is incorporated be a vacancy free mechanism. 4.This means that alumina will only effect the properties of perovskite (compressibility, creep strength, diffusion, etc) in the shallow lower mantle; the deeper lower mantle will have properties in accord with alumina-free perovskite.

Incorporation of Al 3+ in Perovskite from Empirical Potential Calculations 1.At 0 GPa Al 3+ goes into Si site only, charge balanced by oxygen vacancies. 2.But P > 15 GPa, Al substitutes for both Si and Mg with no vacancies. Richmond and Brodholt (1999)

A 2 B 2 O 5 A 3 B 3 O 8 A 4 B 4 O 11 A n B n O 3n-1 ABO 3 Homologous series of compounds along the oxygen deficient axis. Smyth et al. 1996

Computational Details Used commercially available DFT code CASTEP. Ultrasoft pseudopotentials Default planewave cutoff (380 eV). Energy difference between 380 eV and 900 eV was 0.04 eV. K-point grid 0.1 1/Å. Increased sampling gave negligible difference on equation-of- state and reaction enthalpies. All atomic positions were fully relaxed.

(Mg 3 Al)(Si 3 Al)O 12 K = 224 GPa V = Ang**3 Adjust for 5% Al 2 O 3 and assume linear mixing V = <1.001*V(MgSiO 3 ) Vexp = 1.005*V(MgSiO 3 ) K = < 1% smaller Kexp = 10% smaller

Brownmillerite K = 155 GPa V = Ang**3 Adjust for 5% Al 2 O 3 and assume linear mixing V = 1.006*V(MgSiO 3 ) Vexp = 1.005*V(MgSiO 3 ) K = 6% smaller Kexp = 10% smaller

Oxygen Vacancies Enhanced diffusion Weaker – faster creep rates Oxygen ionic conductor – implications for electrical conductivity Seismic velocities might be lower Source for water: V o + O 2- + H 2 O = 2OH -

Oxygen Vacancies Might be an explanation for the mid-mantle discontinuity that is sometimes reported (900 to 1100 km) Might be a zone of mantle below the lower mantle which is weaker than that below.

Cizkova et al., GRL 1999

“ The more progress physical sciences make, the more they tend to enter the domain of mathematics, which is a kind of centre to which they all converge. We may even judge the degree of perfection to which a science has arrived by the facility with which it may be submitted to calculation.” Adolpohe Quetelet

“ Every attempt to employ mathematics in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry. If mathematical sciences should ever hold a prominent place in chemistry – an aberration which is happily almost impossible – it would occasion a rapid and widespread degeneration of that science.” A. Compte

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