Dissociation of MgSiO3 in the Cores of Gas Giants and Terrestrial Exoplanets

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Science  17 Feb 2006:
Vol. 311, Issue 5763, pp. 983-986
DOI: 10.1126/science.1120865


CaIrO3-type MgSiO3 is the planet-forming silicate stable at pressures and temperatures beyond those of Earth's core-mantle boundary. First-principles quasiharmonic free-energy computations show that this mineral should dissociate into CsCl-type MgO cotunnite-type SiO2 at pressures and temperatures expected to occur in the cores of the gas giants + and in terrestrial exoplanets. At ∼10 megabars and ∼10,000 kelvin, cotunnite-type SiO2 should have thermally activated electron carriers and thus electrical conductivity close to metallic values. Electrons will give a large contribution to thermal conductivity, and electronic damping will suppress radiative heat transport.

The transformation of MgSiO3-perovskite into the CaIrO3-type structure near Earth's core-mantle boundary (CMB) conditions (13) invites a new question: What is the next polymorph of MgSiO3? The importance of this question has increased since the discoveries of two new exoplanets: the Earth-like planet with ∼7 Earth masses (4) (Super-Earth hereafter) and the Saturn-like planet with a massive dense core with ∼67 Earth masses (5) (Dense-Saturn hereafter). The extreme conditions at the giants' cores (6) and exoplanet interiors are challenging for first-principles methods. Electrons are thermally excited, and core electrons start to participate in chemical bonds. This requires either all-electron methods or the development of pseudopotentials based on core orbitals. Neither density functional theory (DFT) nor the quasiharmonic approximation (QHA) have been tested at these ultrahigh pressures and temperatures (PTs). Here, we use the Mermin functional (7), i.e., the finite electronic temperature (Tel) version of DFT that includes thermal electronic excitations, and ultrasoft pseudopotentials (8) based on orbitals with quantum number n = 2 and 3 for all three atoms. We studied MgSiO3, MgO, and SiO2 up to 80 Mbar and 20,000 K (figs. S1 and S2, A to C).

MgSiO3 could transform to another ABX3-type silicate or dissociate. We searched systematically for possible ABX3 structures having likely high-pressure coordinations and connectivities, but found none with enthalpy lower than the CaIrO3-type polymorph (see supporting online material). This phase dissociated into CsCl-type MgO and cotunnite-type SiO2 at 11.2 Mbar in static calculations (Fig. 1). Both binary oxides undergo phase transitions below 11.2 Mbar. MgO undergoes the NaCl-type → CsCl-type transformation at 5.3 Mbar, and SiO2 undergoes a series of phase transitions: stishovite → CaCl2-type → α-PbO2-type → pyrite-type → cotunnite-type at 0.48, 0.82, 1.9, and 6.9 Mbar, respectively (Fig. 2). Our static transition pressures agree well with previous first-principles results (912) and experimental transition pressures (13, 14), except for the α-PbO2-type → pyrite-type transition in SiO2, which has been observed once at 2.6 Mbar (15). CsCl-type MgO and cotunnite-type SiO2 have not yet been seen experimentally. Baddeleyite-type and OI-type phases occur as pre-cotunnite phases in TiO2 (16), an analog of SiO2. Our results show that, in agreement with previous calculations (12), these phases are metastable with respect to pyrite-type and cotunnite-type SiO2. Phonon frequencies in the CaIrO3-type phase and in the binary oxides increase with pressure. In our calculations, no soft phonons occurred up to 80 Mbar, the pressure at Jupiter's center. As expected, soft phonons occurred in CsCl-type MgO (∼2 Mbar) and in cotunnite-type SiO2 (∼1.5 Mbar) upon decompression (fig. S2).

Fig. 1.

Crystal structures of the stable phases. (A) CaIrO3-type MgSiO3 at static 10 Mbar. The space group is Cmcm. Lattice constants are (a, b, c) = (2.10, 6.42, 5.26) Å. Atomic coordinates are Mg(4c) (0, 0.749, 0.25), Si(4a) (0, 0, 0), O1(4c) (0, 0.070, 0.25), and O2(8f) (0, 0.353, 0.438). (B) CsCl-type MgO at static 12 Mbar. The space group is Pm3m. The lattice constant is a = 1.870 Å. (C) Cotunnite-type SiO2 at static 12 Mbar. The space group is Pbnm. Lattice constants are (a, b, c) = (4.69, 3.95, 2.08) Å. Atomic coordinates are Si(4c) (0.141, 0.232, 0.25), O1(4c) (0.435, 0.348, 0.25), and O2(4c) (0.666, 0.984, 0.25).

Fig. 2.

Differences between static enthalpies of aggregation of MgO and SiO2 in various forms and CaIrO3-type MgSiO3. Dashed lines denote static transition pressures of NaCl-type → CsCl-type MgO, α-PbO2-type → pyrite-type SiO2, and pyrite-type → cotunnite-type SiO2.

Thermal electronic excitations have negligible effect on the structural, vibrational, and thermal properties of these phases, even at 20,000 K, shifting the phase boundary by less than 1 GPa. Empirically, the QHA should work well until the thermal expansivity α(P,T) becomes nonlinear (17). We find linear T scaling up to the dashed lines in the phase diagram shown in Fig. 3A. The Clapeyron slope (dP/dT = dS/dV) of the dissociation has large negative values at most pressures: –18 MPa/K at 5000 K increasing to –33 MPa/K at 10,000 K. This is caused by an increase in the average bond lengths [2.91 to 3.08 atomic units (au) for Mg-O bond and 2.59 to 2.76 au for Si-O bond] across the dissociation as cation coordination numbers increase. This decreases the average phonon frequencies and increases the entropy (18). At the same time, there is a density increase of 1 to 3% (Fig. 3B; fig. S3 and table S2). Negative Clapeyron slopes occurred also for the NaCl-type to CsCl-type MgO and for the pyrite-type to cotunnite-type SiO2. In both cases, cation coordination numbers and average bond lengths increase through the transition (3.10 to 3.33 au in NaCl-type → CsCl-type MgO; 2.81 to 2.90 au in pyrite-type → cotunnite-type SiO2).

Fig. 3.

(A) Pressure-temperature phase diagram showing the dissociation of CaIrO3-type MgSiO3 into CsCl-type MgO and cotunnite-type SiO2. Red areas denote estimated pressure-temperature conditions at core-envelope boundaries in the solar giants and in Super-Earth. Dashed lines indicate the limit of validity of the quasiharmonic approximation (QHA). The dashed part of the phase boundary is more uncertain. (B) Density increase caused by the dissociation.

This dissociation should affect models of the gas giants' cores (Fig. 3). CaIrO3-type MgSiO3 cannot exist in the cores of Jupiter and Saturn, but should survive in the cores of Uranus and Neptune, unless another phase transition not identified yet occurs at lower pressures. In Jupiter and Saturn, the dissociation occurs at PTs typically expected within the metallic-hydrogen envelope. The importance of this transformation for terrestrial exoplanets is more striking. Super-Earth is predicted to have a temperature of at least ∼4000 K and a pressure of ∼10 Mbar at its CMB (19). This places the dissociation near its CMB. The eventual occurrence of this endothermic transition with a large negative Clapeyron slope just above its CMB would be similar to the occurrence of the endothermic postspinel transition near the core of Mars. Geodynamical modeling suggests that this might be the cause of a proposed large martian superplume (20). Convection in a Dense-Saturn planet could be dramatically affected. PTs in this planet should be higher than in Saturn's interior (Fig. 3), given its smaller size and higher surface Ts. A transformation with such large negative Clapeyron slope in the middle of the silicate core/mantle of terrestrial planets is likely to inhibit convection (21), promote layering, and produce differentiated mantles/cores, with a lower layer consisting primarily of oxides.

At PTs relevant for the giants and exoplanets, major changes in materials properties take place: These minerals become intrinsic semiconductors with electronic gaps (Fig. 4A). Local density approximation (LDA) usually underestimates band gaps by ∼50%, whereas electronphonon interactions cause gaps to narrow by a couple of eVs at elevated Ts (22). The intrinsic carrier [electrons (n) and holes (p)] concentrations, in the range of 10,000 to 20,000 K (Fig. 4B), are typical of semimetals or heavily doped semiconductors. We focus on cotunnite-type SiO2 with the largest carrier concentration.

Fig. 4.

(A) Solid and dashed lines denote the LDA band gaps of the phases involved, E LDAg and 2 E × LDAg, respectively. Actual band gaps should be in this range. (B) Total carrier (n = electrons, p = holes) concentration at 10,000 K (solid lines) ∼20,000 K (dashed lines) assuming Eg = 2 × EgLDA. This should be a lower bound for the carrier concentration.

In evaluating transport coefficients, we treat holes as immobile (23). This model is motivated by the relatively flat valence band edge of cotunnite-type SiO2. Only thermal electrons are free and can carry both electrical and heat currents. The carrier density, n, from Fig. 4B can be represented by Math(1) where the thermal wavelength is Math, and the cell volume V and effective mass me are 276 a 30 and 0.4m, with a0 and m being the Bohr radius and the electron mass. Assuming that band gap Eg = 5 eV and taking T = 104 K, we obtain n ≈ 8 × 1020 cm–3; 0.9% of SiO2 units have one excited electron and hole. This carrier density is typical of semimetals or heavily doped semiconductors.

The electrical conductivity, α, is obtained from (24, 25) Math(2) where μ is the mobility and 〈τ 〉 is the average inverse scattering rate. There are three scattering mechanisms: (i) Coulomb scattering of carriers from each other; (ii) scattering from impurities or defects; and (iii) scattering by phonons, both Fröhlich (F) and optical deformation potential (24). Coulomb scattering is primarily electrons scattering from holes. Because holes are assumed to be localized, this is just a form of charged impurity scattering and likely weaker than scattering from impurities (Al, Fe, OH). Impurity and Fröhlich scattering suffer Debye-Hückel screening (24, 25), with inverse screening length κ given by Math. With our computed dielectric constant ϵ ≈ 4, 1/κ ∼4.7 Å. We obtain the following estimates (see supporting online material): Math(3) Math(4) Math(5) where ximp is the fraction of Si atoms substituted by impurities of effective charge Z = 1. The Fröhlich and impurity rates have additional weak T-dependence (not shown) arising from temperature-dependent screening.

The ratio between impurity and electron-phonon scattering rates is Math, at T = 104 K. If more than 2% of Si atoms are replaced by charged impurities, impurity scattering dominates but falls rapidly at higher T. Minor element partitioning between MgO and SiO2 at these conditions is unknown, and impurity centers may provide thermally inexpensive sources of new carriers. Therefore, any estimate of the influence of impurities has significant uncertainty. Ignoring impurity scattering, the electron-phonon scattering then gives an upper bound for the electronic mobility μ ∼20 cm2/V·s. This exceeds mobilities of typical metals (near 10 cm2/V·s at 300 K and falling as 1/T) but is smaller than μ ∼1400 cm2/V·s for electrons in intrinsic Si at 300 K. The result is σ ≈ 2.6 103 (ohm·cm)–1. The corresponding resistivity, ρ ≈ 380 microhm·cm, is only twice that of liquid iron at atmospheric pressure (26). We believe this is a reliable lower limit and that uncertainties (primarily the value of Eg, the concentration of charged impurity centers, and neglect of some weaker phonon-scattering processes) may increase the resistivity by less than a factor of 5 (see supporting online material). This would still leave the material essentially a metal!

The thermal conductivity can be estimated using an appropriate Weidemann-Franz ratio, Math W/mK. This is large compared to values of 2 to 4 W/mK that are representative of vibrational heat transport in hot anharmonic insulators (27). We also conclude that radiative heat transport should not be significant, despite the high energy content in the black-body spectrum at such high T. Because of free electrons (28), no photons propagate with frequencies less than the plasma frequency Math eV. Above this energy, electronic absorption from deep levels in the gap, resulting from structural defects or impurities, such as iron, is likely to limit the radiative term to values smaller than the electronic contribution. The conductivities, both thermal and electrical, of cotunnite-type SiO2 and most oxides and silicates in terrestrial exoplanets will be large because of carriers activated over the insulating gap.

The dissociation of CaIrO3-type MgSiO3 stands as a challenge to be investigated by ultrahigh-pressure experiments. An alternative, low-pressure analog is NaMgF3. This compound exists in the Pbnm-perovskite structure at ambient pressure and is predicted to transform to the postperovskite structure at 17.5 GPa and to dissociate into NaF (CsCl-type) and MgF2 (cotunnite-type) at ∼40 GPa, after the fluorides undergo transitions similar to those of the oxides (29). The postperovskite transformation appears to have been observed already (30), although the dissociation has not been observed yet.

Supporting Online Material


SOM Text

Figs. S1 to S4

Tables S1 and S2

References and Notes

References and Notes

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