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It is thought that the Earth's outer core consists mainly of liquid iron and that the convection of this metallic liquid gives rise to the Earth's magnetic field. A full understanding of this convection is hampered, however, by uncertainty regarding the viscosity of the outer core. Viscosity estimates from various sources span no less than 12 orders of magnitude(1,2), and it seems unlikely that this uncertainty will be substantially reduced by experimental measurements in the near future. Here we present dynamical first-principles simulations of liquid iron which indicate that the viscosity of iron at core temperatures and pressures is at the low end of the range of previous estimates - roughly 10 times that of typical liquid metals at ambient pressure. This estimate supports the promotion commonly made in magnetohydrodynamic models that the outer core is an inviscid fluid(3-5) undergoing small-scale circulation and turbulent convection(6), rather than large-scale global circulation.

Knowledge of the abundance and distribution of light elements in the core is fundamental to the understanding of the Earth and other planetary systems. Recent studies (Li et al., 2018; Mashino et al., 2019) suggest the particular importance of carbon for explaining core properties, yet knowledge of carbon partitioning between the outer and inner core is unknown. By using the quasiharmonic approximation, ab initio molecular dynamics, and thermodynamic integration techniques, we have computed the chemical potential of carbon in liquid Fe and solid hcp-Fe at core conditions. We find that substitutional carbon is more stable than interstitial carbon and other carbon defect cluster structures in solid Fe. Lattice strain and overcoordination effects lead to a high chemical potential of C in solid Fe compared to the liquid, and consequently carbon partitions almost completely into the liquid. We find that carbon can account for most of the density jump at the inner-core boundary. This provides an alternative mechanism to the necessity of an oxygen-rich outer core and may have significant implications for the composition and structure of the deep Earth.

The Earth's solid inner core consists mainly of iron (Fe), alloyed with lighter elements. such as silicon (Si). Interpretation of seismic anisotropy and layering requires knowledge of the stable crystal structure in the inner core. We report ab initio density functional theory calculations on the free energy and vibrational stability of pure iron and Fe-Si alloys at conditions representative of the Earth's inner core. For pure Fe the stable phase is already known to be hexagonal close-packed (hcp). However, with the addition of similar to 7 wt.% Si at high temperatures, we observe a transition to the face-centred cubic (fcc) phase. We also produce a phase diagram for the Fe-Si system and show that the inner core may exist in the two-phase region, with coexisting fcc and hcp. This may also explain the low S-velocities observed in the inner core. (C) 2009 Elsevier B.V. All rights reserved.

The phase that iron adopts at the conditions of the Earth's inner core is still unknown. The two primary candidates are the hexagonal close packed (hcp) structure and the body centred cubic (bcc) structure polymorphs. Until recently, the former was favoured, but it now seems possible that bcc iron could be present. A remaining uncertainty regarding the latter phase is whether or not bcc iron is stable with respect to tetragonal strain under core conditions. In this paper, therefore, we present the results of high precision ab initio free energy calculations at core pressures and temperatures performed on bcc iron as a function of tetragonal strain. Within the uncertainties of the calculations, direct comparison of free energy values suggests that bcc may be unstable with respect to tetragonal strain at 5500 K; this is confirmed when the associated stresses are taken into account. However, at 6000 K, the results indicate that the bcc phase becomes more stable, although it is unclear as to whether complete stability has been achieved. Nevertheless, it remains distinctly possible that the addition of light elements could stabilise this structure convincingly. Therefore, bcc-Fe cannot be ruled out as a candidate inner core phase. (C) 2008 Elsevier B.V. All rights reserved.

An investigation of the relative stabilities and equations of state of possible Fe3S polymorphs was conducted using first-principles pseudopotential calculations. These calculations were based on density functional theory and performed using ultrasoft Vanderbilt pseudopotentials within the generalized gradient approximation. In accord with experiment, we found that the tetragonal Fe3P-type polymorph is the only stable phase along the 0 K isotherm as a function of pressure. Fe3S exhibits permanent magnetism at ambient conditions (Fei et al., 2000), but magnetism is suppressed by pressure and temperature, and therefore non-magnetic data are appropriate ones to use for modelling planetary interiors. For this reason, and because the Fe3P-type polymorph of Fe3S contains 32 atoms per unit cell it was impractical to incorporate magnetic properties into the simulations of this phase, we studied the behaviour of the non-magnetic phase. We obtained values of 250 GPa for the bulk modulus, K0, and 4.61 for its first derivative withrespect to pressure, K0′, by fitting a 3rd order Birch-Murnaghan equation of state to the calculated internal energy as a function of volume for the non-magnetic Fe3P-type Fe3S. This suggests that a pressure far greater than that expected in the Martian interior would be needed to achieve a density comparable to that of the Martian core. We therefore conclude that it is unlikely that the core of Mars contains significant amounts of solid Fe3S.

A number of melting curves of various materials have recently been measured experimentally and calculated theoretically, but the agreement between different groups is not always good. We discuss here some of the problems which may arise in both experiments and theory. We also report the melting curves of Fe and Al calculated recently using quantum mechanics techniques, based on density functional theory with generalized gradient approximations. For Al our results are in very good agreement with both low pressure diamond-anvil-cell experiments (Boehler and Ross 1997 Earth Planet. Sci. Lett. 153 223, Hanstrom and Lazor 2000 J. Alloys Compounds 305 209) and high pressure shock wave experiments (Shaner et al 1984 High Pressure in Science and Technology ed Homan et al (Amsterdam: North-Holland) p 137). For Fe our results agree with the shock wave experiments of Brown and McQueen (1986 J. Geophys. Res. 91 7485) and Nguyen and Holmes (2000 AIP Shock Compression of Condensed Matter 505 8 1) and the recent diamond-anvil-cell experiments of Shen et al (1998 Geophys. Res. Lett. 25 373). Our results are at variance with the recent calculations of Laio et al (2000 Science 287 1027) and, to a lesser extent, with the calculations of Belonoshko et al (2000 Phys. Rev. Lett. 84 3638). The reasons for these disagreements are discussed.

The Earth has a liquid outer and solid inner core. It is predominantly composed of Fe, alloyed with small amounts of light elements, such as S, O and Si. The detailed chemical and thermal structure of the core is poorly constrained, and it is difficult to perform experiments to establish the properties of core-forming phases at the pressures (ca. 300 GPa) and temperatures (ca. 5000-6000 K) to be found in the core. Here we present some major advances that have been made in using quantum mechanical methods to simulate the high-P/T properties of Fe alloys, which have been made possible by recent developments in high-performance computing. Specifically, we outline how we have calculated the Gibbs free energies of the crystalline and liquid forms of Fe alloys, and so conclude that the inner core of the Earth is composed of hexagonal close packed Fe containing ca. 8.5% S (or Si) and 0.2% O in equilibrium at 5600 K at the boundary between the inner and outer cores with a liquid Fe containing ca. 10% S (or Si) and 8% O.

In order to predict the stable polymorph of iron under core conditions, calculations have been performed on all the candidate phases proposed for inner core conditions, namely, body-centred cubic (bcc), body-centred tetragonal (bct), hexagonal close-packed (hcp), double-hexagonal close-packed (dhcp) and an orthorhombically distorted hcp polymorph. Our simulations are ab initio free energy electronic structure calculations, based upon density functional theory, within the generalised gradient approximation; we use Vanderbilt ultrasoft non-normconserving pseudopotentials to describe the core interactions, and the frozen phonon technique to obtain the vibrational characteristics of the candidate structures. Our results show that under conditions of hydrostatic stress, the orthorhombic, bce and bet structures are mechanically unstable. The relative free energies of the remaining phases indicate that dhcp and fee Fe are thermodynamically less stable than hcp Fe, therefore, we predict that the stable phase of iron at core conditions is hcp-Fe. (C) 2000 Elsevier Science B.V. All rights reserved.

Ab initio finite temperature molecular dynamics simulations have been used to calculate the free energy and elasticity of face-centred cubic (fcc) iron at a state point representative of the Earth's inner core. Whilst the free energy of this phase is found to be higher than that of hexagonal-close-packed (hcp) iron, the difference is only 14 meV/atom. It is possible that this difference might be overcome by the presence of light elements, as previous calculations at zero Kelvin have shown that the addition of elements such as silicon stabilise fec-Fe with respect to hcp-Fe by at least 40 meV/atom. The calculated elastic constants at core pressures and temperatures of pure fec-Fe, and of alloys of Fe with sulphur and nickel (Fe(3)S and Fe(3)Ni) derived from the fee structure, lead to average shear wave velocities that are considerably higher than those inferred from seismology; however, these mineralogical and seismological results could be reconciled by the presence of partial melt in the inner core. The calculated P-wave anisotropy of fec-Fe is comparable with the seismological values, but only if there is a high degree of crystal alignment, although the necessity for alignment can be reduced if a layered model for the inner core is invoked. The results presented in this paper therefore suggest that fcc-Fe cannot be ruled out as a candidate for the dominant phase of the Earth's inner core. (c) 2008 Elsevier B.V. All rights reserved.

Atomic-scale materials modelling based on first-principles quantum mechanics is playing an important role in the science of the Earth and the other planets. We outline the basic theory of this kind of modelling and explain how it can be applied in a variety of different ways to probe the thermodynamics, structure and transport properties of both solids and liquids under extreme conditions. After a summary of the density functional formulation of quantum mechanics and its practical implementation through pseudopotentials, we outline the simplestway of applying first-principles modelling, namely static zero-temperature calculations. We show how calculations of this kind can be compared with static compression experiments to demonstrate the accuracy of first-principles modelling at pressures reached in planetary interiors. Noting that virtually all problems concerning planetary interiors require an understanding of materials at high temperatures as well as high pressures, we then describe how first-principles lattice dynamics gives a powerful way of investigating solids at temperatures not too close to the melting line. We show how such calculations have contributed to important progress, including the recent discovery of the post-perovskite phase of MgSiO3 in the D '' layer at the base of the Earth's mantle. A range of applications of first-principles molecular dynamics are then reviewed, including the properties of metallic hydrogen in Jupiter and Saturn, of water, ammonia and methane in Uranus and Neptune, and of oxides and silicates and solid and liquid iron and its alloys in the Earth's deep interior. Recognizing the importance of phase equilibria throughout the planetary sciences, we review recently developed techniques for the first- principles calculation of solid and liquid free energies, melting curves and chemical potentials of alloys. We show how such calculations have contributed to an improved understanding of the temperature distribution and the chemical composition throughout the Earth's interior. The review concludes with a summary of the present state of the field and with some ideas for future developments.

[1] Using first-principles molecular dynamics (MD) simulations, we study the solubility of argon in molten iron at high pressures and temperatures. In particular we explore whether the low pressure immiscibility of liquid Fe and Ar persists to high pressure (130 GPa) and temperature (4500K), or whether they mix. Starting from a variety of Fe/Ar mixtures we find that they always separate rapidly into two liquids. We conclude that there is no evidence for a significant increase in the solubility of Ar in Fe at these conditions. We cannot, therefore, attribute the lower melting temperatures of Fe obtained from DAC experiments compared to those obtained from ab initio calculations and shock experiments, to eutectic melting between Fe and the Ar pressure medium.

Ab initio calculations of the melting properties of copper in the pressure range 0-100 GPa are reported. The ab initio total energies and ionic forces of systems representing solid and liquid copper are calculated using the projector augmented wave implementation of density functional theory with the generalized gradient approximation for exchange-correlation energy. An initial approximation to the melting curve is obtained using an empirical reference system based on the embedded-atom model, points on the curve being determined by simulations in which solid and liquid coexist. The approximate melting curve so obtained is corrected using calculated free energy differences between the reference and ab initio system. It is shown that for system-size errors to be rendered negligible in this scheme, careful tuning of the reference system to reproduce ab initio energies is essential. The final melting curve is in satisfactory agreement with extrapolated experimental data available up to 20 GPa, and supports the validity of previous calculations of the melting curve up to 100 GPa. (C) 2004 American Institute of Physics.

This thesis reports the results of first-principles computational studies of thirteen crystalline structures in the H2O-NH3 system. This includes eight low- and highpressure polymorphs of pure water ice, two polymorphs of solid ammonia, and three low-pressure stoichiometric ammonia hydrates. These simulations have been used to determine the athermal equation of state (EoS) of each phase. Where empirical data was lacking, experiments have been undertaken. Hence, this thesis also reports the results of time-of-flight neutron scattering studies of deuterated ammonia dihydrate powders down to 4 K, and up to a maximum pressure of 8.6 GPa. In addition, I have developed a flexible and accurate planetary model that can be used to calculate the triaxial shape and gravitational field of any object, regardless of size or composition, given an assumed mineralogical constitution and provided the EoS of said minerals are known. The EoS parameters found in this work have therefore been used to model the structure and thermal evolution of icy moons orbiting Saturn in anticipation of the Cassini spacecraft arriving at Saturn in mid-2004. Models of Rhea, Saturn’s second largest moon, suggest that its volatile component is likely to contain > 3 weight percent ammonia, but that one is unlikely to be able to constrain the bulk chemistry of the ice mantle from Cassini flyby data.

The Earth's core is largely composed of iron (Fe). The phase relations and physical properties of both solid and liquid Fe are therefore of great geophysical importance. As a result, over the past 50 years the properties of Fe have been extensively studied experimentally. However, achieving the extreme pressures (up to 360 GPa) and temperatures (similar to6000 K) found in the core provide a major experimental challenge, and it is not surprising that there are still considerable discrepancies in the results obtained by using different experimental techniques. In the past 15 years quantum mechanical techniques have been applied to predict the proper ties of Fe. Here we review the progress that has been made in the use of first principles methods in the study of Fe, and focus upon (i) the structure of Fe under core conditions, (ii) the high P melting behaviour of Fe, (iii) the thermodynamic properties of hexagonal close-packed (hcp) Fe, and (iv) the rheological and thermodynamic properties of high P liquid Fe. (C) 2003 Elsevier B.V. All rights reserved.

The nature of the stable phase of iron in the Earth's solid inner core is still highly controversial. Laboratory experiments 1 suggest the possibility of an uncharacterized phase transformation in iron at core conditions and seismological observations(2-4) have indicated the possible presence of complex, inner-core layering. Theoretical studies(5,6) currently suggest that the hexagonal close packed (h. c. p.) phase of iron is stable at core pressures and that the body centred cubic (b. c. c.) phase of iron becomes elastically unstable at high pressure. In other h. c. p. metals, however, a high-pressure b. c. c. form has been found to become stabilized at high temperature. We report here a quantum mechanical study of b.c.c.-iron able to model its behaviour at core temperatures as well as pressures, using ab initio molecular dynamics free-energy calculations. We find that b.c.c.-iron indeed becomes entropically stabilized at core temperatures, but in its pure state h.c.p.-iron still remains thermodynamically more favourable. The inner core, however, is not pure iron, and our calculations indicate that the b. c. c. phase will be stabilized with respect to the h. c. p. phase by sulphur or silicon impurities in the core. Consequently, a b.c.c.-structured alloy may be a strong candidate for explaining the observed seismic complexity of the inner core(2-4).

The melting curve of the face-centered cubic (fcc) phase of aluminum has been determined from 0 to $\ensuremath{\sim}150 \mathrm{GPa}$ using first-principles calculations of the free energies of both the solid and liquid. The calculations are based on density functional theory within the generalized gradient approximation using ultrasoft Vanderbilt pseudopotentials. The free energy of the harmonic solid has been calculated within the quasiharmonic approximation using the small-displacement method; the free energy of the liquid and the anharmonic correction to the free energy of the solid have been calculated via thermodynamic integration from suitable reference systems, with thermal averages calculated using ab initio molecular dynamics. The resulting melting curve is in good agreement with both static compression measurements and shock data.

We report phonon densities of states (DOS) of iron measured by nuclear resonant inelastic x-ray scattering to 153 gigapascals and calculated from ab initio theory. Qualitatively, they are in agreement, but the theory predicts density at higher energies. From the DOS, we derive elastic and thermodynamic parameters of iron, including shear modulus, compressional and shear velocities, heat capacity, entropy, kinetic energy, zero-point energy, and Debye temperature. In comparison to the compressional and shear velocities from the preliminary reference Earth model (PREM) seismic model, our results suggest that Earth's inner core has a mean atomic number equal to or higher than pure iron, which is consistent with an iron-nickel alloy.

Ab initio molecular dynamics calculations, based upon density functional theory within the generalised gradient approximation (GGA) using ultrasoft non-norm conserving Vanderbilt pseudopotentials, have been used to predict the transport properties of liquid Fe–S. In order to compare our simulations with experimental data, the simulations were performed for the eutectic composition of Fe–S at the experimentally accessible conditions of 5 GPa and of 1300 and 1500 K. Our results give values for Fe and S diffusion of a few times 10−5 cm2 s−1. Our calculated viscosities, obtained directly from the simulations, are 11±5 and 4±1 mPa s at 1300 and 1500 K, respectively. Our calculated diffusion and viscosity coefficients agree well with recent experiments at similar pressures and temperatures, supporting a high diffusivity and low viscosity in liquid Fe–S at temperatures up to a few hundred Kelvin above the eutectic temperature. Furthermore, an extensive study of the liquid structure shows no evidence for sulphur polymerisation or the existence of any large viscous flow units. These results are in direct conflict with the previously reported experimental results of Le Blanc and Secco [LeBlanc, G.E., Secco, R.A., 1996. Viscosity of an Fe–S liquid up to 1300°C and 5 GPa, Geophys. Res. Lett., 23, 213–216.].

The inferred density of the solid inner core indicates that it is predominantly made of iron. In order to indicates that it is predominantly made of iron. In order to interpret the observed seismic anisotropy and understand the high pressure and temperature behaviour of the core, it is essential to establish the crystal structure of iron under core conditions. On the basis of extrapolated experimental data, a number of candidate structures for the high PIT iron phase have been proposed, namely, body-centred cubic (bcc), body-centred tetragonal (bct), hexagonal close-packed (hcp), double-hexagonal close-packed (dhcp) and an orthorhombically distorted hcp polymorph (Matsui, 1993; Stixrude and Cohen, 1995; Boehler, 1993; Saxena et al., 1996; Andrault et al., 1997). Here we present the results of the first fully ab initio free energy calculations for all of these polymorphs of iron at core pressures and temperatures. Our results show that hcp-Fe is the most stable polymorph of iron under the conditions of the Earth's inner core.