Gross thermodynamics of two-component core convection

We model the inner core by an alloy of iron and 8 per cent sulphur or silicon and the outer core by the same mix with an additional 8 per cent oxygen. This composition matches the densities of seismic model, Preliminary Reference Earth Model (PR-EM). When the liquid core freezes S and Si remain with the Fe to form the solid and excess 0 is ejected into the liquid. Properties of Fe, diffusion constants for S, Si, 0 and chemical potentials are calculated by first-principles methods under the assumption that S, 0, and Si react with the Fe and themselves, however, not with each other. This gives the parameters required to calculate the power supply to the geodynamo as the Earth's core cools. Compositional convection, driven by light O released at the inner-core boundary on freezing, accounts for half the entropy balance and 15 per cent of the heat balance. This means the same magnetic field can be generated with approximately half the heat throughput needed if the geodynamo were driven by heat alone. Chemical effects are significant: heat absorbed by disassociation of Fe and 0 almost nullify the effect of latent heat of freezing in driving the dynamo. Cooling rates below 69 K Gyr(-1) are too low to maintain thermal convection everywhere; when the cooling rate lies between 35 and 69 K Gyr(-1) convection at the top of the core is maintained compositionally against a stabilizing temperature gradient; below 35 K Gyr(-1) the dynamo fails completely. All cooling rates freeze the inner core in less than 1.2 Gyr, in agreement with other recent calculations. The presence of radioactive heating will extend the life of the inner core, however, it requires a high heat flux across the core-mantle boundary. Heating is dominated by radioactivity when the inner core age is 3.5 Gyr. We, also, give calculations for larger concentrations of O in the outer core suggested by a recent estimation of the density jump at the inner-core boundary, which is larger than that of PREM. Compositional convection is enhanced for the higher density jumps and overall heat flux is reduced for the same dynamo dissipation, however, not by enough to alter the qualitative conclusions based on PREM. Our preferred model has the core convecting near the limit of thermal stability, an inner-core age of 3.5 Gyr and a core heat flux of 9 TW or 20 per cent of the Earth's surface heat flux, 80 per cent of which originates from radioactive heating

Can the Earth's dynamo run on heat alone?

The power required to drive the geodynamo places significant constraints on the heat passing across the core-mantle boundary and the Earth's thermal history. Calculations to date have been limited by inaccuracies in the properties of liquid iron mixtures at core pressures and temperatures. Here we re-examine the problem of core energetics in the light of new first-principles calculations for the properties of liquid iron. There is disagreement on the fate of gravitational energy released by contraction on cooling. We show that only a small fraction of this energy, that associated with heating resulting from changes in pressure, is available to drive convection and the dynamo. This leaves two very simple equations in the cooling rate and radioactive heating, one yielding the heat flux out of the core and the other the entropy gain of electrical and thermal dissipation, the two main dissipative processes. This paper is restricted to thermal convection in a pure iron core; compositional convection in a liquid iron mixture is considered in a companion paper. We show that heat sources alone are unlikely to be adequate to power the geodynamo because they require a rapid secular cooling rate, which implies a very young inner core, or a combination of cooling and substantial radioactive heating, which requires a very large heat flux across the core-mantle boundary. A simple calculation with no inner core shows even higher heat fluxes are required in the absence of latent heat before the inner core formed

Thermodynamic stability of Fe/O solid solution at inner-core conditions

We present a new technique which allows the fully {\em ab initio} calculation of the chemical potential of a substitutional impurity in a high-temperature crystal, including harmonic and anharmonic lattice vibrations. The technique uses the combination of thermodynamic integration and reference models developed recently for the {\em ab initio} calculation of the free energy of liquids and anharmonic solids. We apply the technique to the case of the substitutional oxygen impurity in h.c.p. iron under Earth's core conditions, which earlier static {\em ab initio} calculations indicated to be thermodynamically very unstable. Our results show that entropic effects arising from the large vibrational amplitude of the oxygen impurity give a major reduction of the oxygen chemical potential, so that oxygen dissolved in h.c.p. iron may be stabilised at concentrations up a few mol % under core conditions

First principles simulations of direct coexistence of solid and liquid aluminium

First principles calculations based on density functional theory, with generalised gradient corrections and ultrasoft pseudopotentials, have been used to simulate solid and liquid aluminium in direct coexistence at zero pressure. Simulations have been carried out on systems containing up to 1000 atoms for 15 ps. The points on the melting curve extracted from these simulations are in very good agreement with previous calculations, which employed the same electronic structure method but used an approach based on the explicit calculation of free energies [L. Vo\v{c}adlo and D. Alf\`e, Phys. Rev. B, {\bf 65}, 214105 (2002).]Comment: To appear in Phys. Rev.

Comment on 'Molybdenum at High Pressure and Temperature: Melting from Another Solid Phase'

There has been a major controversy over the past seven years about the high-pressure melting curves of transition metals. Static compression (diamond-anvil cell: DAC) experiments up to the Mbar region give very low melting slopes dT_m/dP, but shock-wave (SW) data reveal transitions indicating much larger dT_m/dP values. Ab initio calculations support the correctness of the shock data. In a very recent letter, Belonoshko et al. propose a simple and elegant resolution of this conflict for molybdenum. Using ab initio calculations based on density functional theory (DFT), they show that the high-P/high-T phase diagram of Mo must be more complex than was hitherto thought. Their calculations give convincing evidence that there is a transition boundary between the normal bcc structure of Mo and a high-T phase, which they suggest could be fcc. They propose that this transition was misinterpreted as melting in DAC experiments. In confirmation, they note that their boundary also explains a transition seen in the SW data. We regard Belonoshko et al.'s Letter as extremely important, but we note that it raises some puzzling questions, and we believe that their proposed phase diagram cannot be completely correct. We have calculated the Helmholtz and Gibbs free energies of the bcc, fcc and hcp phases of Mo, using essentially the same quasiharmonic methods as used by Belonoshko et al.; we find that at high-P and T Mo in the hcp structure is more stable than in bcc or fcc.Comment: 1 page, 1 figure. submitted to Phys. Rev. Let

The earth’s core: an approach from first principles

The Earth’s core is largely composed of iron (Fe), alloyed with less dense elements such as sulphur, silicon and/or oxygen. The phase relations and physical properties of both solid and liquid Fe-alloys are therefore of great geophysical importance. As a result, over the past fifty years the properties of Fe and its alloys have been extensively studied experimentally. However, achieving the extreme pressures (up to 360 GPa) and temperatures (~6000K) 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 fifteen years quantum mechanical techniques have been applied to predict the properties of Fe. Here we review the progress that has been made in the use of first principles methods to study Fe and its alloys, and as a result of these studies we conclude: (i) that pure Fe adopts an hexagonal close packed structure under core conditions and melts at ~6200 K at 360 GPa, (ii) that thermodynamic equilibrium and observed seismic data are satisfied by a liquid Fe alloy outer core with a composition of ~10 mole% S (or Si) and 8 mole% O crystallising at ~ 5500 K to give an Fe alloy inner core with ~8 mole% S (or Si) and 0.2 mole % O, and (iii) that with such concentrations of S (or Si), an Fe alloy might adopt a body centred cubic structure in all or part of the inner core. In the future the roles of Ni, C, H and K in the core need to be studied, and techniques to predict the transport and rheological properties of Fe alloys need to be developed

Ab initio statistical mechanics of surface adsorption and desorption: I. H2_2O on MgO (001) at low coverage

We present a general computational scheme based on molecular dynamics (m.d.) simulation for calculating the chemical potential of adsorbed molecules in thermal equilibrium on the surface of a material. The scheme is based on the calculation of the mean force in m.d. simulations in which the height of a chosen molecule above the surface is constrained, and subsequent integration of the mean force to obtain the potential of mean force and hence the chemical potential. The scheme is valid at any coverage and temperature, so that in principle it allows the calculation of the chemical potential as a function of coverage and temperature. It avoids all statistical mechanical approximations, except for the use of classical statistical mechanics for the nuclei, and assumes nothing in advance about the adsorption sites. From the chemical potential, the absolute desorption rate of the molecules can be computed, provided the equilibration rate on the surface is faster than the desorption rate. We apply the theory by {\em ab initio} m.d. simulation to the case of H$_2$O on MgO (001) in the low-coverage limit, using the Perdew-Burke-Ernzerhof (PBE) form of exchange-correlation. The calculations yield an {\em ab initio} value of the Polanyi-Wigner frequency prefactor, which is more than two orders of magnitude greater than the value of $10^{13}$ s$^{-1}$ often assumed in the past. Provisional comparison with experiment suggests that the PBE adsorption energy may be too low, but the extension of the calculations to higher coverages is needed before firm conclusions can be drawn. The possibility of including quantum nuclear effects by using path-integral simulations is noted.Comment: 11 pages + 10 figure

The axial ratio of hcp iron at the conditions of the Earth's inner core

We present ab initio calculations of the high-temperature axial c/a ratio of hexagonal-close-packed (hcp) iron at Earth's core pressures, in order to help interpret the observed seismic anisotropy of the inner core. The calculations are based on density functional theory, which is known to predict the properties of high-pressure iron with good accuracy. The temperature dependence of c/a is determined by minimising the Helmholtz free energy at fixed volume and temperature, with thermal contributions due to lattice vibrations calculated using harmonic theory. Anharmonic corrections to the harmonic predictions are estimated from calculations of the thermal average stress obtained from ab initio molecular dynamics simulations of hcp iron at the conditions of the inner core. We find a very gradual increase of axial ratio with temperature. This increase is much smaller than found in earlier calculations, but is in reasonable agreement with recent high-pressure, high-temperature diffraction measurements. This result casts doubt on an earlier interpretation of the seismic anisotropy of the inner core

Transport properties for liquid silicon-oxygen-iron mixtures at Earth's core conditions

We report on the thermal and electrical conductivities of two liquid silicon-oxygen-iron mixtures (Fe$_{0.82}$Si$_{0.10}$O$_{0.08}$ and Fe$_{0.79}$Si$_{0.08}$O$_{0.13}$), representative of the composition of the Earth's outer core at the relevant pressure-temperature conditions, obtained from density functional theory calculations with the Kubo-Greenwood formulation. We find thermal conductivities $k$ =100 (160) W m$^{-1}$ K$^{-1}$, and electrical conductivities $\sigma = 1.1 (1.3) \times 10^6 \Omega^{-1}$ m$^{-1}$ at the top (bottom) of the outer core. These new values are between 2 and 3 times higher than previous estimates, and have profound implications for our understanding of the Earth's thermal history and the functioning of the Earth's magnetic field, including rapid cooling rate for the whole core or high level of radiogenic elements in the core. We also show results for a number of structural and dynamic properties of the mixtures, including the partial radial distribution functions, mean square displacements, viscosities and speeds of sound.Comment: 16 pages, 12 figure