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

We present a new technique which allows the fully 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 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 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. The thermodynamic stability of oxygen dissolved in iron is a key factor in considering the physics and chemistry of the Earth’s core. We present here a new technique which allows the ab initio calculation of the chemical potential of an impurity in a high-temperature solid solution, including harmonic and anharmonic lattice vibrations. We report the application of the technique to substitutional oxygen dissolved in hexagonal close-packed (h.c.p.) iron at Earth’s core conditions, and we show that the Fe/O solid solution is thermodynamically far more stable than expected from earlier work. The new technique should find wide application to a range of other earth-science problems. It has long been recognised that the liquid outer core must contain a substantial fraction of light impurities, since its density is 6− 10 % less than that estimated for pure liquid Fe (Birch 1964, Poirier 1994); similar arguments suggest that the inner core contains a smaller, but still appreciable impurity fraction (Stixrude et al. 1997, Jephcoat and Olson 1987). The leading impurity candidates are S, Si and O, and arguments have been presented for and against each of them (Poirier 1994). Ringwood (Ringwood 1977) and others (Dubrovskiy and Pan’kov 1972) have argued strongly on grounds of geochemistry that oxygen must account for a large part of the impurity content. However, it has proved difficult to assess these ideas, because the Fe/O phase diagram is so poorly known at Earth’s core conditions. (For Copyright 2000 by the American Geophysical Union. Paper number 2000GL011567. 0094-8276/00/2000GL011567$05.00 reference, we note that the pressures at the core-mantle boundary, the inner-core boundary (ICB) and the centre of the Earth are 136, 330 and 364 GPa respectively; the temperatures at the core-mantle boundary and the ICB are poorly established, but are believed to be in the region of 4000 and 6000 K respectively.) The thermodynamic stability of dissolved oxygen is governed by the free energy change in the reaction (n− 1)Fe(solid) + FeO(solid)→ FenO(solid solution) (1) Let ∆G be the increase of Gibbs free energy as this reaction goes from left to right, excluding the configurational contribution associated with the randomness of the lattice sites occupied by dissolved O. Then the maximum concentration (number of O atoms per crystal lattice site) at which dissolved O is thermodynamically stable with respect to precipitation of FeO is cmax = exp(−∆G/kBT ). (At equilibrium the Gibbs free energy (G) of the left-hand-side (lhs) of Eq. (1) must be equal to that of the right-hand-side (rhs). On the rhs G = kBT ln c + G̃, where c is the O concentration and the kBT ln c term is the configurational contribution. The configurational term is absent in the lhs because Fe and FeO are separated perfect crystals.) Several years ago, Sherman (Sherman 1995) used ab initio calculations based on density functional theory (DFT) to calculate the zero-temperature limit of ∆G, i.e. the enthalpy ∆H of reaction (1). He found that ∆H is very large (∼ 5 eV at the ICB pressure of 330 GPa), and concluded that the concentration of dissolved O in the inner core must be completely negligible. His argument has been widely cited. However, these were static, zero-temperature calculations, which entirely ignored entropic effects. We shall show here that the hightemperature entropy of dissolved O produces such a large reduction of free energy that Sherman’s argument should be treated with caution when considering core temperatures. Our ab initio calculations are based on the well established DFT methods used in virtually all ab initio investigations of solid and liquid Fe (Stixrude et al. 1994, Soderlind et al. 1996, de Wijs et al. 1998a, Alfè et al. 1999a, 2000a), including Sherman’s (Sherman 1995). We employ the generalised gradient approximation for exchangecorrelation energy, as formulated by Perdew et al. (Perdew et al. 1992), which is known to give very accurate results for the low-pressure elastic, vibrational and magnetic properties of body-centred cubic (b.c.c.) Fe, the b.c.c. → h.c.p. transition pressure, and the pressure-volume relation for h.c.p. Fe up to over 300 GPa (Stixrude et al. 1994, Alfè et al. 2000a). We use the ultra-soft pseudopotential implementa-