Elasticity and anisotropy of iron-nickel phosphides at high pressures

Authors


Abstract

[1] Geochemical estimates indicate that around 90% of the planet's inventory of phosphorus is likely to be sequestered in the Earth's core. Iron phosphides such as scheirbisites (Fe3P) are commonly found in iron meteorites. Recently, melliniite (Fe,Ni)4P with 12.2 wt% phosphorus has been reported in iron-meteorites. Using static electronic structure calculations, we predict that Fe4P is unlikely to dissociate into Fe3P and hcp Fe at inner core conditions. Among the different structural varieties of Fe4P, we find the cubic polymorph with P213 space group symmetry to be stable over a wide range of geophysically relevant pressures. We have determined the equation of state and the full elastic constant tensor of the stable (Fe,Ni)4P phase at pressures up to 400 GPa. Upon compression, Fe4P undergoes a ferromagnetic (fm) to nonmagnetic (nm) transition at 80 GPa. In nonmagnetic (Fe,Ni)4P, nickel incorporation results in reduction of the P- and S-wave velocities. However, incorporation of nickel enhances the P- and S-wave anisotropy.

1. Introduction

[2] “Light” elements (elements that have a lower atomic number), which are cosmically abundant and soluble in Fe, are required to explain the “missing” 8–10% and 4–5% of the outer and inner core, respectively. Thus H, O, C, S, P and Si are the most likely “light” elements in the core [Poirier, 1994]. Early models for the Earth's core consisted of considerable amount (up to ∼5% or so) of phosphides (schreibersite, (Fe,Ni)3P), carbides (cohenite, Fe3C), sulfides (troilite, FeS) and carbon (diamond and graphite) [Washington, 1925]. Based on geochemical estimates, sulfur, carbon and phosphorus are among the 10 most common elements in the Earth. Together with the major and refractory elements (Fe, O, Si, Mg, Ni, Ca and Al) sulfur, carbon and phosphorus account for >99% of the total mass [McDonough, 2003]. The concentration of phosphorus in the Earth's mantle is around 0.009 wt% [McDonough et al., 1985] whereas the phosphorus content of the bulk Earth is estimated to be around 0.10 wt%. Although phosphorus is a volatile element, it plots below the volatility trend in the plot of the ratios of elemental concentrations in the bulk silicate Earth normalized by those in CI carbonaceous chondrites [McDonough, 2003]. This indicates that phosphorus is likely to be partitioned into the metallic phase. Current estimates of the phosphorus content of the Earth's core speculate that around 0.20 wt%, i.e., 90% of the planet's inventory of phosphorus is in the Earth's core [McDonough, 2003].

[3] Experimental studies on Fe-P(-S) system at 1 atm pressure [Baker, 1992; Zaitsev et al., 1995] and up to 23 GPa [Stewart and Schmidt, 2007] provide valuable insights on the incorporation of phosphorus in metallic melts. Up to 4 wt% of phosphorus is found to dissolve in crystalline iron at 23 GPa [Stewart and Schmidt, 2007]. In the Fe-P system, towards the iron rich portion of the eutectic composition, Fe and Fe3P are found to crystallize. In addition, Fe3P and Fe3S are iso-structural and are likely to form extensive solid solution in Fe-P-S system [Stewart and Schmidt, 2007]. Thus it is not unlikely to have iron phosphides in the solid inner core in equilibrium with the dissolved phosphorus in the liquid outer core.

[4] Iron meteorites provide additional clues regarding the composition of the Earth's core. Iron phosphide phases are quite common in iron meteorites [Wasson, 1974], and the amount of phosphorus in the meteorites ranges between 0.2 to 1.1 wt% [Buchwald, 1975]. Various iron phosphides in meteorites include schreibersite [Skála and Císařová, 2005], allabogdanite ((Fe,Ni)2P) [Britvin et al., 2002], barringerite ((Fe0.58Ni0.42Co0.003)1.95P) [Buseck, 1969], florenskyite (FeTiP) [Ivanov et al., 2000], and Perryite ((Ni,Fe)8(Si,P)3) [Reed, 1968]. More recently a metal rich variety, melliniite ((Fe,Ni)4P) [Pratesi et al., 2006] has also been reported.

[5] So far, most of the high-pressure investigations on iron phosphides have been restricted on Fe3P [Scott et al., 2007], Fe2P [Dera et al., 2008; Wu and Qin, 2010] and FeP [Gu et al., 2011]. Under compression, Fe3P undergoes first-order transition at around 15 GPa [Scott et al., 2007]. However, our understanding of the role of phosphorus as a light element in the core remains incomplete without understanding the high-pressure behavior of the metal rich variety, Fe4P, given that the bulk phosphorus content of the core is of the order of 0.2 wt% [McDonough, 2003]. The relative stability of Fe3P and Fe4P at high pressure needs to be evaluated. Density and elasticity of Fe4P will provide crucial information to evaluate the role of phosphorus in the Earth's core. In order to address these issues, we undertake first principle calculations to evaluate the energetics, structure, and elasticity of iron-nickel phosphides under high pressure.

2. Method

[6] We use density functional theory calculations with highly accurate projector augmented wave method (PAW) [Kresse and Joubert, 1999] as implemented in Vienna ab-initio simulation package (VASP) [Kresse and Hafner, 1993; Kresse and Furthmüller, 1996]. The generalized gradient approximation (PBE-GGA) [Perdew et al., 1996] is used for all calculation as it yields good results for metals and iron alloys [Zhang and Oganov, 2010; Mookherjee et al., 2011; Mookherjee, 2011a]. Geophysically important physical properties such as elasticity and compressibility depend on the specific volume of the material. The calculated physical properties are shifted in static calculations by a few percent relative to the experimental values at higher temperature. But the shift arises in a consistent manner [Mookherjee and Capitani, 2011; Mookherjee, 2011a]. This makes the static (0 K) elastic data extremely useful. Single crystal elastic constants of cubic (Fe,Ni)4P were computed from the stress-strain relations as outlined by Karki et al. [2001]. For cubic symmetry we derive three independent elastic constants (c11, c12, and c44). We applied positive and negative strains of magnitude 1% in order to accurately determine the stresses in the appropriate limit of zero strain.

[7] Iron phosphide, Fe4P has at least three candidates with distinct space group symmetry, Pmmm (and Z = 1) [Hornbogen, 1961], Pm3m (Z = 1, based on iron nitride, γ-Fe4N [Adler and Williams, 2005]), and P213 (Z = 4 [Pratesi et al., 2006]). Fe4P with P213 symmetry has two distinct metal site, Fe-I(12b) and Fe-II(4a). The phosphorus atoms are located in P(4a) sites (Figure S1 in the auxiliary material).

3. Results

[8] Static calculations indicate that compared to the two other polymorph with Pmmm and Pm3m symmetry, Fe4P with P213 has the lowest energy across the wide range of volume explored (Figure 1a). In order to evaluate the relative stability of Fe4P and Fe3P at high pressures, we have considered the following reaction:

equation image

At the pressures corresponding to the Earth's core, the Fe4P, Fe3P and Fe phases are non magnetic. Hence, we have estimated the enthalpy based on the nonmagnetic energies at static conditions. For Fe we have considered the hcp phase which is predicted to be stable compared to bcc Fe at 0 K [Oganov and Glass, 2006, and references therein]. The predicted bulk modulus for hcp Fe and bcc Fe is considerably stiffer compared to the experimental results (Table S1). The discrepancy between measured and computed values of K0 is consistent with previous ab initio studies [Mookherjee et al., 2011, and references therein]. Hence, we have used the experimentally determined bulk modulus for Fe for our calculations (Table S1). Our calculations predict that Fe3P undergoes a transition from tetragonal (space group I4) to orthorhombic (space group Pnma) structure similar to Fe3C [Mookherjee, 2011a] at 60 GPa. The transition from tetragonal to orthorhombic structure is consistent with the experimental observations, although the transition pressure is over-predicted [Scott et al., 2008]. Upon further compression, orthorhombic Fe3P transforms to hexagonal symmetry (space group P63mmc) structure similar to Sc3In [Compton and Matthias, 1962] at 260 GPa (Figure S2). At static condition, the free energy change associated with the reaction, ΔG is equivalent to the enthalpy change, ΔH of the reaction, the entropy term, TΔS could be ignored since T ∼ 0 K. Based on the enthalpy difference, we conclude that Fe4P is likely to be stable till 360 GPa (inner core conditions). We have considered representative structures for Fe3P based on iron alloys such as Fe3S and Fe3C. It is possible that other stable crystal structures might exist, and these could be predicted from the evolutionary structure algorithm [Oganov and Glass, 2006]. However, simulations predict that the bulk modulus (K0) for the non-magnetic Fe3P and Fe4P irrespective of the structures are of the similar order (Table S1). The lower bulk modulus of Fe will lead to a larger volume at higher pressure and as a result higher enthalpy for the Fe3P + Fe assemblage at inner core conditions, i.e., favoring Fe4P at inner core conditions. It is to be noted that the enthalpy difference at the inner core condition is of the order of 0.16 eV (∼1800 K) (Figure 1b). Temperature experienced by meteorites on entering the Earth's atmosphere is likely to be of similar order might transform Fe4P to Fe3P. Higher inner core temperatures (∼6000 K) and presence of other minor elements (e.g., sulfur) in the core are likely to affect the relative stability of Fe4P versus Fe3P. However, the limited phosphorus budget in the Earth's interior might favor Fe4P over Fe3P.

Figure 1.

(a) The total energy vs. volume for the Fe4P with P213 (green rhombs), Pmmm (blue triangles) and Pm3m (red circles); (b) the calculated enthalpy difference as a function of pressure, for the reaction equation (1), (tet.), (orth.) and (hex.) refers to the symmetry of the Fe3P phase, the green line refers to the likely Fe3P + Fe assemblage, the dashed lines refers to the energetically unstable extensions for the Fe3P + Fe assemblage, the “zero” enthalpy refers to Fe4P; the Fe4P is becomes stable at around 100 GPa and remains stable with respect to the Fe3P + Fe assemblage till 400 GPa. (c) Energy vs. volume plot for fm (green circles) and nm (green rhombs) Fe4P, the lines represent finite strain fits. The inset refers to the magnetic moment for Fe-I (golden circles) and Fe-II (green circles) atoms vs. unit cell volume, in the right hand axis, the magnetization energy (green line), i.e., the energy difference between nm and fm phase is also plotted for comparison.

[9] Ferromagnetic Fe4P is found to be energetically stable over the nonmagnetic solution for all volumes larger than 9.2 Å3/atom, where the magnetic moment is lost from the structure on all Fe sites and the magnetization energy drops to zero at around 80 GPa (Figure 1c). High-pressure experiments such as X-ray emission spectroscopic or Mössbauer studies are required to constrain the environment of Fe atoms and the pressures induced magnetic collapse in Fe4P. The calculated energy-volume relations for the fm and nm Fe4P are shown in (Figure 1c). For the nm phase we find that a third-order Birch-Murnaghan finite strain equation of state (BM-EoS) [Birch, 1947] adequately describes energy volume results with V0nm = 10.35 Å3/atom, K0nm = 269 GPa and K0nm = 4.6 (Table S1). Owing to lack of experimental data on Fe4P, we compare first principle results of ferromagnetic tetragonal Fe3P with that of experimental results. Birch-Murnaghan finite strain equation of state data on our simulation results- V0fm = 11.25 Å3/atom, K0fm = 179 GPa and K0fm = 3.9- nicely agree with the experimental data of V0fm = 11.54 Å3/atom, K0fm = 159 GPa and K0fm = 4.0 [Scott et al., 2007]. Similar to other iron alloys such as Fe3C and Fe7C3 [Mookherjee et al., 2011; Mookherjee, 2011a], there is a significant difference in the bulk moduli of the ferromagnetic and nonmagnetic phases, with the nm Fe4P being considerably stiffer than the fm phase, as illustrated by the energy difference between the nm and fm E-V EoS (Figure 1c).

[10] The elastic constants of the nm phase increase monotonically with pressure (Figure 2 and Table S2). For densities corresponding to the inner core, the bulk modulus (K) and the shear modulus (G) for Fe4P are greater than PREM [Dziewonski and Anderson, 1981] whereas the K for Ni4P is slightly smaller than PREM, and the G for Ni4P remains greater than PREM (Figure 2). FeNi−1 substitution tends to reduce the primary VP and shear VS wave velocities (Figure 2).

Figure 2.

Pressure dependence of single crystal elastic constants of cubic phase (a) Fe4P and (b) Ni4P with c11 (green “upward” pointing triangle); c12 (green “downward” pointing triangle); c44 (green “rightward” pointing triangle); K (grey circles); G (grey rhombs) and PREM: crosses [Dziewonski and Anderson, 1981]; (c) VP (blue circles) and VS (red rhombs) as a function of nickel content [Ni%]; (d) single crystal velocities- VP (blue), VS1 (red), and VS2 (grey) for Fe4P (solid line) and Ni4P (dashed line) phases as a function of direction.

4. Discussion

[11] At the inner core conditions (P ∼ 329 GPa; T ∼ 6000 K), the extrapolated density of Fe4P is around, image ∼ 12.50 g/cm3. We have assumed a thermal expansion α0 ∼ 1.27 × 10−5 K−1 based on the thermal expansion of Ni3P [Samsonov et al., 1971] and its high-pressure extrapolation based on the relation α(P, T) = image We have assumed the Anderson-Grüneisen parameter, δT ∼ 4, based on cubic solid metals [Perrin and Delannoy-Coutris, 1988]. We determined the maximum volume fraction of Fe4P in the inner core boundary pressure and temperatures using the relation

equation image

where, ρFe ∼ 12.9 g/cm3, and ρPREM ∼ 12.76 g/cm3 [Dziewonski and Anderson, 1981], and the volume fraction x. We find a volume fraction, x ∼ 35% or a maximum possible phosphorus content of 4.2 wt% in the inner core. However, the phosphorus content is likely to be lower in presence of other light elements. In addition, the thermoelastic parameters at inner core conditions are not well constrained and certainly warrants further experimental and theoretical investigations.

[12] We also note that the compressional velocity, VP, is the fastest along the [111] direction of the Fe4P phase (Figure 2). If the iron phosphides are among the likely inner core candidates, then in order to explain the inner core anisotropy, the [111] direction of cubic Fe4P needs to be oriented along the polar direction. However, the knowledge of the slip systems and polycrystalline textural information, the pressure-temperature dependence of the slip-system needs to be explored before definitive conclusions could be drawn. Hence, any meaningful interpretation of inner core anisotropy requires high-temperature extrapolations. Nickel tends to enhance the anisotropy. However, the fast direction remains unaltered by the FeNi−1 substitution.

Acknowledgments

[13] This project is supported by grants from the National Natural Science Foundation of China (grants 40972029 and 41072027). Authors thank anonymous reviewer and Artem Oganov for their constructive criticism, which significantly improved the paper. We would like to thank Daniel J Frost for his comments on the science and presentation of the article.

[14] The Editor thanks Artem Oganov and an anonymous reviewer for their assistance in evaluating this paper.