Electrical resistivity and thermal conductivity of hcp Fe–Ni alloys under high pressure: Implications for thermal convection in the Earth’s core

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Abstract

We measured the electrical resistivity of Fe–Ni alloys (iron with 5, 10, and 15 wt.% nickel) using four-terminal method in a diamond-anvil cell up to 70 GPa at 300 K. The results demonstrate that measured resistivity increases linearly with increasing nickel impurity concentration, as predicted by the Matthiessen’s rule. The impurity resistivity is predominant at ambient temperature; the incorporation of 5 wt.% nickel into iron doubles the electrical resistivity at 60 GPa. Such impurity effect becomes minor at high temperature of the Earth’s core because of the resistivity “saturation”. We also calculated that >0.9 TW heat flow is necessary at the top of the inner core for thermal convection in the inner core. It requires the CMB heat flow of ∼30 TW, which is much higher than recent estimates of 5–15 TW. This means that purely thermal convection does not occur in the inner core.

Introduction

The estimates of the electrical resistivity of the Earth’s core started from Elsasser (1946) (Fig. 1, Table 1). He considered that the resistivity is proportional to temperature and inversely proportional to square of the Debye temperature, neglecting the effect of alloying. On the other hand, Bullard (1948) calculated the resistivity considering temperature, pressure, and nickel impurity effect. The values found by these two pioneer studies, however, differed by a factor of three. Bullard (1949) therefore adopted 3.0 × 10−6 Ω m for core resistivity by averaging Elsasser’s (1946) value and his own value. This value had been most widely used before Gardiner and Stacey (1971), who reported 2.59–3.19 × 10−6 Ω m for core-mantle boundary (CMB) and 2.62–3.25 × 10−6 Ω m for inner core boundary (ICB), respectively, with uncertainty of 1–6 × 10−6 Ω m, based on the resistivity of both liquid iron alloys under 1 bar (Baum et al., 1967) and solid iron alloys at high pressure (Bridgman, 1957) as well as theory for pressure dependence of the resistivity (Lawson, 1956). On the other hand, Evans and Jain (1972) first computed the electrical resistivity of liquid iron as a function of density, and concluded that the resistivity should not exceed 3.3 × 10−6 Ω m. Jain and Evans (1972) further calculated the resistivity of pure liquid iron to be 1.04 ± 0.06 × 10−6 Ω m for the CMB and suggested that the resistivity of the Earth’s core lies between 1.0 and 2.0 × 10−6 Ω m considering the effect of impurity. Johnston and Strens (1973) observed the insulator to metal transition in graphite-saturated (Fe90Ni10)3S2 liquid above 1100 °C and concluded that the core resistivity must be higher than 2.0 × 10−6 Ω m. The shock compression experiments on iron-silicon alloys (Fe with 7.7–34.2 at.% Si) performed by Matassov (1977) at 5–140 GPa and corresponding Hugoniot temperature of 670–2700 K estimated the resistivity of the core to be 1.15 × 10−6 Ω m. Static high-pressure experiments were also carried out by Secco and Schloessin (1989). The resistivity of iron was measured up to 7 GPa and above melting temperature in a large-volume press, and the change in resistivity was found to be very little upon melting, consistent with measurements at ambient pressure. And they empirically estimated the core resistivity to be 1.2–1.5 × 10−6 Ω m. In addition, various experimental measurements have been reported. Both static (Balchan and Drickamer, 1961, Bargen and Boehler, 1990, Garg et al., 2004, Jaccard et al., 2002, Reichlin, 1983) and shock wave experiments (Keeler and Mitchell, 1969) observed sharp resistivity change across the phase transition from body-centered cubic (bcc) to hexagonal-close-packed (hcp) structure.

Following these measurements, especially shock wave experiments by Matassov (1977) performed in wide pressure, temperature, and compositional ranges, analytical modeling of core resistivity has been frequently reported (e.g. Anderson, 1998, Stacey and Anderson, 2001). Stacey and Anderson (2001) hypothesized that the resistivity of iron is constant along its melting curve, based on the idea that the resistivity of pure metal is determined by atomic vibration and melting occurs when the magnitude of such vibration reaches a given critical value as predicted by Lindemann’s melting law (Gilvarry, 1956). Consequently the resistivity of iron was calculated to be 1.22 × 10−6 Ω m at the CMB, and 1.12 × 10−6 Ω m at the ICB. Adding the impurity resistivity of silicon and nickel, Stacey and Anderson (2001) gave 2.12 × 10−6 Ω m and 2.02 × 10−6 Ω m at the CMB and the ICB, respectively. Then, new shock compression experiment was reported up to 200 GPa (Bi et al., 2002). Bi and others suggested the possibility that resistivity was underestimated in earlier shock wave experiments because of the melting of insulating epoxy at high temperature. Considering the new measurement (Bi et al., 2002), both Davies (2007) and Stacey and Loper (2007) revised the estimate by Stacey and Anderson (2001) downwardly. Davies (2007) calculated 1.25–1.9 × 10−6 Ω m for pure iron and 2.15–2.80 × 10−6 Ω m for Fe–Si alloy at the CMB. He also mentioned that the impurity resistivity of oxygen and/or sulfur may be larger than that of silicon by a factor of 2 or more. Stacey and Loper (2007) reconsidered the pressure dependence of pure metal, based on both new measurements (Bi et al., 2002) and geophysical arguments, and obtained the core resistivity of 3.62 × 10−6 Ω m and 4.65 × 10−6 Ω m for the CMB and ICB, respectively. These values are the highest estimates ever made.

Sha and Cohen (2011) conducted first-principles calculation of hcp iron and reported the resistivity of inner core to be 7.5 × 10−7Ω m–1.5 × 10−6 Ω m from the Boltzmann equation, which are significantly lower than those obtained by earlier analytical works (Davies, 2007, Stacey and Loper, 2007). More recently, the first-principles molecular dynamics (MD) simulations demonstrated the electrical resistivity by means of Kubo–Greenwood equation; de Koker et al. (2012) for liquid iron, iron-silicon, and iron-oxygen alloys, Pozzo et al. (2012) for pure liquid iron, and Pozzo et al. (2013) for liquid iron–silicon–oxygen alloy. These three calculations used very similar methods and thus found similar resistivity for liquid outer core around 9 × 10−7 Ω m, which is the lowest value among previous estimates. Additionally, Pozzo et al. (2014) computed the resistivity of solid iron–silicon mixture at the inner core conditions. Zhang et al. (2015) calculated the resistivity originated from electron–electron scattering, which has not been considered by previous estimates, and found that it is comparable to that by electron–phonon scattering at high temperature. They argued that the resistivity of iron at core density and temperature is significantly higher than previous MD simulations (de Koker et al., 2012, Pozzo et al., 2012, Pozzo et al., 2013) and consistent with the values given by Stacey and Anderson (2001), assuming that both Bloch–Grüneisen law and Matthiessen’s rule are valid.

Gomi et al. (2013) measured the resistivity of iron and iron–silicon alloy in a diamond-anvil cell (DAC) up to 100 GPa at ambient temperature and also conducted first-principles calculations with pseudo-potential methods. They first considered the effect of resistivity “saturation” (e.g. Bohnenkamp et al., 2002) in the modeling and obtained the core resistivity ranging from 7.04 × 10−7 Ω m to 1.09 × 10−6 Ω m for the CMB and from 5.52 × 10−7 Ω m to 8.89 × 10−7 Ω m for the ICB, depending on the choice of light alloying component whose effect is suggested by the Linde’s rule. These values are indeed consistent with previous first-principles MD calculations (de Koker et al., 2012, Pozzo et al., 2012, Pozzo et al., 2013). Seagle et al. (2013) also measured the resistivity of iron and iron-silicon alloy up to 60 GPa and 300 K in a DAC, and reported smaller resistivity below 40 GPa but similar results at higher pressures. Additionally, Kiarasi (2013) reported the electrical resistivity measurements of solid and liquid Fe + 17 wt.% Si alloy in a cubic anvil press up to 5 GPa, and estimated the core resistivity to be 9.0–9.4 × 10−7 Ω m.

It is widely accepted that the Earth’s iron-based core is alloying with 5–10 at.% nickel. The impurity resistivity of nickel, however, has not been measured under static high pressure yet. This is partly because the impurity resistivity of transition metal hosted in transition metal has been found to be small in general (Mertig, 1999, Tsiovkin et al., 2005, Tsiovkin et al., 2006). Stacey and Anderson (2001) estimated the impurity resistivity for 10 at.% Ni to be 1.5 × 10−7 Ω m. Here we performed the electrical resistivity measurements of Fe–Ni alloys (5, 10, and 15 at.% Ni) at high pressures to 70 GPa in a diamond-anvil cell (DAC). With the impurity of nickel obtained, we model the thermal conductivity of the Earth’s core considering the resistivity saturation originally proposed by Gomi et al. (2013).

Section snippets

Methods

The electrical resistivity was measured at high pressure in a DAC with flat 300 μm or beveled 150 μm culet diamond anvils (Fig. 2). We used rhenium and Al2O3 composite as a gasket. A rhenium gasket was pre-indented to a thickness of about 30 μm. A hole whose diameter was 80% of the culet size was drilled on a pre-indented gasket. Al2O3 powder filled the hole and covered the surface of the indent in order to isolate the rhenium gasket from a sample and electrical leads. They were then re-compressed

Results

We obtained the electrical resistivity of iron–nickel alloys up to 70 GPa in this study (Fig. 3). Large drops in resistivity were observed across the phase transition from hexagonal-close-packed (hcp) to body-centered cubic (bcc) structure with decreasing pressure. The nickel impurity resistivity is found to be significant at room temperature. At 60 GPa where magnetic fluctuations are suppressed, the incorporation of 5 at.% Ni into Fe doubles its resistivity.

The present resistivity data provide

Difference in resistivity between bcc and hcp phases

The difference in resistivity between the bcc and the hcp phases of iron has been inconsistent in the literature (Balchan and Drickamer, 1961, Garg et al., 2004, Gomi et al., 2013, Jaccard et al., 2002, Reichlin, 1983, Seagle et al., 2013). The resent measurements by Gomi et al. (2013) reported a jump by ∼1 × 10−7 Ω m at the bcc to hcp transition around 15 GPa (Fig. 3), consistent with earlier static (Jaccard et al., 2002) and shock compression experiments (Keeler and Mitchell, 1969) (see Fig. 2 of

Acknowledgments

We thank two anonymous reviewers for their helpful comments to the manuscript.

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