The high conductivity of iron and thermal evolution of the Earth’s core
Introduction
Earth’s magnetic field is re-generated by dynamo action via convection currents in the liquid metal outer core, which are in turn driven by a combination of thermal buoyancy associated with secular cooling (along with possible radioactive heating) and buoyant release of incompatible light alloying components upon inner core solidification. Prior to the crystallization of an inner core, if thermal buoyancy alone drives convection, then the power for maintaining a geodynamo must be in excess of the heat conducted down the isentropic gradient that develops in the presence of convection, placing tight constraints upon the core’s thermal evolution (Stevenson, 2003, Labrosse, 2003).
Electrical conduction in metals is impeded by coupling between mobile electrons and the atomic lattice, a process that also dominates heat transfer in metals. This link between electrical resistivity and the electronic contribution to thermal conductivity of metallic iron is expressed through the Wiedemann–Franz law, k = 1/ρ × L × T, where k is the thermal conductivity, ρ is the electrical resistivity, L is the Lorentz number (L = 2.44 × 10−8 W Ω/K2), and T is the absolute temperature (Anderson, 1998, Poirier, 2000). While it is a lower bound, this relation provides a good estimate of the total thermal conductivity of metals because other heat transport mechanisms are thought to be small in comparison to electronic heat transport. The effects of pressure, temperature, and impurities on metal resistivity are, however, still poorly constrained by experiments (Stacey and Anderson, 2001, Stacey and Loper, 2007, Bi et al., 2002, Keeler and Mitchell, 1969, Matassov, 1977), although many recent theoretical predictions have been proposed (Sha and Cohen, 2011, de Koker et al., 2012, Pozzo et al., 2012, Pozzo et al., 2013). The resistivity of iron has been examined by static experiments only to ∼40 GPa (Balchan and Drickamer, 1961, Reichlin, 1983), while the core is subject to pressures of more than 135 GPa.
In this study, we measured the electrical resistivity of pure Fe and Fe–Si alloy (3.90 at.% Si) to 100 GPa in a diamond-anvil cell (DAC). The resistivity of iron was also calculated to core pressures based on density-functional theory. In addition to the impurity resistivity and temperature effects described by the Bloch–Grüneisen formula, we consider the effect of resistivity saturation in the estimates of core resistivity (see Appendix A). While the saturation of metal resistivity is well known in metallurgy, it has never been included in the geophysical modeling of core metals. The thermal conductivity of the core, calculated from present estimates of the electrical resistivity with the Wiedemann–Franz law, yields much higher values than conventional estimates (Stacey and Anderson, 2001, Stacey and Loper, 2007) but are generally consistent with those recently predicted by theoretical calculations (de Koker et al., 2012, Pozzo et al., 2012, Pozzo et al., 2013). We also discuss implications for the possibility of a thermal stratification of the core and its thermal evolution with an approach rather different from these previous studies, based on global energy and entropy balances.
Section snippets
High-pressure resistivity measurements
The electrical resistivity was measured at high pressure in a DAC with flat 300 μm or beveled 200 μm culet diamonds. Foils of iron (99.99% purity) and iron–silicon alloy (3.90 at.% Si) with initial thickness of ∼10 μm were used as samples. Pressure was determined from the Raman spectrum of the diamond anvil at room temperature (Akahama and Kawamura, 2004). The sample resistance was obtained by the four-terminal method under a constant DC current of 10 mA with a digital multi-meter (ADCMT 6581),
Results
We measured the electrical resistivity of iron at high-pressure and room-temperature up to 100 GPa (Fig. 2). A jump in resistivity around 15 GPa is attributed to a phase transition from body-centered-cubic (bcc) to hexagonal-close-packed (hcp) structure (Balchan and Drickamer, 1961). The jump in resistivity at the bcc–hcp phase transition observed in this study is much larger than that predicted by Sha and Cohen (2011) but consistent with the results of earlier static experiments (Balchan and
Thermal conductivity of the core
The resistivity of iron alloy at core pressure and temperature is estimated in this study from (1) resistivity of iron at high pressure, (2) impurity resistivity of silicon and other light alloying elements, (3) temperature effect following the Bloch–Grüneisen formula, and (4) saturation resistivity. Thermal conductivity is then calculated from the electrical resistivity based on the Wiedemann–Franz law.
Thermal structure, dynamics, and evolution of the core
We now consider the implications of the large and depth-increasing thermal conductivity results for the structure, dynamics and evolution of the core and deep mantle. The possibility that a large thermal conductivity of the core could make the heat flow down the isentropic temperature gradient overwhelm the amount taken away by mantle convection has been discussed for a long time. Most earlier studies (e.g., Loper, 1978a, Loper, 1978b, Gubbins et al., 1979, Stevenson, 1983) proposed that the
Conclusions
We measured the electrical resistivity of iron at room temperature up to 100 GPa in a DAC. While a sharp resistivity increase was observed during the bcc to hcp phase transition, the resistivity diminished with increasing pressure above 20 GPa in the stability range of the hcp phase (Fig. 2). A heating experiment was also conducted, which confirmed the Bloch–Grüneisen law up to 383 K at 65 GPa (Fig. 3a). We also performed first-principles calculations up to 360 GPa and 1000 K. These experimental and
Acknowledgments
We thank T. Komabayashi for discussions and Francis Nimmo and an anonymous reviewer for valuable comments. S.L. and R.C. are grateful to the LABEX Lyon Institute of Origins (ANR-10-LABX-0066) of the Université de Lyon for its financial support within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) of the French government operated par the National Research Agency (ANR). S.L. has been supported by the Agence Nationale de la Recherche under the grant ANR-08-JCJC-0084-01. J.H. was
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Present address: Center for Quantum Science and Technology under Extreme Conditions, Osaka University, Toyonaka, Osaka 560-8531, Japan.