Direct measurement of thermal conductivity in solid iron at planetary core conditions

Abstract

The conduction of heat through minerals and melts at extreme pressures and temperatures is of central importance to the evolution and dynamics of planets. In the cooling Earth’s core, the thermal conductivity of iron alloys defines the adiabatic heat flux and therefore the thermal and compositional energy available to support the production of Earth’s magnetic field via dynamo action1,2,3. Attempts to describe thermal transport in Earth’s core have been problematic, with predictions of high thermal conductivity4,5,6,7 at odds with traditional geophysical models and direct evidence for a primordial magnetic field in the rock record8,9,10. Measurements of core heat transport are needed to resolve this difference. Here we present direct measurements of the thermal conductivity of solid iron at pressure and temperature conditions relevant to the cores of Mercury-sized to Earth-sized planets, using a dynamically laser-heated diamond-anvil cell11,12. Our measurements place the thermal conductivity of Earth’s core near the low end of previous estimates, at 18–44 watts per metre per kelvin. The result is in agreement with palaeomagnetic measurements10 indicating that Earth’s geodynamo has persisted since the beginning of Earth’s history, and allows for a solid inner core as old as the dynamo.

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Figure 1: Temperature of Fe foils during flash heating at high initial temperature and pressure.
Figure 2: Thermal conductivity of Fe at high pressure and temperature.
Figure 3: Thermal conductivity of Fe versus temperature.

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Acknowledgements

We acknowledge experimental assistance from H. Marquardt. This work was supported by the NSF (grant numbers DMR-1039807, EAR-1015239, EAR-1520648 and EAR/IF-1128867), the Army Research Office (grant 56122-CH-H), the Carnegie Institution of Washington, the National Natural Science Foundation of China (grant number 21473211), the Chinese Academy of Science (grant number YZ201524), the University of Edinburgh, and the British Council Researcher Links Programme. Portions of this research were carried out at the light source Petra III at DESY, a member of the Helmholtz Association (HGF).

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Contributions

Z.K., R.S.M. and A.F.G. designed and conducted experiments. R.S.M. reduced raw data. Z.K. and N.G.P. performed finite-element modelling. N.G.P. performed error analysis and geophysical calculations. All authors wrote the manuscript.

Corresponding authors

Correspondence to Zuzana Konôpková or R. Stewart McWilliams or Natalia Gómez-Pérez or Alexander F. Goncharov.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 High-temperature transport properties of Fe.

a, Graph of the electrical conductivity5 as a function of temperature of ε Fe at 65 GPa and model fit (to equation (7)). b, Thermal conductivity temperature dependence at 112 GPa. The model fit (to equation (2), solid line) and a 20% uncertainty envelope are in blue; the model fit without linear term (to equation (10)) is a dashed blue line. Present data are solid circles and data derived from prior electrical resistivity measurements5,14,15 are open symbols (see Fig. 3). The red band is the minimum thermal conductivity assuming resistivity saturation5. c, Electrical resistivity at several pressures, for multiple phases at 15 GPa (blue)21, and the ε phase at 65 GPa (red)5 and 112 GPa (this study, black).

Extended Data Figure 2 Comparison between measurements and models for different values of thermal conductivity.

Data for pulsed and opposite sides of the foil are dots; the larger temperature excursion is on the pulsed side. Green, magenta and cyan curves are simulations with different values of sample k, all other parameters being held constant. The data sets at 112 GPa (a) and 130 GPa (b) have been measured using 3-μs and 10-μs sweep windows, respectively.

Extended Data Figure 3 Tests of the sensitivity of finite-element model results to input parameters for an example run at 112 GPa.

This experiment shows a large amplitude of temperature modulation that accentuates the effects of parameter changes. A best-fit value of k = 30 W m−1 K−1, obtained using parameters listed in Extended Data Table 1, is obtained from these model fits unless stated otherwise. a, Effect of heat capacity of the Ar pressure medium. Uncertainty in medium CP has no effect on k of the sample. b, Effect of heat capacity of the sample. Temperature profiles for two values of CP of Fe (500 J kg−1 K−1 and 700 J kg−1 K−1) indicate that results are only weakly affected by the uncertainty in CP for Fe. c, Change in the thermal conductivity value of diamond anvils from 1,500 W m−1 K−1 to 2,000 W m−1 K−1 requires an increase in thermal conductivity of the sample from 30 W m−1 K−1 to 31 W m−1 K−1. d, Effect of using a T-dependent k of the medium. After ref. 49, a dependence k(T) = k300(300/T)m is used, where k300 is the 300-K conductivity, T is in kelvin, and m is an exponent (of order 1); k300 (300 W m−1 K−1) is extrapolated from prior results at lower pressure49 and m (0.7) is fitted to the present data. No change in sample k is indicated using this or any other k(T) model we tested for the media. e, Laser beam radius change of ±13% does not affect the temperature noticeably. f, A sample thinner by 23% (reduced from 2.6 μm to 2.0 μm) would require a lower sample k of 22 W m−1 K−1. g, A sample thicker by 15% (increased from 2.6 μm to 3.0 μm) would require an increased sample k of 37 W m−1 K−1. h, The insulation layer was decreased on both sides by 38%, from 1.6 μm to 1.0 μm. Sample k had to increase to 39 W m−1 K−1. i, The insulation layer was increased on both sides by 25%, from 1.6 μm to 2.0 μm. Sample k had to decrease to 27 W m−1 K−1. j, Effect of including T dependence of sample k in models. The temperature profile calculated using our global fit at 112 GPa (equation (2)) is shown as a magenta line; this dependence scaled within its uncertainty (reduced by a factor of 0.83) to improve the fit is shown as a cyan line. The resulting sample k varies between 24 W m−1 K−1 and 35 W m−1 K−1 in the T range of the experiment; the estimate assuming constant sample k is the average of these values.

Extended Data Figure 4 Comparison of data on Fe and Pt at 48 GPa for an identical sample configuration.

The data clearly show slower propagation of heat across the Fe foil compared to Pt (ref. 11), as given by the half-rise time τ. This observation directly shows that thermal diffusivity κ = (k/ρCP) of Fe is much less than Pt, since11,36 κ 1/τ. Similarly, the smaller amplitude of the perturbation upon opposite surface arrival indicates a smaller k in Fe than in Pt.

Extended Data Figure 5 Comparison between manual and automatic optimization results for an experiment at 130 GPa.

The manual approach, used as our primary fitting method, was based on an adjustment of model parameters by hand within a precision of ~5 W m−1 K−1, giving k = 45 W m−1 K−1 and kAr = 60 W m−1 K−1 as the best fit. The automatic result is the best fit based on a Levenberg–Marquardt least-squares minimization of model parameters, yielding k = 38.6 W m−1 K−1 and kAr = 50.4 W m−1 K−1. The automatic optimization obtained a better least-squares fit (χ2 improved by 23%); however, the difference in k is not statistically significant.

Extended Data Figure 6 Monte Carlo analysis of error coupling in thickness uncertainties and effect on thermal conductivities, for the 130-GPa data set shown in Extended Data Fig. 5.

a, Histogram showing randomly sampled thicknesses (upper and lower medium, and foil) in Gaussian probability distributions with standard deviation 30%. b, Thermal conductivities for Ar and Fe for 64 samples. The greyscale refers to the value of the coupler thickness, showing the correlation between high values for k and thicker coupler. The results of fits shown in Extended Data Fig. 5 are blue and red triangles, while the mean and one standard deviation found from the spread of sampled thermal conductivities is the orange triangle. c and d are histograms showing the distribution of thermal conductivities in b.

Extended Data Table 1 Input parameters used for the finite-element modeling
Extended Data Table 2 Coefficients for the Grüneisen parameter and the isothermal bulk modulus used to estimate pressure variation of thermal conductivity

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Konôpková, Z., McWilliams, R., Gómez-Pérez, N. et al. Direct measurement of thermal conductivity in solid iron at planetary core conditions. Nature 534, 99–101 (2016). https://doi.org/10.1038/nature18009

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