Geophysical Research Letters

High pressure and temperature electrical resistivity of iron and implications for planetary cores

Authors

  • Liwei Deng,

    Corresponding author
    1. Now at Key Laboratory of the Earth's Deep Interior, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China
    • Geophysical Laboratory, Carnegie Institution of Washington, Washington DC, USA
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  • Christopher Seagle,

    1. Geophysical Laboratory, Carnegie Institution of Washington, Washington DC, USA
    2. Smithsonian Institution, National Museum of Natural History, Washington DC, USA
    3. Now at Sandia National Laboratories, Albuquerque, USA
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  • Yingwei Fei,

    1. Geophysical Laboratory, Carnegie Institution of Washington, Washington DC, USA
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  • Anat Shahar

    1. Geophysical Laboratory, Carnegie Institution of Washington, Washington DC, USA
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Corresponding author: L. Deng, No. 19, Beitucheng Western Road, Chaoyang District, 100029, Beijing, China.P.R. (dengliwei@mail.iggcas.ac.cn)

Abstract

[1] Electrical resistivity measurements of polycrystalline iron have been performed at 5, 7, and 15 GPa and in the temperature range 293–2200 K by employing a four-wired method. The kinks in electrical resistivity associated with solid iron phase transitions and the solid to liquid transition were clearly observed upon increasing temperature. Geometry corrections due to volume variations with pressure and temperature were applied to the entire data set. High pressure and temperature thermal conductivity were calculated by fitting resistivity data through the Wiedemann-Franz law. The temperature dependences of electrical resistivity and thermal conductivity for α, γ, and ε solid iron have been determined at high-pressure conditions. Our study provides the first experimental constraint on the heat flux conducted at Mercury's outmost core, estimated to be 0.29–0.36 TW, assuming an adiabatic core. Extrapolations of our data to Martian outer core conditions yield a series of heat transport parameters (e.g., electrical resistivity, thermal conductivity, and heat flux), which are in reasonable comparison with various geophysical estimates.

1 Introduction

[2] Heat flow conducted at the terrestrial planet outmost core is directly coupled to potential vigorous convection within the cores, generating magnetic field with different magnitudes [de Wijs et al., 1998]. It is also a key process which reveals the interior thermal structure of the planet body. In addition, heat conducted from the deep interior of a planetary body may contribute significantly to the total heat loss at the planetary surface (e.g., 11–33% heat loss on the Earth's surface comes from the core) [Lay, 2008]. Therefore, knowledge of the thermal conductivity of core material under high-pressure and temperature conditions is crucial to understanding the fundamental operations of the dynamo, determining the interior thermal state, and constraining the heat budget within the deep interior. Unfortunately, direct measurements of high pressure and temperature thermal conductivity are technically challenging and such experimental data are rarely reported. As an alternative approach, thermal conductivity can be calculated from electrical resistivity, which is experimentally measurable, through the Wiedemann-Franz law:

display math(1)

where κ, L, and ρ are thermal conductivity (W/m∙K), Lorenz number (2.44 × 10−8 WΩK−2), and electrical resistivity (Ω-m), respectively. There are limited resistivity measurements, most are for α phase metallic iron, conducted by static compressions [Secco and Schloessin, 1989; Yousuf et al., 1986; Fulkerson et al., 1966] at 5.85 GPa and 2000 K [Secco and Schloessin, 1989]. It is important to note most static experiments to date have failed to measure ρ for the high pressure iron phase (γ and ε), which are believed to be the dominate core materials in terrestrial planets. High pressure and temperature resistivity measurements for γ and ε phase iron have been reported by shock wave studies [Keeler and Mitchell, 1969, 1971; Bi et al., 2002]. However, resistivity data above 50 GPa shown by Keeler et al. [1969, 1971] might be biased because the epoxy resin around the sample became conductive under compression [Bi et al., 2002]. Bi et al. [2002] reported three electrical resistivity shock data between 100 and 208 GPa; the lowest pressure datum scatters apart from the other two for an unknown reason. In light of large scatter in the Bi et al. [2002] data set, and the serious disagreement between Bi et al. [2002] and Keeler et al. [1969, 1971], the high pressure and temperature behavior of ρ for metallic iron, especially for γ and ε phases, need to be better constrained.

[3] In this study, we report electrical resistivity measurements for α, γ, and ε iron phases at pressures between 5 and 15 GPa up to 2100 K. To the best of our knowledge, this is the first report of the pressure and temperature dependences of electrical resistivity and thermal conductivity for γ and ε iron. Thermal conductivity and heat flux were calculated at characteristic planetary core pressure and temperature conditions assuming an adiabatic temperature profile in the core. This heat flux is then compared with the observed heat loss on planetary surfaces, with the aim of constraining the core heat contribution and understanding the thermal structure of metallic cores in planetary bodies.

2 Experimental Details

[4] High-pressure, high-temperature electrical resistivity measurements were performed on 99.995% iron wire, procured from AlfaAesar, using the multi-anvil apparatus at the Geophysical Laboratory. The iron wire, stored under vacuum to avoid oxidation, was taken out just before assembling the components of the high-pressure cell. Each run utilized a cylindrical sample (lengths ranged from 0.79 to 2.68 mm).

[5] A four-probe method was employed to measure the resistance of the cylindrical iron wire during heating cycles at high pressures (see experimental details and recovered sample image in the auxiliary material). A Keithley 2400 series SourceMeter was employed to generate direct current (DC) in the circuit and measure voltage across the sample. DC was sourced, and voltage was measured along different pairs of platinum wires to eliminate the effect of contact resistances at the Pt-Fe junction; hence, resistances measured are due solely to the iron wire itself. The platinum wires were insulated in Al2O3 tubes to avoid electrical shorting with the surrounding heater. In each run, pressure was increased to a level and resistance data were collected upon increasing temperature. A minimal current (typically ~500 mA) was sourced to generate a measurable voltage in each run. Temperatures were estimated by calibrated power-temperature curves using W5Re-W26Re thermocouple at high pressures in identical assemblies without platinum electrical leads.

3 Results and Discussions

[6] The measured resistance was used to calculate electrical resistivity. Geometry corrections due to volume variations with pressure and temperature were applied to the entire data set (see geometry corrections details in the auxiliary material). The temperature dependence of electrical resistivity at 5, 7, and 15 GPa for solid and liquid iron is plotted in Figure 1. At 5 and 7 GPa, our measured electrical resistivities at ambient temperature are 9.06 ± 0.27 and 8.85 ± 0.27 μΩ-cm, respectively. These results are in good agreement with reported 9.26 and 9.05 μΩ-cm for α iron [Yousuf et al., 1986]. Kinks in electrical resistivity associated with the iron phase transitions were clearly observed (Figure 1a). At 5 and 7 GPa, kinks in the electrical resistivity at 950 K and 925 K are due to the α to γ phase transition [Komabayashi and Fei, 2010]. At 5 GPa and 1960 K, melting led to an abrupt change in electrical resistivity.

Figure 1.

Temperature dependence of electrical resistivity for solid and liquid iron. (a) At 5 GPa, structural transitions of α to γ and γ to liquid lead to discontinuous changes in ρ corresponding to the temperatures T = 1000 K and T = 1950 K. The similar kink due to the α to γ transition was also observed at 7 GPa, T = 976 K. Up and down arrows indicate the solid-liquid and α to γ transitions, respectively. The grey dashed and dotted lines are for PR957 and PR959, respectively. (b) In run PR957, heating duration at each temperature step was shorted by recording the sample voltage as soon as a stable value could be recorded from the SourceMeter. In PR959, voltage data were recorded after several seconds of thermal annealing. Two preheating cycles were performed below 800 K. The dotted line indicates the transition from mixture phases of α and ε to ε and γ.

[7] Empirical relationships for the temperature dependence of electrical resistivity for α and γ phase iron were determined from the data in Figure 1 (Table S1 in auxiliary material). For α phase, temperature-resistivity data are best fitted by quadratic curves. For γ phase, the temperature derivatives at 5 and 7 GPa seem to be almost equal and slightly smaller than that at ambient pressure. The similar behaviors, decreasing electrical resistivity with increasing pressure or decreasing temperature, were observed in both α and γ phases. This is typical of the metallic resistivity of a transition metal [Secco and Schloessin, 1989]. Resistivity is manifested by scattering of electrons and phonons; a temperature increase results in an escalation of the number of electron-phonon and electron-electron scattering events leading to an increase in electrical resistivity. However, pressure has the opposite effect; pressure reduces the inter-atomic spacing and the atomic vibrational amplitude resulting in decreasing electrical resistivity with increasing pressure. The temperature derivative of resistivity for α phase iron is larger than that of γ phase under the same pressure, which shows that the electrical resistivity of α phase has stronger temperature dependence than γ phase.

[8] Two independent runs at 15 GPa, with different heating strategies, were conducted and the results presented in Figure 1b. Sample voltage was recorded immediately after heating power was applied in PR957, while in PR959 thermal annealing (several seconds) was performed at each temperature step. Moreover, in PR959, temperature was cycled twice under 900 K before heating the sample up to 1500 K. As shown in Figure 1b, at temperature lower than 850 K, resistivity-temperature slope remains almost flat in PR957 while it is much steeper in PR959. Based on the high pressure iron phase diagram [Komabayashi and Fei, 2010], ε should be the stable phase at 15 GPa and temperature lower than 900 K. The sluggish phase change occurs in PR957 because of the energy barrier of α to ε phase transition. However, such phase transformation was enhanced in PR959 due to the multiple thermal treatments including annealing and preheating. In fact, the sluggish high-pressure phase transformation was also observed in our synchrotron thermal expansion measurement (see details in geometry corrections of auxiliary material) and previous diamond-anvil cell runs [Reichlin, 1983]. At 12 GPa, X-ray diffraction peaks indicated the coexistence of α and ε phases at ambient temperature and low-pressure α phase persisted upon heating to 600 K. With temperature increasing further, we found ε and γ phases coexisted over an extended temperature range between 673 and 721 K before all traces of ε phase was finally eliminated at above 721 K. Although the resistivity-temperature derivative (~0.01462 μΩ-cm/K) in PR959 below 900 K is much larger than that in PR957, data sets above 900 K in both runs merge into a consistent trajectory due to the ε to γ phase transition. It is important to note resistivity data at temperature above 900 K are unexpectedly larger than their counterparts at 7 GPa of γ phase. This observation is inconsistent with anticipated decreased resistivity with increased pressure. We suggest that our electrical resistivity results above 900 K in fact represent a mixture of ε and γ phases due to the sluggish phase transformation. Therefore, in PR957 and PR959 conducted at 15 GPa, the temperature dependence of electrical resistivity (dρ/dT = 0.044 μΩ-cm/K) above 900 K bounds the lower value for ε phase.

[9] Thermal conductivity was calculated via fitting electrical resistivity data to the Wiedemann-Franz law. Figure 2 illustrates the thermal conductivity against temperature for different solid iron phases. Temperature has a negative effect on thermal conductivity of α and ε phases while positive for γ phase. The temperature coefficients of the thermal conductivity (dκ/dT)P were calculated for α and γ phases (Table S2 in auxiliary material). For α phase, (dκ/dT)P decreases slightly with increasing pressure. On the other hand, it is likely pressure does not have significant effect on (dκ/dT)P for γ phase, especially under high pressure. In all, α displays stronger temperature dependence on thermal conductivity than γ phase.

Figure 2.

Temperature dependence of thermal conductivity for (a) α and (b) γ and ε phases under high pressures.

[10] The pressure dependences of thermal conductivity (dκ/dP)T for α and γ phase are nonlinear. Taking phase α as an example, the average value of (dκ/dP)T between 600 and 900 K is 3.82 W/m · K · GPa between 1 atm and 5 GPa. However, this value is 22.08 W/m · K · GPa between 5 GPa and 7 GPa. Considering the reported 2.8 W/m · K · GPa at room temperature [Sundqvist, 1982], (dκ/dP)T may increase significantly with pressure within α stability field. On the contrary, simulation study [de Koker et al., 2012] suggests that (dκ/dP)T is less pronounced in liquid iron. However, it should be noted that theoretical models were constrained at 2000 K and above, which is much higher than our experimental temperature. Besides, we also noted that the simulated κ at 5 GPa and 2000 K (55 W/m · K) is obviously smaller than our measured value (65 W/m · K). Plus, ambient pressure ρ calculated at 2000 K from de Koker et al. [2012] (~85 μΩ · cm) is also inconsistent with experimental result of liquid iron (~140 μΩ · cm) reported by Yousuf et al. [1986].

4 Discussions

[11] Heat flow conducted at the top of a planetary core is the most important energy source to sustain dynamo action. Heat flow from the core to the mantle may also create a thermal layer at the base of mantle, which fundamentally alters the internal thermal structure of the planet.

[12] The heat flux (Q) along the adiabatic temperature can be expressed by

display math(2)

where z and (dT/dz) are, respectively, the radius and adiabatic temperature gradient. (dT/dz) can be written in term of

display math(3)

[13] Where γ, g, d, and Ks are Gruneisen parameter, gravity (m/s2), density (g/cm3) and adiabatic bulk modulus (GPa), respectively [Stacey and Davis, 2008].

[14] Mercury has the weakest intrinsic magnetic field in the solar system with a mean surface strength of only about 0.3 μT, which is 1% of Earth's. A self-sustained dynamo in Mercury's iron core is the most plausible source of the weak magnetic field [Christensen, 2006]. The estimated pressure and temperature at the top of Mercury's core are ~7 GPa and 1800 ~ 2200 K, respectively. γ phase Fe should represent the dominate chemical component for the outmost core according to high pressure iron phase relations; however, impurities in Mercury's core could lower the melting point relative to pure iron creating a molten or partially molten core. We assume that the resistivity of liquid and solid metal phases is similar, which is known to be the case for Fe at low pressure [Zytveld, 2001; Secco and Schloessin, 1989]. The electrical resistivity at 7 GPa for γ phase Fe is ~36 and ~44 μΩ · cm at 1800 and 2200 K, respectively. Assuming the thermal profile is adiabatic in the core, our measurements suggest the thermal conductivity at Mercury's outmost core (~1800 km) is 113–125 W/m · K, which yields an adiabatic heat flux of 0.29–0.36 TW based on equations ((2)) and ((3)). This is the first experimental constraint on the heat flux conducted along Mercury's outmost core. Thermal evolution models [Schubert et al., 1988; Hauck et al., 2004] predict a heat flow at Mercury's core-mantle boundary (CMB) of several mW/m2. This is translated into 0.04–0.4 TW, which is comparable with our estimation at the outmost core, assuming an 1800 km radius of the core. [Riner et al., 2008] Current constraints on the heat flux conducted across Mercury's CMB are still very loose. Thus, a better understanding of Mercury's core thermal structure and dynamo mechanism requires further study of heat transfer at the bottom of the deep mantle and dynamic properties of the core. It should be noted that numerical simulations also provide some other alternative explanations for Mercury's weak magnetic field. For example, a negative feedback between the magnetospheric and the internal magnetic fields would saturate Mercury's field strength at much lower level. However, those theories are needed to be testified by more accurate field models [Glassmeier et al., 2007, Heyner et al., 2011, Stanley et al., 2005].

[15] Analyses of the measured tidal effect on spacecraft orbits suggest that at least the outer portion of the Martian core is liquid [Yoder et al., 2003; Balmino et al., 2006; Konopliv et al., 2006, 2011; Marty et al., 2009]. The Martian core is mainly composed of γ phase iron based on high pressure mineralogy studies [Bertka and Fei, 1997; Fei and Bertka 2005 2004; Komabayashi and Fei, 2010]. If we take ρ(7 GPa, 2000 K) = 40 μΩ-cm, which is extrapolated from 7 GPa measurement for γ phase, as the resistivity for Martian outmost core. The estimated heat flow at the Martian outmost core (T = 2000 K, dT/dz = 0.001 K/m, z = 1900 Km) is ~6 TW. Figure 1 shows that the temperature effect on the resistivity is negligible at high pressures for γ. Thus, ρ at the real Martian outmost core pressure (~24 GPa) condition should be smaller than ρ(7 GPa, 2000 K) as ρ decreases with pressure increasing. Therefore, our calculated 6 TW bounds the lower value of heat flow at Martian outmost core assuming an adiabatic temperature gradient. Estimated heat loss values from the Martian surface scatter in a wide range between 5 and 50 mW/m2 with large uncertainties [McGovern et al., 2002, 2004]. This is corresponding to 0.7–7 TW total heat flow considering a 3376 km Martian radius. Our estimated conducted heat at the outmost core hits the upper bound of heat released at the Martian surface. Model calculation indicates the radioactive heat flow in present Martian interior is between 10 and 25 mW/m2, which is translated into total heat flow of 0.3–0.9 TW at the surface [Ruiz et al., 2011]. Therefore, heat conducting from the deep Martian interior contributes significantly to the total heat loss at the surface compared with other heat sources, such as radioactive contribution.

[16] Light elements are likely present in the metallic cores of planetary bodies (such as Earth, Mercury, and Mars), especially in the liquid portions of those cores. It has been suggested that iron resistivity shows a strong dependence on light element concentrations and types [de Koker et al., 2012; Stacey and Anderson, 2001]. Data for a variety of iron alloys at characteristic planetary core pressure and temperature are urgently needed to better constrain the energy budget from planetary deep interiors and understand thermal evolution of planetary cores.

Acknowledgments

[17] The synchrotron radiation experiments were performed at the BL04B1 of Spring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2011B1550). We thank T. Komabayashi at Tokyo Institute of Technology and Y. Higo at the Spring 8 for experiment assistance. We are indebted to Viktor Struzhkin and Haijun Huang for resistance measurements assisting. This research was supported by NASA grant to YF (NNX11AC68G), NSF grant to AS (EAR0948131), and the Carnegie Postdoctoral Fellowship.

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