GEOPHYSICAL RESEARCH LETTERS, VOL. 40, 5377–5381, doi:10.1002/2013GL057930, 2013
Electrical and thermal transport properties of iron and iron-silicon alloy at high pressure Christopher T. Seagle,1,2,3 Elizabeth Cottrell,1 Yingwei Fei,2 Daniel R. Hummer,2,4 and Vitali B. Prakapenka5 Received 6 September 2013; revised 1 October 2013; accepted 1 October 2013; published 21 October 2013.
[1] The efficiency of heat transfer by conduction in the Earth’s core controls the dynamics of convection and limits the power available for the geodynamo. We have measured the room temperature electrical resistivity of iron and iron-silicon alloy to 60 GPa and present a new model of the resistivity at high pressures and temperatures relevant to the Earth’s core. The model is compared with available shock wave data and theoretical studies. For a power law and linear temperature dependence of electrical resistivity, the calculated thermal conductivity at the core-mantle boundary is ~67–145 W/m/K for pure Fe and ~41–60 W/m/K for Fe–9 wt % Si. Impurities in the core have a strong effect on the transport properties of iron that could significantly impact core thermal models. The models describing the data indicate higher thermal conductivity at core pressure than previously suggested, requiring additional energy sources in the past to operate the geodynamo. Citation: Seagle, C. T., E. Cottrell, Y. Fei, D. R. Hummer, and V. B. Prakapenka (2013), Electrical and thermal transport properties of iron and iron-silicon alloy at high pressure, Geophys. Res. Lett., 40, 5377–5381, doi:10.1002/2013GL057930.
1. Introduction [2] The Earth’s core consists of a molten iron alloy outer core surrounding a solid inner core of hexagonal closepacked (hcp) iron [Tateno et al., 2010]. Over geologic time, secular cooling causes the boundary of the solid inner core to encroach toward the surface, and in the process, light elements are fractionated preferentially into the liquid [Stevenson, 2008]. This fractionation, in addition to cooling itself, drives turbulent convection resulting in large-scale patterns of fluid flow that creates the geodynamo. An inventory of known energy sources to power the geodynamo compared to an inventory of energy loss suggests a shortage of the former, resulting in an energy imbalance especially pronounced
Additional supporting information may be found in the online version of this article. 1 National Museum of Natural History, Smithsonian Institution, Washington, District of Columbia, USA. 2 Geophysical Laboratory, Carnegie Institution of Washington, Washington, District of Columbia, USA. 3 Now at Sandia National Laboratories, Albuquerque, New Mexico, USA. 4 Now at the Department of Earth and Space Sciences, University of California, Los Angeles, California, USA. 5 Center for Advanced Radiation Sources, University of Chicago, Chicago, Illinois, USA. Corresponding author: C. T. Seagle, Sandia National Laboratories, PO Box 5800, MS 1189, Albuquerque, NM 87185, USA. (
[email protected]) ©2013. American Geophysical Union. All Rights Reserved. 0094-8276/13/10.1002/2013GL057930
in the deep past [Nimmo, 2007]. In order to understand the extent of this imbalance, we need to have not only a better estimation of the temperature and cooling of the overlying mantle but also knowledge of the thermal conductivity of the core fluid because this physical property is directly proportional to the maximum amount of energy that can be conductively lost from the core [Stacey, 2010]. [3] Electrical resistivity is inversely proportional to thermal conductivity through the Wiedemann-Franz law. Previous studies [Balchan and Drickamer, 1961; Reichlin, 1983] have measured the electrical resistivity of iron as a function of pressure; these studies are not in good agreement above ~13 GPa, and the effect of light elements on the resistivity of iron at static high pressure had not been investigated. Therefore, to better constrain the core energy balance, we have measured the electrical resistivity of hcp iron and Fe–9 wt % Si (4.7 mol % Si) using the diamond anvil cell (DAC). We then model the thermal and electrical transport properties of iron and iron-silicon alloy as a function of volume and temperature and show that the heat flux has been underestimated in previous studies. We also show that light elements can dramatically lower the thermal conductivity of iron, and this may reduce the core energy imbalance.
2. Methods [4] Three sets of DAC experiments were conducted at room temperature: The first (Experiment 1, pure Fe) was performed at the GeoSoilEnviroCARS facility of the Advanced Photon Source, Argonne National Laboratory [Prakapenka et al., 2008; Shen et al., 2005], using a sodium chloride (NaCl) pressure medium. Pressure was monitored with the equation of state of platinum [Zha et al., 2008] and iron [Dewaele et al., 2006]. The second and third experiments (Experiment 2, pure iron; Experiment 3, Fe–9 wt % Si) had no pressure medium, and pressure was monitored by the laser-induced fluorescence of the ruby R1 line [Chijioke et al., 2005]. Resistivity was measured with the four-point (van der Pauw) method [van der Pauw, 1958] (Figure 1). The measured resistivity of iron was identical within error for both pure Fe samples on decompression (Figure 2) and over an order of magnitude higher in the sample with 9 wt % Si (Figure 3). Additional details of the experimental methods are available in Text S1 and Figures S1–S3 in the supporting information. [5] A model was developed to describe the pressure and temperature dependence of the electrical resistivity, ρe, of hcp iron. The model is derived from two basic assumptions: ρe ∝ T α, α = 1, is usually observed for metals above ~40% n= 3 , where of the Debye temperature (θD), and ρe ∝θn D V n = 2 describes an ideal metal for the free-electron model,
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3. Results and Discussion [6] Figure 2a shows the experimental resistivity data for the pure iron samples at room temperature on decompression as a function of pressure with the model fits, recent theoretical calculations [Sha and Cohen, 2011], and previous DAC studies on compression [Balchan and Drickamer, 1961; Reichlin, 1983]. At the expected bcc-hcp phase transition at ~13 GPa, the resistivity increases abruptly. It is preferable to measure resistivity during decompression because sample thickness can change nonlinearly upon compression due to compaction and plastic flow of the sample chamber, thus complicating the effort to accurately measure resistivity values below ~20 GPa. The magnitude of the resistivity of hcp iron derived from the Balchan and Drickamer [1961] experiments is higher than that we measured, likely due to
Figure 1. (a) Experimental geometry of Experiment 1. (b) Photomicrograph of Experiment 1 viewed through the top anvil in Figure 1a in reflected and transmitted light at ~50 GPa. A cubic boron nitride insert was transparent at this pressure, and the Pt electrical leads are the opaque wedges connecting to the iron sample in the center.
n = 3 for s-d electron scattering, and n = 5 for electron phonon scattering [White and Woods, 1959]. From these assumptions, we derive a model (see supporting information) for the temperature and pressure dependence of the electrical resistivity, i.e., ρe ¼ ρ0e
T T0
α
θD θ0D
!n
V V0
n=3
;
(1)
where T is temperature, V is volume, and zero subscript/superscript refers to the reference state (1 bar, 300 K). This equation is similar to the traditional high-temperature, freeelectron Bloch-Grüneisen relation. With α = 1, rearranging and taking the natural logarithm of equation (1) results in a linear equation which describes the volume and temperature dependence of the resistivity with slope n and intercept ln ρ0e . To check, we show that this model accurately predicts the high pressure/temperature resistivity for gold (see supporting information). The best fit to the hcp iron data produced n = 2.7 ± 0.2 with ρ0e = 17.6 ± 0.9 μΩ cm (R2 = 0.927), indicating the importance of s-d electron scattering in the manifestation of electrical resistance for hcp iron. Previous estimates of the electrical resistivity of iron in the Earth’s core have used n = 2 [Stacey and Anderson, 2001; Stacey and Loper, 2007].
Figure 2. (a) The resistivity of iron as a function of pressure with the best fit model of equation (1) to the hcp iron data. Both Experiments 1 and 2 are on decompression. Recent theoretical calculations [Sha and Cohen, 2011] and previous DAC studies on compression [Balchan and Drickamer, 1961; Reichlin, 1983] are shown for comparison. (b) Comparison of pure iron resistivity Hugoniot data [Bi et al., 2002; Keeler and Mitchell, 1969] to the α = 1 and α = 1.3 models as well as theoretical results interpolated along the Hugoniot [de Koker et al., 2012] (those results are extrapolated below ~2000 K or ~1 Mbar).
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A combination of static data and theoretical studies from room temperature to well above the Debye temperature would be very useful to help discriminate between the various models available. [8] For the Fe–9 wt % Si data (Figure 3), we enforce Matthiessen’s rule [Matthiessen and Vogt, 1864] at highpressure conditions. This rule states that the resistivity of an alloy is the sum of two components: the intrinsic temperature-dependent resistivity, ρi(T), and the residual or impurity resistivity, ρc(x), which is independent of temperature but depends on composition. Although the rule is untested at high pressure, the Fe-Si system does closely follow this rule at 1 bar [Domenicali and Otter, 1955]. Empirically, we found that the residual resistivity varies as an exponential function of volume. The total resistivity of the Fe-Si alloy takes the following form: !n T θD V Fe ðPÞ n=3 ρe ¼ T0 V 0Fe ðPÞ θ0D h i 0 12 exp V Fe-Si ðPÞ V 0 ðPÞ 1 Fe-Si A þ ρ0c @ exp 1
ρ0e
Figure 3. Resistivity of Fe–9 wt % Si as a function of pressure with the best fit model of equation (2) to the data above 22 GPa compared to shock data [Matassov, 1977] and 1 bar data [Schwerer et al., 1969]. their unknown sample thickness during compression, their use of only two electrical leads, and uncertainties in their calibration method. Reichlin [1983] improved the sample design and measured lower resistivity, but the resistivity increase occurs at a pressure inconsistent with the bcc-hcp phase transition. Reichlin [1983] measured only one of eight possible source-current configurations, likely adding geometrical uncertainties to the calculation of resistivity. Unknown sample thickness, electrode placement, or electrical shorts of the electrodes are possible sources of errors in resistivity measurements [Koon, 1989; Peng et al., 2010]. We have taken steps to reduce or eliminate these sources of error (see section 2 and supporting information) and find that our results are more consistent with recent theoretical calculations of the high-pressure resistivity of hcp iron at room temperature [Sha and Cohen, 2011] than in previous studies. [7] Available shock resistivity data [Bi et al., 2002; Keeler and Mitchell, 1969] do not satisfactorily fit the α = 1 model. Keeping the room temperature model parameters fixed and regressing the shock data to a free parameter α model describe both the shock and static data fairly well with α = 1.3 (Figure 2b). However, shock waves are known to generate a significant density of vacancies, which could be responsible for increased resistivity and, hence, a deviation from linear temperature dependence. Theoretical calculations of the resistivity of hcp iron at high pressure [Sha and Cohen, 2011] agree well with the room temperature compression data but tend to underestimate the resistivity at high temperatures relative to the shock data [Bi et al., 2002; Keeler and Mitchell, 1969], while the calculations of de Koker et al. [2012] match the shock data well over only a limited pressure range. Extrapolation of the α = 1 model to core pressures and temperature results in good agreement with recent theoretical studies [de Koker et al., 2012; Pozzo et al., 2012]; however, this may be fortuitous as the α = 1 model converges to the theoretical studies in the 1–3 Mbar and 4000–5000 K region of phase space. This is the only region where the theoretical models agree well with the model presented here.
(2)
[9] The first term in equation (2) is the resistivity of iron as in equation (1). The parameter ρc0 is the compositionally dependent residual resistivity for Fe-Si alloy at standard pressure. Evaluating the volumes in equation (2) at constant pressure ensures that the temperature derivative of the electrical resistivity for pure iron and iron-silicon alloy will be equal for a given pressure and temperature; hence, Matthiessen’s rule is enforced at constant pressure. The best fit to the Fe–9 wt % Si data yielded ρc0 = 178 ± 1 μΩ cm. At 1 bar, the resistivity of bcc Fe–9 wt % Si alloy is ~115 μΩ cm [Schwerer et al., 1969], close to the value determined here (121 ± 1 μΩ cm). We have not attempted to interpolate the residual resistivity between pure iron and 9 wt % silicon. The measured resistivity of Fe–9 wt % Si along with the best fit of equation (2) to the data above 22 GPa is shown in Figure 3, along with a comparison to the shock data for 10 wt % Si alloy [Matassov, 1977]. Resistivity normally increases with temperature for metals; thus, some of the shock data that have a lower measured resistivity than the room temperature resistivity are likely in error. However, slight differences in composition and/or grain structure make direct comparison difficult. [10] Thermal conductivity, k, is related to electrical resistivity through the Wiedemann-Franz law, approximately valid for most metals: kρe = LT, where L = 2.44 × 108 WΩ/K2 is the theoretical value of the Lorentz number. For simplicity and lack of experimental data at high-pressure conditions, we chose to use the theoretical value of the Lorentz number for the calculation of thermal conductivity. Additional uncertainty exists with the application of the Wiedemann-Franz law to alloys [Klemens, 1958]; however, for consistency, the theoretical value of the Lorentz number is used for Fe-Si alloy as well. The α = 1 model assumes that the resistivity of hcp iron (and silicon alloy) is linear in T (at constant volume), and therefore, the thermal conductivity is independent of temperature with the Lorentz number constant. Unresolved issues remain regarding the temperature dependence of electrical resistivity based on the measured thermal conductivity and electrical conductivity for pure metals such as Cu, Al, and Pb [Madelung and White, 1991]; and this matter requires
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further study under simultaneous high-temperature and high-pressure conditions. The data of Hust and Lankford [1984] provide measurements of both electrical conductivity and thermal conductivity as well as the Lorentz number for iron as a function of temperature up to 1000 K. The magnitude of the Lorentz number for iron at 1 bar ranges from 1.6 to 3 × 108 WΩ/K2 from zero temperature to 1000 K. A lower-than-theoretical [de Koker et al., 2012] or a neartheoretical [Pozzo et al., 2012] Lorentz number is predicted for high-pressure, high-temperature conditions. [11] With the α = 1 model, we estimate the thermal conductivity for pure iron at core-mantle boundary (CMB) conditions to be 145 W/m/K, essentially identical to theoretical studies [de Koker et al., 2012; Pozzo et al., 2012]. The α = 1.3 model (power law temperature dependence) predicts a thermal conductivity for pure iron at CMB conditions of ~67 W/m/K, somewhat higher than the results of Stacey [2010] and Stacey and Anderson [2001], who used the shock data as a primary constraint. The presence of 9 wt % dissolved silicon reduces the conductivity substantially to 60 W/m/K under the same conditions with α = 1 or to 41 W/m/K with α = 1.3. [12] Previous estimates of the thermal conductivity of the Earth’s core based on experimental shock wave resistivities of Fe and Fe-Si alloys are typically much lower because of the higher resistivity measured in those studies [Bi et al., 2002; Keeler and Mitchell, 1969; Matassov, 1977]. The shock studies are likely overestimating the magnitude of the resistivity due to vacancy and defect generation during shock transit [Dick and Styris, 1975; Murr et al., 1976]; however, the magnitude of this overestimate is highly uncertain. The temperature dependence of the model presented here is uncertain as described above. We have not attempted to account for the ~1–3% change in resistivity associated with melting [Deng et al., 2013; Yousuf et al., 1986]; however, the change in thermal conductivity associated with the solid-liquid transition in iron at 1 bar is only ~3% [Nishi et al., 2003; Powell et al., 1966]. The pressure effect of melting on thermal conductivity is unknown and adds additional uncertainty when applying the model to the Earth’s core. Even with these considerations, however, the data we present here in concert with available shock wave data and theoretical studies all suggest that the conductivity of pure iron at core conditions is too high to allow the geodynamo to operate throughout the Earth’s history using the known inventory of available power supplies. Static high-temperature data and theoretical studies spanning large temperature and compositional ranges are needed to strengthen this suggestion. [13] Energy sources within the core include primordial heat from accretion, compositional buoyancy associated with fractionation of light elements during inner core growth, tidal coupling, radioactivity, and others. At early times in the Earth’s history, estimates of the energy available for the geodynamo after taking into account the conducted heat typically fall short of that required; this imbalance becomes greater in the past for higher estimates of the thermal conductivity of the Earth’s core [Nimmo, 2007]. Despite their quantitative differences, recent theoretical studies and extrapolated experimental models have consistently predicted higher thermal conductivity than models for geodynamo evolution allow, and the present contribution is no exception. We have shown here, however, that light elements have a strong effect on the transport properties of iron such that further studies of the thermal conductivity of iron alloys at core conditions are clearly needed.
4. Conclusions [14] The resistivity of pure hcp iron decreases as a function of pressure, becoming less than 6 μΩ cm as pressure increases above 60 GPa at room temperature. Extrapolated to the pressure and temperature of the core, the resistivity is ~60–130 μΩ cm, where the range relates to the uncertainty in the temperature dependence of electrical resistivity. This corresponds to a thermal conductivity on the order of ~67–145 W/m/K, far higher than that needed to stem conductive energy losses from the core to power the geodynamo throughout the known extent of paleomagnetism. The Earth’s core is certain to contain some percentage of light element(s), however, and we show that addition of 9 wt % Si can increase resistivity by an order of magnitude above that of pure iron, lowering the conductivity to ~41–60 W/m/K under core conditions. The core energy imbalance becomes less significant at early times in the Earth’s history compared to pure iron if Si is the light element in the Earth’s core or if other light elements have a similarly dramatic effect on the conductivity of iron. Further studies of the thermal conductivity of iron alloys at core conditions are clearly needed. [15] Acknowledgments. Portions of this work were performed at GeoSoilEnviroCARS (Sector 13), Advanced Photon Source (APS), Argonne National Laboratory. GeoSoilEnviroCARS is supported by the National Science Foundation (Earth Sciences) under grant EAR-0622171, the Department of Energy (Geosciences) under grant DE-FG02-94ER14466, and the State of Illinois. Use of the APS was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under contract DE-AC02-06CH11357. This research was supported by NSF grant EAR0738654 to E.C. and NSF grant EAR-0738741 to Y.F. [16] The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.
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