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Thermal conductivity of dense hcp iron: Direct measurements using laser heated diamond anvil cell
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Research Paper
Thermal conductivity of dense hcp iron: Direct measurements using laser heated diamond anvil cell Pinku Saha, Aritra Mazumder, Goutam Dev Mukherjee * National Centre for High Pressure Studies, Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur Campus, Mohanpur, 741246, Nadia, West Bengal, India
A R T I C L E I N F O
A B S T R A C T
Handling Editor: Richard M Palin
Thermal conductivity (k) of Iron is measured up to about 134 GPa. The measurements are carried out using the single sided laser heated diamond anvil cell, where the power absorbed by a Fe metal foil at hotspot is calculated using a novel thermodynamical method. Thermal conductivity of fcc (γ) Fe increases up to a pressure of about 46 GPa. We find thermal conductivity values in the range of 70–80 Wm1K1 (with an uncertainty of 40%), almost constant with pressure, in the hcp (ε) phase of Fe. We attribute the pressure independent k above 46 GPa to the strong electronic correlation effects driven by the electronic topological transition (ETT). We predict a value of thermal conductivity of ε -Fe of about 40 16 Wm1K1 at the outer core of Earth.
Keywords: Laser heated diamond anvil cell Thermal conductivity High pressure effects Geodynamo
1. Introduction It is well known that Fe is the main component of Earth’s core and hence it has been subjected to extensive studies at extreme conditions of pressure and temperature. Experimental and theoretical investigations related to the physical properties of dense Fe at very high pressures revealed the presence of anomalies due to electron correlation effects leading to a possibility of electronic topological transition (Merkel et al., 2000; Steinle-Neumann et al., 2004; Glazyrin et al., 2013). Heat loss at the Earth’s surface strongly depends on the heat flux conducted at the outer core of the planet. Therefore thermal conductivity (k) of Fe and its alloys with lighter elements at extreme pressures and temperatures can provide important input for the understanding of the dynamics of Earth’s interior. Theoretical calculations on Fe Ni Si alloy by Stacey and Anderson (2001) predicted k value to be about 46 Wm1K1 at the core-mantle boundary (CMB), which was later revised to a lower value (28–29 Wm1K1) by Stacey and Loper (2007). Sha and Cohen (2011) estimated the thermal conductivity of Fe to be in the range of 160–162 Wm1K1 at inner core conditions from the first principles theoretical simulation studies, in which, only electron-phonon coupling contributions were considered and contributions from electron-electron correlations, anharmonicity, etc. were neglected. First principles calculations using density functional theory (DFT) estimated the value of k to be around 200 Wm1K1 at inner core conditions (de Koker et al., 2012;
Pozzo et al., 2012). Four probe electrical resistivity measurements using large volume multi-anvil high pressure cells estimated thermal conductivity of γ Fe from Wiedemann-Franz-Lorenz law (using the theoretical Lorentz number) in the range of 40–100 Wm1K1 at the pressure and temperature range of 5–7 GPa and 1000–1600 K, respectively (Deng et al., 2013). Electrical resistivity measurements using externally heated diamond anvil cell combined with theoretical calculations by Gomi et al. (2013) predicted the thermal conductivity value of Iron to a higher range > 90 Wm1K1 at Earth’s outer core conditions. Ohta et al. (2016) estimated the thermal conductivity value of Iron to a higher range of 226 (þ72/31) Wm1K1 close to core mantle boundary conditions from four probe resistivity measurements in laser heated diamond anvil cell. The electrical conductivity of iron measured by shock compression varied from 1.45 104 Ω1 cm1 at 101 GPa and 2010 K to 7.65 103 Ω 1 cm1 at 208 GPa and 5220 K (Keeler and Royce, 1971; Bi et al., 2002), much smaller than measured by Ohta et al. (2016). Considering the Sommerfeld value of Lorentz number, thermal conductivity values are calculated to be about 71 and 97 Wm1K1 at 101 GPa and 208 GPa, respectively. Direct measurements of thermal conductivity of Fe at high pressures and temperatures are extremely rare. Thermal conductivity experiments using continuous wave (CW) IR laser, combined with finite-element numerical simulations by Kon^ opkova et al. (2011) estimated a value of 32 7 Wm1K1 at 78 GPa and 2000 K. A very recent study by the same
* Corresponding author. E-mail address: [email protected] (G.D. Mukherjee). Peer-review under responsibility of China University of Geosciences (Beijing). https://doi.org/10.1016/j.gsf.2019.12.010 Received 27 August 2019; Received in revised form 16 October 2019; Accepted 22 December 2019 Available online xxxx 1674-9871/© 2020 China University of Geosciences (Beijing) and Peking University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: Saha, P. et al., Thermal conductivity of dense hcp iron: Direct measurements using laser heated diamond anvil cell, Geoscience Frontiers, https://doi.org/10.1016/j.gsf.2019.12.010
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diamond anvil cell set-up is shown in Supplementary Fig. S3. Taking into account of beam divergence and the total path of the laser, we have estimated a beam diameter of about 22 μm at the sample surface with the best focusing of the laser. This produces a hotspot of 16–18 μm in diameter, which is maintained for the localized heating of the sample. Temperature of the hot spot is controlled by monitoring the laser output power. We have shown the schematic drawing of the sample chamber in Supplementary Fig. S4. For temperature measurements, the incandescent light from the surface of the metal is collected using achromats. It is magnified by about 16 times and focused on the 40 μm entrance pinhole of SP150 Acton series spectrometer with back-illuminated PIXIS 100BR (pixel size: 1340 100) camera in the wavelength range of 565–900 nm. The temperature measurement system is calibrated with the tungsten filament light source (NPL, UK) having known intensity vs. wavelength distribution with temperature. Temperature of the sample surface is measured by spectroradiometry technique (Boehler et al., 1990; Mukherjee and Boehler, 2007) by fitting Planck’s radiation function (Planck, 1901) to the flatfield corrected spectrum. With appropriate flatfield, temperature of the hotspot can be estimated from fitting within an error of about 20 K. However, for determination of actual temperature error in DAC, we have carried out measurements of melting temperatures of Ar and Pt by increasing as well as decreasing laser power, which we have determined within an error of 100 K (Kavner and Jeanloz, 1998; Boehler, 2001). Therefore, we attribute error in our temperature estimation to be within 100 K. Temperatures at different positions are measured by translating the 40 μm pin-hole attached to the spectrometer across the magnified image of the sample surface with a resolution of 1 μm. Supplementary Fig. S5 represents flatfield corrected spectrum with its fitting by Planck radiation function at the hotspot and r2 distance away from the hotspot. For calculation of thermal conductivity we have used a model that simulates the steady-state temperature distribution in a cylindrical Fe plate heated by a small heat source at its center. We continuously pump the laser power on then sample and temperature measurements are carried out only when a constant temperature gradient is observed. Therefore the assumption considered in our model seems to be satisfied. However, to test the above assumption, we have carried out time dependent temperature measurements at two different positions on the sample surface at various pressures and temperatures and the data are shown in Fig. 1. The temperature at the same position with time is found to remain constant within the error limit and gives us enough confidence
group using a dynamically laser-heated diamond anvil cell showed a low value of thermal conductivity of Iron: (i) about 35 Wm1K1 at 48 GPa and 2000 K and (ii) in the range of 18–44 Wm1K1 up to pressure 130 GPa and temperature range of 2000–3000 K respectively (Kon^ opkova et al., 2016). In both the experiments thermal conductivity was estimated from finite element simulation analysis and considering the reflectance of laser power from interfaces. All the above experiments and theoretical simulations do show large variations in thermal conductivity of Fe at high pressures and temperatures. The large discrepancy in the value of thermal conductivity of Fe at extreme pressures and temperatures lead to radically different values of estimating the age of the Earth’s solid inner core (Biggin et al., 2015; Davies et al., 2015; Tarduno et al., 2015; Kon^ opkov a et al., 2016). High thermal conductivity values of Fe at Earth’s core conditions estimated from indirect measurements and theoretical simulations indicate to a young core less than a billion years (Davies et al., 2015). However, presence of ancient magnetic field from full-vector paleointensity measurements indicate to the presence of geodynamo at much earlier stage (Tarduno et al., 2015) and supports the low value of thermal conductivity of Fe. Dobson (2016) raised the points that: (i) high thermal conductivity of Fe at CMB estimated by Ohta et al. (2016) from resistivity measurements may be an artifact due to the underestimate of heat loss through the electrodes and; (ii) very low thermal conductivity estimated using laser heated diamond anvil cell by Kon^ opkov a et al. (2016) may arise from the melting of Fe surface due to the very short laser pulses. These observations obviously open up the controversies regarding the measurement of thermal conductivity values at the conditions of CMB and hence it is important to have new direct measurements. In the present work we have carried out measurements of thermal conductivity of Fe, both in solid as well as liquid phases in its fcc(γ)-phase and hcp(ε)-phase up to about the pressures of Earth’s outer core using a single sided laser heated diamond anvil cell (LHDAC). 2. Materials and methods Thermal conductivity measurements at high-pressures and hightemperatures are carried out in LHDAC using plate type diamond anvil cells (Almax-Boehler design). Diamond anvils of culet diameter 300 μm and 100 μm are used for pressure ranges from ambient to 50 GPa and 50–134 GPa, respectively. Steel gasket of thickness 225 μm is preindented to a thickness of 50 μm. A hole of diameter ~110 μm is drilled at the center of diamond imprint of 300 μm culet. For 100 μm DAC culet the steel gasket pre-indented to a thickness of 45 μm followed by drilling a sample hole of diameter ~75 μm. Thin plate of iron of approximate thickness of about 15–20 μm is made by compacting polycrystalline iron powder using a 300 ton hydraulic press. Thin pieces of Feplates of desired size (approximate diameter of about 70–90 μm) are cut to load them in the LHDAC. A scan of the compressed sample surface is taken using SEM to find out about the grain boundary distribution and uniform sample surface and is shown in Supplementary Fig. S1. Supplementary Fig. S2 is the EDS of the compressed iron powder before loading taken using IISER-Kolkata FESEM facility. Sample is sandwiched in between thin layers of pressure transmitting medium (PTM) of approximate thickness of about 15 μm inside the central hole of the gasket. Two different sets of experiments are carried out using NaCl and Al2O3 as pressure transmitting medium. Both NaCl and Al2O3 are dried at a temperature of 393 K for 6 h to remove any trace of moisture. Pressure in the LHDAC before and after heating is determined using the ruby fluorescence method (Mao et al., 1986) by placing a few ruby chips (approximate sizes of about 3–4 μm) at the edge of the iron plate. Heating is carried out using a diode-pumped Ytterbium fiber optic laser (YLR100-SM-AC-Y11) with central emission wavelength, λ ¼ 1.070 μm (maximum power 100 W). Laser beam is focused on the sample inside the diamond anvil cell at an angle of approximately 23 with respect to the normal to the diamond surface using 100 mm focal length anti-IR reflection coated fused silica lens. A schematic of our laser heated
Fig. 1. Time dependent temperature at hotspot (filled symbols) and at a distance r2 (~70 μm) from the hotspot (open symbols). Different symbols represent different experimental runs. 2
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The 3D geometry of our sample chamber is shown in the Fig. 2. The laser is focused on the sample and then the temperatures are measured from various positions of the sample surface radially in a single line. This is possible due to the fact that the laser is focused off-axis and the temperature is measured on-axis. The hot spot has a diameter 16–18 μm and the sample has a diameter of 70–90 μm and the gasket hole is of about 75–110 μm. We have used finite-element software COMSOL Multiphysics (COMSOL) to simulate the temperature distribution in the laser heated diamond anvil cell. The main aim of the simulation is to match the experimental temperature profile by varying the thermal conductivity of the sample and the gasket boundary condition (Tg). We have listed the material properties and geometry in Table 3.
using the steady state condition. In the computational model we have considered the heat to be constant at a value noted by Q inside the hotspot diameter. However, we have considered a distribution of heat inside the sample (f(r)) as given by Δ⋅(k ΔT) ¼ f(r)
(1)
where, k is thermal conductivity of iron, ΔT is the temperature gradient across the sample surface. We have used the boundary conditions for calculation of thermal conductivity of iron in COMSOL; f(r) ¼ Q for r Rhotspot and T ¼ Tg at gasket boundary
(2)
The heat energy Q absorbed by the Fe plate is calculated by Q ¼ mCp(Thotspot TRoom)v
3. Results and discussion
(3)
where, m is the mass of the sample at the hotspot, Cp is the specific heat capacity of iron at constant pressure, and (Thotspot TRoom) is the temperature difference between hot spot and room temperature and v is the modulation frequency (50 kHz) of the 1.070 μm wavelength laser. Exposure time of collected spectrum for temperature measurement is about 100 msec, which is much larger than the modulation period. The specific heat c of the Iron is taken to be 450 JKg1K1 up to 40 GPa and 708 JKg1K1 from 46 to 134 GPa (Gubbins et al., 2003; Hirose et al., 2013). Mass of hot spot m in the Eq. (3) is calculated as m ¼ π R2hotspot hρ
The temperature gradient on the sample surface at four different pressure points (6, 31, 46 and 60 GPa) are measured by heating the sample at the center and it is shown in Fig. 3. The filled scattered symbols are the measured temperature across the sample surface and the solid lines are the computed temperature profile obtained from the COMSOL software. The best match of the experimental data points to the computed lines is obtained by varying the thermal conductivity of the Fe plate taking all the other parameters as mentioned in the Table 3. Other measurements of thermal conductivity are carried out by heating the sample at one end and measuring the temperatures at a distance of 70 μm away. Thermal conductivity is measured in fcc (γ)-phase and hcp (ε)-phase by following the high pressure and the high temperature phase diagram of Fe (Boehler, 1993; Boehler et al., 2008; Anzellini et al., 2013). Since our experiments are carried out with accurate temperature measurements of the sample surface, we could follow the phase diagram precisely. We have carried out some measurements on the melting line of Fe to see the effect of liquid Fe on the thermal conductivity value. Temperature dependence of thermal conductivity values estimated by taking the absorbed power following Eq. (3) at a lower pressure of 10 GPa are shown in Fig. 4. A sharp fall in the thermal conductivity is observed at 2146 K and is attributed to the melting (Liu and Bassett, 1975; Anderson, 1986; Ezenwa and Secco, 2019) at the hotspot while the sample is still at solid state at the edge opposite to the hotspot. The drop in the thermal conductivity across melting at high pressures is observed to much sharper from our experiments compared to that observed by Ho (1972) and Nishi et al. (2003) at ambient pressure. Possibly liquid and solid interface at the boundary of the hotspot gives rise to the above effect. A sudden jump to higher electrical resistivity values have been observed during melting of Fe in high pressure experiments indicating a sudden decrease in thermal conductivity (Deng et al., 2013; Ohta et al., 2016; Silber et al., 2018; Basu et al., 2019). Thermal conductivity due to molten iron at the hot spot is found to be 25%–30% lower than that of solid iron (Fig. 4). Pressure dependence of thermal conductivity of Fe in fcc(γ)-phase, hcp(ε)-phase and their mixed phase are shown in Fig. 5 and compared with other reported values. Filled symbols depict our data (black filled square for γ-Fe and olive filled triangle for ε-Fe using NaCl as pressure transmitting medium and blue filled triangle for ε-Fe using Al2O3 as pressure transmitting medium and red filled circle for mixed phase) and open symbols show the reported values in literature (Kon^ opkova et al., 2011, 2016; Deng et al., 2013; Xu et al., 2018). The value of k in fcc(γ)-phase of Fe is consistent with the same phase of Fe with the literature (Deng et al., 2013). k value of γ-Fe seems to increase with pressure. It shows a minimum value 93(35) Wm1K1 at 6 GPa to a maximum value 125(50) Wm1K1 at 46 GPa in temperature range 1700–1900 K. Thermal conductivity of the mixed phase (hotspot at γ-phase and edge at ε-phase as per the phase diagram (Boehler, 1993) has a value in the range 48–65 Wm1K1 in the pressure and hotspot temperature range of 18.5–46 GPa and 1400–1700 K, respectively. Pressure dependence of thermal conductivity of mixed phase is also found to increase up to 46 GPa (Fig. 5). These values of the thermal conductivity are seems to be
(4)
where, Rhotspot is the radius of the hot spot, h is the thickness of compacted iron powder before loading and ρ is the density of compacted iron powder. Initial density of the compacted iron powder is determined by preparing a pellet of weighed amount of powder and measuring its dimensions. The mass of the hotspot is assumed to remain constant with pressure in each experiment as the sample chamber is quasi-hydrostatic. We have carried out the temperature measurements on the both side of the sample at the hotspot. Two representative values of the temperature are tabulated in Table 1. Very small difference in the temperature at the hotspot on both sides of the sample indicates to uniform heating of the Feplate at the hotspot. By taking into account the estimation of mass at the hotspot and uncertainty in its value, the final uncertainty in absorbed power is found to be about 31%. Estimated heat absorbed by the Fe-surface using Eq. (3) along with error estimation at various stages is given in a tabular form in Table 2. In support of the above, we have tried to estimate the total laser power dumped on the sample surface by taking into account the reflections from the optics and different interfaces (Benedetti and Loubeyre, 2004; Seagle et al., 2009) and same is given in Supplementary Table S1. By taking into account of errors in every step, we find an uncertainty of about 40% in determination of thermal conductivity values in Fe (Table 2). Table 1 Temperature measured at the front and back of the sample surface corresponding to the incident laser spot. Temperatures at a particular pressure are measured by changing the laser power. P (GPa), Phase, PTM
Front side T (K)
Rear side T (K)
Difference in T (K)
39, γ-phase, NaCl
1609 1676 1734 1791 1818 1982 1863 1914 1986 1991 2047 2129
1593 1672 1734 1776 1811 1970 1851 1905 1079 1991 2044 2117
16 4 0 5 7 12 12 9 7 0 3 12
46, γ-phase, NaCl
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Table 2 Detailed error assessment in measuring thermal conductivity of ε-phase of Iron at 134 GPa. Here r1 is the radius of the hotspot, h is the thickness of the Fe-plate before loading, m is the mass of the hotspot, c is the specific heat of Iron, T1 is the temperature of the hotspot, T2 is the temperature at a distance r2 from the hotspot, Q is the absorbed power at hotspot measured using Eq. (3). The estimated beam waist of the laser has larger diameter than the hot spot. Therefore some part of the laser power is also incident outside the hotspot and hence we estimate the error in r1 from the difference of the half of beam waist and the radius of the hotspot. We have measured the thickness (h) of the compressed Fe-plate before loading several times using micrometer and we assign the error in h from our measurements. The mass (m) of the hotspot is measured from the error measurements of density of the Fe-plate and volume of the hotspot. The error in c is assigned from the literature (Gubbins et al., 2003; Hirose et al., 2013) corresponding to the Earth’s outer core condition. The error in T1 and T2 is already explained above. The error in Q is assigned from the propagation of the errors. The error in r2 is assigned from the resolution of the motion of spectrometer pinhole. The error in k is assigned from all the propagated errors. A total error is estimated to be 40% in the thermal conductivity values.
Meassured value Error
Rhotspot (μm)
h (μm)
m (Kg)
c (JKg-1K-1)
T1 (K)
T2 (K)
Q (Watt)
r2 (μm)
k (Wm-1K-1)
9 2
15 0.2
5.5 10-11 1.6 10-11
708 8
1823 100
1567 100
1.2 0.3
70 1
78 31
Fig. 2. (a) Schematic drawing of the sample chamber in laser heated diamond anvil cell facility. Number 1 represents the hotspot at the center of the sample, 2 represents the sample, and 3 represents pressure transmitting medium. (b) Cross sectional view of the sample chamber with gasket. (c) Cross sectional view of computed temperature distribution in the sample chamber and gasket.
sample much more accurately. This probably results in thermal conductivity values, which are not affected by the type of pressure transmitting medium used. It can be seen from Fig. 5 that our thermal conductivity values are found to be 2 to 3.5 times higher than those determined using CW and pulsed LHDAC route (Kon^ opkov a et al., 2011, 2016). Recent computational study by taking account of both electron-phonon (eph) and electron-electron (ee) scattering contributions to the thermal conductivity of solid iron have reported k to be in the range of 90–110 Wm1K1 in the pressure range of 110–140 GPa at 2000 K (Xu et al., 2018), which are in agreement with this study. Our results of k in ε-phase are found to remain almost constant with pressure from 46 to 134 GPa, the highest pressure of our study. To check the diffusion of carbon in the sample (Prakapenka et al., 2004), Raman measurements have been carried out with a micro-Raman spectrometer LabRAM HR from Horiba Jobin Yvon with 1800 grooves mm1 grating using the excitation wavelengths of 488 and 632.8 nm with spectral resolution of 1 cm1. All Raman spectra have been collected on quenched samples after cleaning using distilled water outside DAC and are shown in the Supplementary Fig. S6. No observable peak of carbon phase is detected in the Raman signals from the samples quenched at different pressures and temperatures. Also, no chemical reaction is reported under high-temperature and high-pressure by Errandonea (2009) during laser heating on other transition metals. For estimation of thermal conductivity of Earth’s interior it is also important to look into the temperature variation at high pressures. We observe a slight decrease with increase in temperature. We have fitted the temperature dependence of k at various pressures using the relation
Table 3 The used parameters in COMSOL for the determination of thermal conductivity of iron. Material
Dimensions Thickness & Diameter (μm)
Density (Kgm-3)
Thermal Conductivity (Wm-1K-1)
Fe NaCl
15–20 & 70–100 15 & 75–110
7538 2160
Al2O3 Gasket (steel)
15 & 75 45–50 & 106
3950 8050
variable 6 (Hakansson and Andersson, 1986) 5 (Hofmeister, 1999) 20 (Keifer and Duffy, 2005)
30%–40% higher than that of the direct measurements by Kon^ opkova et al. (2011). These low values of thermal conductivity with respect to the γ-phase in this study may be due to the phase boundary effects. All our thermal conductivity measurements in ε-phase are carried out above 46–134 GPa around 1800–2000 K. Measured temperature values of the this study and computed thermal conductivity values for nine different runs are shown in Supplementary Table S2. Our thermal conductivity measurements using two types of pressure transmitting media, NaCl and Al2O3 do not show much of a scatter and are comparable within the error-bars. In the spectroradiometric technique of temperature measurement we focus the sample surface at the entrance pinhole of the monochromator. Therefore, only the incandescent light from the sample surface is used for temperature measurement and we are able to measure the surface temperature gradient of the 4
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Fig. 4. Temperature dependent thermal conductivity of iron at pressures at 10 GPa. The sharp fall in the thermal conductivity indicated inside green dotted box is attributed to the part of the Fe sample that is in the liquid phase. Melting of the iron sample was confirmed from the laser speckle motion of a blue laser incident on the sample.
Fig. 3. Measure and computed temperature distribution on iron foil heated at different pressures. Temperatures were measured by translating the pinhole across the magnified image (magnified by 16 times) of the sample surface. Inset (a) shows the image of hotspot about diameter 18 μm on iron at 46 GPa under transmitting light and (b) shows the magnified image of hotspot on Iron at 46 GPa with radial temperature distribution.
(Kon^ opkov a et al., 2016) as given below k ¼ aT þ
b Tn
(5)
We have included the values of k in the temperature range of 300–500 K in the fit by estimating them from the high pressure resistivity data by Gomi et al. (2013) at 102 GPa. We have used Wiedemann-Franz-Lorenz law for the conversion of resistivity value to thermal conductivity and taken the Lorenz number to be 2.40 108 WΩK2. This Lorenz number was estimated using our thermal conductivity values and electrical conductivity data under shock compression (Bi et al., 2002). Due to the dynamic and fast measurements of resistance using shock experiments, the controversy on heat leak through the electrodes can be avoided. Therefore, we have chosen resistance measurements under shock compression for the estimation of the Lorentz number. We observe that best fit at high pressures are obtained using a value of n ¼ 0.23. The parameter a is found to be in the order of ~ 103. The leading term in the fit comes from the parameter b and is found to be around 600. In Fig. 6 we have shown a representative fit of k with respect to temperature at 98 GPa along with its 95% confidence band and k values determined at various pressures and temperatures. The cluster of k values determined in our experiments seems to be described well by the temperature dependent fit at 98 GPa along with the confidence band within the error-bars. Our direct experimental measurement of k values gives rise to an interesting fact that in ε-phase, k seems to saturate at high pressures and only depends on temperature with a power law behavior having a small
Fig. 5. Our results of pressure dependent thermal conductivity of iron are compared with those in literature. Filled symbols are the results of our study and open symbols are reported values. Black filled squares for γFe and olive filled triangles for εFe using NaCl as pressure transmitting medium and blue filled triangles for ε Fe using Al2O3 as pressure transmitting medium and red filled circles are for mixed phase.
exponent of 0.23. High pressure studies have shown a transition in
ε-phase of Fe above 40 GPa with anomalies in Raman spectroscopy (Merkel et al., 2000) and first principles calculations indicating a
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4. Conclusions To summarize, a new technique has been employed to measure the thermal conductivity in laser heated DAC. In this technique thermodynamical equation has been used to calculate absorbed power by the metal plate. The thermal conductivity is estimated from the temperature differences across the sample and using the COMSOL software. The k values of iron are measured up to Earth’s outer core pressure ~134 GPa with an uncertainty of about 40%. Value of thermal conductivity of γ-phase and mixed phase of iron is seemed to increase with pressure. Thermal conductivity values in the ε-phase of iron lies in the range of 70–80 Wm1K1, almost constant with pressure. We believe our new measurements will be a solid step towards solving the discrepancy about the time period of generation of Earth’s magnetic field as well the source of heat flow from core to the surface. Author contributions All the authors have equal contribution. Both authors reviewed the manuscript.
Fig. 6. Thermal conductivity of ε Fe versus temperature. Solid filled squares depict the value of k estimated from resistivity measurements by Gomi et al. (2013). All the open symbols data are of this study. Red and equally spaced symmetric blue solid lines represent the model fit of thermal conductivity data at 98 GPa using Eq. (5) and 95% Confidence band, respectively.
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
hyperfine field splitting (Steinle-Neumann et al., 2004). Nuclear in-elastic scattering experiments and Mossbauer studies have predicted a pressure induced electronic topological transition (ETT) in ε-phase of Fe above 40 GPa (Glazyrin et al., 2013). Using dynamical mean field theory simulations, the same group has shown that the ETT is mediated by strong electron correlation effect. At the range of pressures of our measurements, Debye temperature of pure iron is found to be much smaller than our experimental temperature range (Anderson et al., 2001). This means that lattice contribution to specific heat capacity of iron will almost saturate at high pressures and high temperatures. From Eq. (3), it is clear that both pressure and temperature dependence of k will then be affected mostly by the behavior of electronic correlations. In classical electronic theory of metals, thermal conductivity is proportional to the electron mean free path and inversely proportional to the effective mass. In the dense hexagonal phase of Fe many-electron correlation effects may increase the effective mass, which may in turn lead to lower constant value of thermal conductivity in ε-phase compared to γ-phase. The above may be an outcome of large overlap of 3d and 4s electron states in pure iron (Stacey and Loper, 2007). Slight power law decrease in k with respect to temperature is probably due to increase in temperature induced scattering. Hence, the saturation effect with respect to pressure in k may be related to ETT in pure iron. Temperature dependent model fit of our data predicts a value of k of solid iron to be 55 22 Wm1K1 at CMB conditions. Our results obviously limit thermal conductivity value in ε-phase to the temperature effects only and can provide important inputs for the question over age of Earth’s inner core. As discussed earlier our thermal conductivity measurement on a Fe sample which had both molten and solid regions is found to be 25%–30% less than the solid iron at same pressure. Using the above result one can estimate thermal conductivity of molten iron at CMB to be about 40 16 Wm1K1. This result is in good agreement with the thermal conductivity estimation of Stacey and Loper (2007) and as well as recent direct measurements by Kon^ opkova et al. (2016). This moderate low value of thermal conductivity indicates slow cooling of Earth’s core from its initial state. Therefore, presence of core convection can not be ruled out at the early stage resulting in generation of magnetic field as the dynamo entropy is positive as per the estimation of Davies et al. (2015). Our results strongly support the conclusions and results of palaeomagnetic measurements (Biggin et al., 2015; Tarduno et al., 2015), which estimates the presence of ancient terrestrial magnetic field may be due to presence of geodynamo at early stages of Earth.
Acknowledgement(s) The authors also gratefully thank Prof. Isaac F. Silvera, Harvard University for useful discussions and his constructive suggestions. GDM wishes to thank Ministry of Earth Sciences, Government of India for financial support under the project grant no. MoES/16/25/10-RDEAS. PS wishes to thank DST, INSPIRE program by Department of Science and Technology, Government of India for financial support. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.gsf.2019.12.010. References Anderson, O.L., 1986. Properties of iron at the Earth’s core conditions. Geophys. J. Int. 84 (3), 561–579. https://doi.org/10.1111/j.1365-246X.1986.tb04371.x. Anderson, O.L., et al., 2001. Experimental vibrational grüneisen ratio values for ε-iron up to 330 GPa at 300 K. Geophys. Res. Lett. 28 (2), 399–402. https://doi.org/10.1029/ 2001GL013085. Anzellini, S., et al., 2013. Melting of iron at Earth’s inner core boundary based on fast Xray diffraction. Science 340, 464–466. Basu, A., et al., 2019. New measurement of melting and thermal conductivity of iron close to outer core conditions. Geosci. Front. https://doi.org/10.1016/j.gsf.2019.06.007. Benedetti, L.R., Loubeyre, P., 2004. Temperature gradients, wavelength-dependent emissivity, and accuracy of high and very-high temperatures measured in the laserheated diamond cell. High Press. Res. 24 (4), 423–445. https://doi.org/10.1080/ 08957950412331331718. Bi, Y., et al., 2002. Electrical conductivity of iron under shock compression up to 200 GPa. J. Phys. Condens. Matter 14 (44), 10849. https://doi.org/10.1088/0953-8984/14/ 44/389. Biggin, A.J., et al., 2015. Palaeomagnetic field intensity variations suggest mesoproterozoic inner-core nucleation. Nature 526 (7572), 245. https://doi.org/ 10.1038/nature15523. Boehler, R., 1993. Temperatures in the Earth’s core from melting-point measurements of iron at high static pressures. Nature 363 (6429), 534. https://doi.org/10.1038/ 363534a0. Boehler, R., et al., 1990. Melting, thermal expansion, and phase transitions of iron at high pressures. J. Geophys. Res.: Solid Earth 95 (B13), 21731–21736. https://doi.org/ 10.1029/JB095iB13p21731. Boehler, R., et al., 2001. High-pressure melting curves of argon, krypton, and xenon: deviation from corresponding states theory. Phys. Rev. Lett. 86, 5731. https://do i:10.1103/PhysRevLett.86.5731. 6
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Research Paper
Thermal conductivity of dense hcp iron: Direct measurements using laser heated diamond anvil cell Pinku Saha, Aritra Mazumder, Goutam Dev Mukherjee * National Centre for High Pressure Studies, Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur Campus, Mohanpur, 741246, Nadia, West Bengal, India
A R T I C L E I N F O
A B S T R A C T
Handling Editor: Richard M Palin
Thermal conductivity (k) of Iron is measured up to about 134 GPa. The measurements are carried out using the single sided laser heated diamond anvil cell, where the power absorbed by a Fe metal foil at hotspot is calculated using a novel thermodynamical method. Thermal conductivity of fcc (γ) Fe increases up to a pressure of about 46 GPa. We find thermal conductivity values in the range of 70–80 Wm1K1 (with an uncertainty of 40%), almost constant with pressure, in the hcp (ε) phase of Fe. We attribute the pressure independent k above 46 GPa to the strong electronic correlation effects driven by the electronic topological transition (ETT). We predict a value of thermal conductivity of ε -Fe of about 40 16 Wm1K1 at the outer core of Earth.
Keywords: Laser heated diamond anvil cell Thermal conductivity High pressure effects Geodynamo
1. Introduction It is well known that Fe is the main component of Earth’s core and hence it has been subjected to extensive studies at extreme conditions of pressure and temperature. Experimental and theoretical investigations related to the physical properties of dense Fe at very high pressures revealed the presence of anomalies due to electron correlation effects leading to a possibility of electronic topological transition (Merkel et al., 2000; Steinle-Neumann et al., 2004; Glazyrin et al., 2013). Heat loss at the Earth’s surface strongly depends on the heat flux conducted at the outer core of the planet. Therefore thermal conductivity (k) of Fe and its alloys with lighter elements at extreme pressures and temperatures can provide important input for the understanding of the dynamics of Earth’s interior. Theoretical calculations on Fe Ni Si alloy by Stacey and Anderson (2001) predicted k value to be about 46 Wm1K1 at the core-mantle boundary (CMB), which was later revised to a lower value (28–29 Wm1K1) by Stacey and Loper (2007). Sha and Cohen (2011) estimated the thermal conductivity of Fe to be in the range of 160–162 Wm1K1 at inner core conditions from the first principles theoretical simulation studies, in which, only electron-phonon coupling contributions were considered and contributions from electron-electron correlations, anharmonicity, etc. were neglected. First principles calculations using density functional theory (DFT) estimated the value of k to be around 200 Wm1K1 at inner core conditions (de Koker et al., 2012;
Pozzo et al., 2012). Four probe electrical resistivity measurements using large volume multi-anvil high pressure cells estimated thermal conductivity of γ Fe from Wiedemann-Franz-Lorenz law (using the theoretical Lorentz number) in the range of 40–100 Wm1K1 at the pressure and temperature range of 5–7 GPa and 1000–1600 K, respectively (Deng et al., 2013). Electrical resistivity measurements using externally heated diamond anvil cell combined with theoretical calculations by Gomi et al. (2013) predicted the thermal conductivity value of Iron to a higher range > 90 Wm1K1 at Earth’s outer core conditions. Ohta et al. (2016) estimated the thermal conductivity value of Iron to a higher range of 226 (þ72/31) Wm1K1 close to core mantle boundary conditions from four probe resistivity measurements in laser heated diamond anvil cell. The electrical conductivity of iron measured by shock compression varied from 1.45 104 Ω1 cm1 at 101 GPa and 2010 K to 7.65 103 Ω 1 cm1 at 208 GPa and 5220 K (Keeler and Royce, 1971; Bi et al., 2002), much smaller than measured by Ohta et al. (2016). Considering the Sommerfeld value of Lorentz number, thermal conductivity values are calculated to be about 71 and 97 Wm1K1 at 101 GPa and 208 GPa, respectively. Direct measurements of thermal conductivity of Fe at high pressures and temperatures are extremely rare. Thermal conductivity experiments using continuous wave (CW) IR laser, combined with finite-element numerical simulations by Kon^ opkova et al. (2011) estimated a value of 32 7 Wm1K1 at 78 GPa and 2000 K. A very recent study by the same
* Corresponding author. E-mail address: [email protected] (G.D. Mukherjee). Peer-review under responsibility of China University of Geosciences (Beijing). https://doi.org/10.1016/j.gsf.2019.12.010 Received 27 August 2019; Received in revised form 16 October 2019; Accepted 22 December 2019 Available online xxxx 1674-9871/© 2020 China University of Geosciences (Beijing) and Peking University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: Saha, P. et al., Thermal conductivity of dense hcp iron: Direct measurements using laser heated diamond anvil cell, Geoscience Frontiers, https://doi.org/10.1016/j.gsf.2019.12.010
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diamond anvil cell set-up is shown in Supplementary Fig. S3. Taking into account of beam divergence and the total path of the laser, we have estimated a beam diameter of about 22 μm at the sample surface with the best focusing of the laser. This produces a hotspot of 16–18 μm in diameter, which is maintained for the localized heating of the sample. Temperature of the hot spot is controlled by monitoring the laser output power. We have shown the schematic drawing of the sample chamber in Supplementary Fig. S4. For temperature measurements, the incandescent light from the surface of the metal is collected using achromats. It is magnified by about 16 times and focused on the 40 μm entrance pinhole of SP150 Acton series spectrometer with back-illuminated PIXIS 100BR (pixel size: 1340 100) camera in the wavelength range of 565–900 nm. The temperature measurement system is calibrated with the tungsten filament light source (NPL, UK) having known intensity vs. wavelength distribution with temperature. Temperature of the sample surface is measured by spectroradiometry technique (Boehler et al., 1990; Mukherjee and Boehler, 2007) by fitting Planck’s radiation function (Planck, 1901) to the flatfield corrected spectrum. With appropriate flatfield, temperature of the hotspot can be estimated from fitting within an error of about 20 K. However, for determination of actual temperature error in DAC, we have carried out measurements of melting temperatures of Ar and Pt by increasing as well as decreasing laser power, which we have determined within an error of 100 K (Kavner and Jeanloz, 1998; Boehler, 2001). Therefore, we attribute error in our temperature estimation to be within 100 K. Temperatures at different positions are measured by translating the 40 μm pin-hole attached to the spectrometer across the magnified image of the sample surface with a resolution of 1 μm. Supplementary Fig. S5 represents flatfield corrected spectrum with its fitting by Planck radiation function at the hotspot and r2 distance away from the hotspot. For calculation of thermal conductivity we have used a model that simulates the steady-state temperature distribution in a cylindrical Fe plate heated by a small heat source at its center. We continuously pump the laser power on then sample and temperature measurements are carried out only when a constant temperature gradient is observed. Therefore the assumption considered in our model seems to be satisfied. However, to test the above assumption, we have carried out time dependent temperature measurements at two different positions on the sample surface at various pressures and temperatures and the data are shown in Fig. 1. The temperature at the same position with time is found to remain constant within the error limit and gives us enough confidence
group using a dynamically laser-heated diamond anvil cell showed a low value of thermal conductivity of Iron: (i) about 35 Wm1K1 at 48 GPa and 2000 K and (ii) in the range of 18–44 Wm1K1 up to pressure 130 GPa and temperature range of 2000–3000 K respectively (Kon^ opkova et al., 2016). In both the experiments thermal conductivity was estimated from finite element simulation analysis and considering the reflectance of laser power from interfaces. All the above experiments and theoretical simulations do show large variations in thermal conductivity of Fe at high pressures and temperatures. The large discrepancy in the value of thermal conductivity of Fe at extreme pressures and temperatures lead to radically different values of estimating the age of the Earth’s solid inner core (Biggin et al., 2015; Davies et al., 2015; Tarduno et al., 2015; Kon^ opkov a et al., 2016). High thermal conductivity values of Fe at Earth’s core conditions estimated from indirect measurements and theoretical simulations indicate to a young core less than a billion years (Davies et al., 2015). However, presence of ancient magnetic field from full-vector paleointensity measurements indicate to the presence of geodynamo at much earlier stage (Tarduno et al., 2015) and supports the low value of thermal conductivity of Fe. Dobson (2016) raised the points that: (i) high thermal conductivity of Fe at CMB estimated by Ohta et al. (2016) from resistivity measurements may be an artifact due to the underestimate of heat loss through the electrodes and; (ii) very low thermal conductivity estimated using laser heated diamond anvil cell by Kon^ opkov a et al. (2016) may arise from the melting of Fe surface due to the very short laser pulses. These observations obviously open up the controversies regarding the measurement of thermal conductivity values at the conditions of CMB and hence it is important to have new direct measurements. In the present work we have carried out measurements of thermal conductivity of Fe, both in solid as well as liquid phases in its fcc(γ)-phase and hcp(ε)-phase up to about the pressures of Earth’s outer core using a single sided laser heated diamond anvil cell (LHDAC). 2. Materials and methods Thermal conductivity measurements at high-pressures and hightemperatures are carried out in LHDAC using plate type diamond anvil cells (Almax-Boehler design). Diamond anvils of culet diameter 300 μm and 100 μm are used for pressure ranges from ambient to 50 GPa and 50–134 GPa, respectively. Steel gasket of thickness 225 μm is preindented to a thickness of 50 μm. A hole of diameter ~110 μm is drilled at the center of diamond imprint of 300 μm culet. For 100 μm DAC culet the steel gasket pre-indented to a thickness of 45 μm followed by drilling a sample hole of diameter ~75 μm. Thin plate of iron of approximate thickness of about 15–20 μm is made by compacting polycrystalline iron powder using a 300 ton hydraulic press. Thin pieces of Feplates of desired size (approximate diameter of about 70–90 μm) are cut to load them in the LHDAC. A scan of the compressed sample surface is taken using SEM to find out about the grain boundary distribution and uniform sample surface and is shown in Supplementary Fig. S1. Supplementary Fig. S2 is the EDS of the compressed iron powder before loading taken using IISER-Kolkata FESEM facility. Sample is sandwiched in between thin layers of pressure transmitting medium (PTM) of approximate thickness of about 15 μm inside the central hole of the gasket. Two different sets of experiments are carried out using NaCl and Al2O3 as pressure transmitting medium. Both NaCl and Al2O3 are dried at a temperature of 393 K for 6 h to remove any trace of moisture. Pressure in the LHDAC before and after heating is determined using the ruby fluorescence method (Mao et al., 1986) by placing a few ruby chips (approximate sizes of about 3–4 μm) at the edge of the iron plate. Heating is carried out using a diode-pumped Ytterbium fiber optic laser (YLR100-SM-AC-Y11) with central emission wavelength, λ ¼ 1.070 μm (maximum power 100 W). Laser beam is focused on the sample inside the diamond anvil cell at an angle of approximately 23 with respect to the normal to the diamond surface using 100 mm focal length anti-IR reflection coated fused silica lens. A schematic of our laser heated
Fig. 1. Time dependent temperature at hotspot (filled symbols) and at a distance r2 (~70 μm) from the hotspot (open symbols). Different symbols represent different experimental runs. 2
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The 3D geometry of our sample chamber is shown in the Fig. 2. The laser is focused on the sample and then the temperatures are measured from various positions of the sample surface radially in a single line. This is possible due to the fact that the laser is focused off-axis and the temperature is measured on-axis. The hot spot has a diameter 16–18 μm and the sample has a diameter of 70–90 μm and the gasket hole is of about 75–110 μm. We have used finite-element software COMSOL Multiphysics (COMSOL) to simulate the temperature distribution in the laser heated diamond anvil cell. The main aim of the simulation is to match the experimental temperature profile by varying the thermal conductivity of the sample and the gasket boundary condition (Tg). We have listed the material properties and geometry in Table 3.
using the steady state condition. In the computational model we have considered the heat to be constant at a value noted by Q inside the hotspot diameter. However, we have considered a distribution of heat inside the sample (f(r)) as given by Δ⋅(k ΔT) ¼ f(r)
(1)
where, k is thermal conductivity of iron, ΔT is the temperature gradient across the sample surface. We have used the boundary conditions for calculation of thermal conductivity of iron in COMSOL; f(r) ¼ Q for r Rhotspot and T ¼ Tg at gasket boundary
(2)
The heat energy Q absorbed by the Fe plate is calculated by Q ¼ mCp(Thotspot TRoom)v
3. Results and discussion
(3)
where, m is the mass of the sample at the hotspot, Cp is the specific heat capacity of iron at constant pressure, and (Thotspot TRoom) is the temperature difference between hot spot and room temperature and v is the modulation frequency (50 kHz) of the 1.070 μm wavelength laser. Exposure time of collected spectrum for temperature measurement is about 100 msec, which is much larger than the modulation period. The specific heat c of the Iron is taken to be 450 JKg1K1 up to 40 GPa and 708 JKg1K1 from 46 to 134 GPa (Gubbins et al., 2003; Hirose et al., 2013). Mass of hot spot m in the Eq. (3) is calculated as m ¼ π R2hotspot hρ
The temperature gradient on the sample surface at four different pressure points (6, 31, 46 and 60 GPa) are measured by heating the sample at the center and it is shown in Fig. 3. The filled scattered symbols are the measured temperature across the sample surface and the solid lines are the computed temperature profile obtained from the COMSOL software. The best match of the experimental data points to the computed lines is obtained by varying the thermal conductivity of the Fe plate taking all the other parameters as mentioned in the Table 3. Other measurements of thermal conductivity are carried out by heating the sample at one end and measuring the temperatures at a distance of 70 μm away. Thermal conductivity is measured in fcc (γ)-phase and hcp (ε)-phase by following the high pressure and the high temperature phase diagram of Fe (Boehler, 1993; Boehler et al., 2008; Anzellini et al., 2013). Since our experiments are carried out with accurate temperature measurements of the sample surface, we could follow the phase diagram precisely. We have carried out some measurements on the melting line of Fe to see the effect of liquid Fe on the thermal conductivity value. Temperature dependence of thermal conductivity values estimated by taking the absorbed power following Eq. (3) at a lower pressure of 10 GPa are shown in Fig. 4. A sharp fall in the thermal conductivity is observed at 2146 K and is attributed to the melting (Liu and Bassett, 1975; Anderson, 1986; Ezenwa and Secco, 2019) at the hotspot while the sample is still at solid state at the edge opposite to the hotspot. The drop in the thermal conductivity across melting at high pressures is observed to much sharper from our experiments compared to that observed by Ho (1972) and Nishi et al. (2003) at ambient pressure. Possibly liquid and solid interface at the boundary of the hotspot gives rise to the above effect. A sudden jump to higher electrical resistivity values have been observed during melting of Fe in high pressure experiments indicating a sudden decrease in thermal conductivity (Deng et al., 2013; Ohta et al., 2016; Silber et al., 2018; Basu et al., 2019). Thermal conductivity due to molten iron at the hot spot is found to be 25%–30% lower than that of solid iron (Fig. 4). Pressure dependence of thermal conductivity of Fe in fcc(γ)-phase, hcp(ε)-phase and their mixed phase are shown in Fig. 5 and compared with other reported values. Filled symbols depict our data (black filled square for γ-Fe and olive filled triangle for ε-Fe using NaCl as pressure transmitting medium and blue filled triangle for ε-Fe using Al2O3 as pressure transmitting medium and red filled circle for mixed phase) and open symbols show the reported values in literature (Kon^ opkova et al., 2011, 2016; Deng et al., 2013; Xu et al., 2018). The value of k in fcc(γ)-phase of Fe is consistent with the same phase of Fe with the literature (Deng et al., 2013). k value of γ-Fe seems to increase with pressure. It shows a minimum value 93(35) Wm1K1 at 6 GPa to a maximum value 125(50) Wm1K1 at 46 GPa in temperature range 1700–1900 K. Thermal conductivity of the mixed phase (hotspot at γ-phase and edge at ε-phase as per the phase diagram (Boehler, 1993) has a value in the range 48–65 Wm1K1 in the pressure and hotspot temperature range of 18.5–46 GPa and 1400–1700 K, respectively. Pressure dependence of thermal conductivity of mixed phase is also found to increase up to 46 GPa (Fig. 5). These values of the thermal conductivity are seems to be
(4)
where, Rhotspot is the radius of the hot spot, h is the thickness of compacted iron powder before loading and ρ is the density of compacted iron powder. Initial density of the compacted iron powder is determined by preparing a pellet of weighed amount of powder and measuring its dimensions. The mass of the hotspot is assumed to remain constant with pressure in each experiment as the sample chamber is quasi-hydrostatic. We have carried out the temperature measurements on the both side of the sample at the hotspot. Two representative values of the temperature are tabulated in Table 1. Very small difference in the temperature at the hotspot on both sides of the sample indicates to uniform heating of the Feplate at the hotspot. By taking into account the estimation of mass at the hotspot and uncertainty in its value, the final uncertainty in absorbed power is found to be about 31%. Estimated heat absorbed by the Fe-surface using Eq. (3) along with error estimation at various stages is given in a tabular form in Table 2. In support of the above, we have tried to estimate the total laser power dumped on the sample surface by taking into account the reflections from the optics and different interfaces (Benedetti and Loubeyre, 2004; Seagle et al., 2009) and same is given in Supplementary Table S1. By taking into account of errors in every step, we find an uncertainty of about 40% in determination of thermal conductivity values in Fe (Table 2). Table 1 Temperature measured at the front and back of the sample surface corresponding to the incident laser spot. Temperatures at a particular pressure are measured by changing the laser power. P (GPa), Phase, PTM
Front side T (K)
Rear side T (K)
Difference in T (K)
39, γ-phase, NaCl
1609 1676 1734 1791 1818 1982 1863 1914 1986 1991 2047 2129
1593 1672 1734 1776 1811 1970 1851 1905 1079 1991 2044 2117
16 4 0 5 7 12 12 9 7 0 3 12
46, γ-phase, NaCl
3
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Table 2 Detailed error assessment in measuring thermal conductivity of ε-phase of Iron at 134 GPa. Here r1 is the radius of the hotspot, h is the thickness of the Fe-plate before loading, m is the mass of the hotspot, c is the specific heat of Iron, T1 is the temperature of the hotspot, T2 is the temperature at a distance r2 from the hotspot, Q is the absorbed power at hotspot measured using Eq. (3). The estimated beam waist of the laser has larger diameter than the hot spot. Therefore some part of the laser power is also incident outside the hotspot and hence we estimate the error in r1 from the difference of the half of beam waist and the radius of the hotspot. We have measured the thickness (h) of the compressed Fe-plate before loading several times using micrometer and we assign the error in h from our measurements. The mass (m) of the hotspot is measured from the error measurements of density of the Fe-plate and volume of the hotspot. The error in c is assigned from the literature (Gubbins et al., 2003; Hirose et al., 2013) corresponding to the Earth’s outer core condition. The error in T1 and T2 is already explained above. The error in Q is assigned from the propagation of the errors. The error in r2 is assigned from the resolution of the motion of spectrometer pinhole. The error in k is assigned from all the propagated errors. A total error is estimated to be 40% in the thermal conductivity values.
Meassured value Error
Rhotspot (μm)
h (μm)
m (Kg)
c (JKg-1K-1)
T1 (K)
T2 (K)
Q (Watt)
r2 (μm)
k (Wm-1K-1)
9 2
15 0.2
5.5 10-11 1.6 10-11
708 8
1823 100
1567 100
1.2 0.3
70 1
78 31
Fig. 2. (a) Schematic drawing of the sample chamber in laser heated diamond anvil cell facility. Number 1 represents the hotspot at the center of the sample, 2 represents the sample, and 3 represents pressure transmitting medium. (b) Cross sectional view of the sample chamber with gasket. (c) Cross sectional view of computed temperature distribution in the sample chamber and gasket.
sample much more accurately. This probably results in thermal conductivity values, which are not affected by the type of pressure transmitting medium used. It can be seen from Fig. 5 that our thermal conductivity values are found to be 2 to 3.5 times higher than those determined using CW and pulsed LHDAC route (Kon^ opkov a et al., 2011, 2016). Recent computational study by taking account of both electron-phonon (eph) and electron-electron (ee) scattering contributions to the thermal conductivity of solid iron have reported k to be in the range of 90–110 Wm1K1 in the pressure range of 110–140 GPa at 2000 K (Xu et al., 2018), which are in agreement with this study. Our results of k in ε-phase are found to remain almost constant with pressure from 46 to 134 GPa, the highest pressure of our study. To check the diffusion of carbon in the sample (Prakapenka et al., 2004), Raman measurements have been carried out with a micro-Raman spectrometer LabRAM HR from Horiba Jobin Yvon with 1800 grooves mm1 grating using the excitation wavelengths of 488 and 632.8 nm with spectral resolution of 1 cm1. All Raman spectra have been collected on quenched samples after cleaning using distilled water outside DAC and are shown in the Supplementary Fig. S6. No observable peak of carbon phase is detected in the Raman signals from the samples quenched at different pressures and temperatures. Also, no chemical reaction is reported under high-temperature and high-pressure by Errandonea (2009) during laser heating on other transition metals. For estimation of thermal conductivity of Earth’s interior it is also important to look into the temperature variation at high pressures. We observe a slight decrease with increase in temperature. We have fitted the temperature dependence of k at various pressures using the relation
Table 3 The used parameters in COMSOL for the determination of thermal conductivity of iron. Material
Dimensions Thickness & Diameter (μm)
Density (Kgm-3)
Thermal Conductivity (Wm-1K-1)
Fe NaCl
15–20 & 70–100 15 & 75–110
7538 2160
Al2O3 Gasket (steel)
15 & 75 45–50 & 106
3950 8050
variable 6 (Hakansson and Andersson, 1986) 5 (Hofmeister, 1999) 20 (Keifer and Duffy, 2005)
30%–40% higher than that of the direct measurements by Kon^ opkova et al. (2011). These low values of thermal conductivity with respect to the γ-phase in this study may be due to the phase boundary effects. All our thermal conductivity measurements in ε-phase are carried out above 46–134 GPa around 1800–2000 K. Measured temperature values of the this study and computed thermal conductivity values for nine different runs are shown in Supplementary Table S2. Our thermal conductivity measurements using two types of pressure transmitting media, NaCl and Al2O3 do not show much of a scatter and are comparable within the error-bars. In the spectroradiometric technique of temperature measurement we focus the sample surface at the entrance pinhole of the monochromator. Therefore, only the incandescent light from the sample surface is used for temperature measurement and we are able to measure the surface temperature gradient of the 4
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Fig. 4. Temperature dependent thermal conductivity of iron at pressures at 10 GPa. The sharp fall in the thermal conductivity indicated inside green dotted box is attributed to the part of the Fe sample that is in the liquid phase. Melting of the iron sample was confirmed from the laser speckle motion of a blue laser incident on the sample.
Fig. 3. Measure and computed temperature distribution on iron foil heated at different pressures. Temperatures were measured by translating the pinhole across the magnified image (magnified by 16 times) of the sample surface. Inset (a) shows the image of hotspot about diameter 18 μm on iron at 46 GPa under transmitting light and (b) shows the magnified image of hotspot on Iron at 46 GPa with radial temperature distribution.
(Kon^ opkov a et al., 2016) as given below k ¼ aT þ
b Tn
(5)
We have included the values of k in the temperature range of 300–500 K in the fit by estimating them from the high pressure resistivity data by Gomi et al. (2013) at 102 GPa. We have used Wiedemann-Franz-Lorenz law for the conversion of resistivity value to thermal conductivity and taken the Lorenz number to be 2.40 108 WΩK2. This Lorenz number was estimated using our thermal conductivity values and electrical conductivity data under shock compression (Bi et al., 2002). Due to the dynamic and fast measurements of resistance using shock experiments, the controversy on heat leak through the electrodes can be avoided. Therefore, we have chosen resistance measurements under shock compression for the estimation of the Lorentz number. We observe that best fit at high pressures are obtained using a value of n ¼ 0.23. The parameter a is found to be in the order of ~ 103. The leading term in the fit comes from the parameter b and is found to be around 600. In Fig. 6 we have shown a representative fit of k with respect to temperature at 98 GPa along with its 95% confidence band and k values determined at various pressures and temperatures. The cluster of k values determined in our experiments seems to be described well by the temperature dependent fit at 98 GPa along with the confidence band within the error-bars. Our direct experimental measurement of k values gives rise to an interesting fact that in ε-phase, k seems to saturate at high pressures and only depends on temperature with a power law behavior having a small
Fig. 5. Our results of pressure dependent thermal conductivity of iron are compared with those in literature. Filled symbols are the results of our study and open symbols are reported values. Black filled squares for γFe and olive filled triangles for εFe using NaCl as pressure transmitting medium and blue filled triangles for ε Fe using Al2O3 as pressure transmitting medium and red filled circles are for mixed phase.
exponent of 0.23. High pressure studies have shown a transition in
ε-phase of Fe above 40 GPa with anomalies in Raman spectroscopy (Merkel et al., 2000) and first principles calculations indicating a
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4. Conclusions To summarize, a new technique has been employed to measure the thermal conductivity in laser heated DAC. In this technique thermodynamical equation has been used to calculate absorbed power by the metal plate. The thermal conductivity is estimated from the temperature differences across the sample and using the COMSOL software. The k values of iron are measured up to Earth’s outer core pressure ~134 GPa with an uncertainty of about 40%. Value of thermal conductivity of γ-phase and mixed phase of iron is seemed to increase with pressure. Thermal conductivity values in the ε-phase of iron lies in the range of 70–80 Wm1K1, almost constant with pressure. We believe our new measurements will be a solid step towards solving the discrepancy about the time period of generation of Earth’s magnetic field as well the source of heat flow from core to the surface. Author contributions All the authors have equal contribution. Both authors reviewed the manuscript.
Fig. 6. Thermal conductivity of ε Fe versus temperature. Solid filled squares depict the value of k estimated from resistivity measurements by Gomi et al. (2013). All the open symbols data are of this study. Red and equally spaced symmetric blue solid lines represent the model fit of thermal conductivity data at 98 GPa using Eq. (5) and 95% Confidence band, respectively.
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
hyperfine field splitting (Steinle-Neumann et al., 2004). Nuclear in-elastic scattering experiments and Mossbauer studies have predicted a pressure induced electronic topological transition (ETT) in ε-phase of Fe above 40 GPa (Glazyrin et al., 2013). Using dynamical mean field theory simulations, the same group has shown that the ETT is mediated by strong electron correlation effect. At the range of pressures of our measurements, Debye temperature of pure iron is found to be much smaller than our experimental temperature range (Anderson et al., 2001). This means that lattice contribution to specific heat capacity of iron will almost saturate at high pressures and high temperatures. From Eq. (3), it is clear that both pressure and temperature dependence of k will then be affected mostly by the behavior of electronic correlations. In classical electronic theory of metals, thermal conductivity is proportional to the electron mean free path and inversely proportional to the effective mass. In the dense hexagonal phase of Fe many-electron correlation effects may increase the effective mass, which may in turn lead to lower constant value of thermal conductivity in ε-phase compared to γ-phase. The above may be an outcome of large overlap of 3d and 4s electron states in pure iron (Stacey and Loper, 2007). Slight power law decrease in k with respect to temperature is probably due to increase in temperature induced scattering. Hence, the saturation effect with respect to pressure in k may be related to ETT in pure iron. Temperature dependent model fit of our data predicts a value of k of solid iron to be 55 22 Wm1K1 at CMB conditions. Our results obviously limit thermal conductivity value in ε-phase to the temperature effects only and can provide important inputs for the question over age of Earth’s inner core. As discussed earlier our thermal conductivity measurement on a Fe sample which had both molten and solid regions is found to be 25%–30% less than the solid iron at same pressure. Using the above result one can estimate thermal conductivity of molten iron at CMB to be about 40 16 Wm1K1. This result is in good agreement with the thermal conductivity estimation of Stacey and Loper (2007) and as well as recent direct measurements by Kon^ opkova et al. (2016). This moderate low value of thermal conductivity indicates slow cooling of Earth’s core from its initial state. Therefore, presence of core convection can not be ruled out at the early stage resulting in generation of magnetic field as the dynamo entropy is positive as per the estimation of Davies et al. (2015). Our results strongly support the conclusions and results of palaeomagnetic measurements (Biggin et al., 2015; Tarduno et al., 2015), which estimates the presence of ancient terrestrial magnetic field may be due to presence of geodynamo at early stages of Earth.
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