Thermal conductivity of hcp iron at high pressure and temperature

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Thermal conductivity of hcp iron at high pressure and temperatureZ. Konôpkováa; P. Lazora; A. F. Goncharovb; V. V. Struzhkinb

a Department of Earth Sciences, Uppsala University, Uppsala, Sweden b Geophysical Laboratory,Carnegie Institution of Washington, Washington, DC, USA

First published on: 18 February 2011

To cite this Article Konôpková, Z. , Lazor, P. , Goncharov, A. F. and Struzhkin, V. V.(2011) 'Thermal conductivity of hcpiron at high pressure and temperature', High Pressure Research, 31: 1, 228 — 236, First published on: 18 February 2011(iFirst)To link to this Article: DOI: 10.1080/08957959.2010.545059URL: http://dx.doi.org/10.1080/08957959.2010.545059

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High Pressure ResearchVol. 31, No. 1, March 2011, 228–236

Thermal conductivity of hcp iron at high pressure andtemperature

Z. Konôpkováa*, P. Lazora, A.F. Goncharovb and V.V. Struzhkinb

aDepartment of Earth Sciences, Uppsala University, Villavägen 16, 75236 Uppsala, Sweden;bGeophysical Laboratory, Carnegie Institution of Washington, 5251 Broad Branch Road N.W.,

Washington, DC 20015-1305, USA

(Received 4 October 2010; final version received 29 November 2010 )

Results of steady-state heat transfer experiments on iron in laser-heated diamond anvil cell, combined withnumerical simulation using finite-element method are reported. Thermal boundary conditions, dimensionsof sample assemblage, heating-laser beam characteristics and relevant optical properties have been welldefined in the course of experiments. The thermal conductivity of the polycrystalline hexagonal-iron foilhas been determined up to pressure 70 GPa and temperature 2000 K. At these conditions, the conductivityvalue of 32 ± 7 W/m K was found. Sources of errors arising from uncertainties in input parameters andapplied experimental procedures are discussed. Considering results of earlier preferred-orientation studiesin diamond anvil cell, an averaging effect of polycrystalline texture on the intrinsic anisotropy is assumed.The obtained conductivity is interpreted as an effective value, falling in between the upper and lowerbounds on the average conductivity of a random aggregate of uniaxial crystals.

Keywords: thermal conductivity; laser heated diamond anvil cell; iron

1. Introduction

Thermal conductivity of materials in planetary interiors belongs to key parameters ruling overtheir thermal evolution and dynamics. Yet it is insufficiently constrained, in particular under thedeep mantle and core conditions. Thermal conductivity of iron alloy in Earth’s core has beenrecently revised downward by a factor of 2, to 28–29 W/m K [1].

Transport properties, describing non-equilibrium phenomena are infamously difficult to mea-sure. Thermal conductivities are often determined with uncertainties as high as 20% even atambient pressure [2]. At high pressures and temperatures, various techniques have been appliedin order to evaluate thermal conductivity and diffusivity of mantle minerals, e.g. the Ångströmmethod in the multi-anvil apparatus [3,4], laser flash heating in the diamond anvil cell (DAC) [5],and a model-based approach utilizing IR reflectivity and thermoelastic data [6]. Theoretical calcu-lations at high pressures are rare and mostly focused on simple single crystals, such as periclase,where the heat is conducted by lattice vibrations [7,8].

*Corresponding author. Email: [email protected]

ISSN 0895-7959 print/ISSN 1477-2299 online© 2011 Taylor & FrancisDOI: 10.1080/08957959.2010.545059http://www.informaworld.com

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High Pressure Research 229

Unlike the insulators, in metals, the heat is carried predominantly by electrons and the con-duction is limited by scattering processes of both electrons and phonons. Moreover, the mobilityof free electrons is also affected by a magnetic state, and/or coordination number, which fur-ther complicates the predictions [9]. This is readily observed in ambient-pressure data on certainmetals such as nickel or iron where the magnetic/structural transitions manifest themselves withchange in slope and magnitude of the thermal conductivity [10].

In metals, common carriers of the heat and charge makes the thermal and electrical conduc-tivities interrelated; at a given temperature, their ratio is constant (Wiedemann–Franz law). Athigh pressures, the electrical resistance measurements are more convenient to perform than mea-suring the thermal conductivity; the electronic contribution of the latter can be then inferredfrom the Wiedemann–Franz law. It should be noted, however, that deviations from the constantLorentz number were reported at intermediate temperatures for some metals [11–13]. More-over, the pressure coefficient of the thermal conductivity was measured significantly larger thanthat of the electrical conductivity [14]. Nevertheless, electrical resistance data bear importantinformation on the heat conductivity and its changes through any structural, magnetic or otherphase transition.

At ambient pressure, iron undergoes a series of phase transitions with temperature, each ofthem affecting the heat conduction. At high pressures, the measurements of thermal and electricalconductivities of bcc iron revealed a deviation from the Wiedemann–Franz law [14]. The effect ofpressure on the thermal conductivity of bcc iron at room temperature reported by Sundqvist [14]corresponds to dk/dP = 2.8 Wm−1 K−1 GPa−1. The electrical resistivity at simultaneously highpressure and temperature of bcc and liquid iron has been measured up to 7 GPa [15]. A largeincrease (2.5 − 3.5×) in the electrical resistance of iron has been reported from a static and shock-wave compression experiments at pressures of bcc–hcp phase transition at room temperature[16–19]. On the other hand, a much smaller increase (≈ 30%) in the resistivity of across thetransition is reported in a recent first-principles study by Sha and Cohen [20]. Experiments onshock-compressed iron above 200 GPa suggest that the electrical conductivity of hcp iron isproportional to 1/T according to the Bloch–Grüneisen expression [21].

In this study, we report a novel method for the determination of thermal conductivity of metalsat high pressures and temperatures, which is based on heat transfer experiments in laser-heatedDAC, combined with numerical simulation using the finite-element method.

2. Experiment

The sample was prepared from a bulk piece of iron, first flattened between two anvils to athin foil of thickness 2.3(1) μm, then cut by laser into small pieces fitting the sample cham-ber. A two-stepped gasket hole was drilled in the indented rhenium gasket. The recessed holehelps to position the sample in about the same distance from the diamond culets, ideally par-allel to them. One standard diamond anvil with 300 μm culet and one beveled diamond with100 μm were used. Argon gas sealed in the sample chamber at pressure 0.2 GPa served as apressure medium.

Pressures were determined from the ruby fluorescence [22] at ambient temperature. These alsorepresent the pressures reported in the study; the effect of thermal pressure was not taken intoaccount. Using this method, the lower bound on the experimental pressure is around 35 GPa,where the temperatures are high enough to be measured by spectroradiometry, and yet do notexceed the temperature of the phase hcp–fcc transition in iron [23]. This also represents a prac-tical low pressure limit from the points of view of heating stability and heat transfer analysis;at lower pressures, the melting of argon makes the heat transfer mechanism no longer solelyconductive [24].


230 Z. Konôpková et al.

Table 1. Dimensions of the sample assemblage.

Thickness (μm) 0 GPa 38 GPa 70 GPa Note

Iron foil 2.30 1.87 ± 0.20 1.73 ± 0.20 EoS [26]1. Argon layer 4.63 ± 0.50 3.80 ± 0.50 Fringes2. Argon layer 5.45 ± 0.50 3.29 ± 0.50 FringesDiamond–diamond 13.75 ± 1.0 9.55 ± 1.0 FringesRefractive index of Ar 1.60 1.66, 1.50 [27,28]Refractive index of diamond 2.42 2.60 2.75 [31]Refractive index of air 1.00Refractive index of Fe 4.00 2.50 [32]Coef. of ext. Fe 4.50 1.00 [32]

Note: Refractive indices used for calculation of reflection and absorption of the laser power.

The thicknesses of argon–iron layers in the sample chamber were calculated using the methodof interference fringes [25], combined with the information on volumetric changes obtained fromthe equation of state of iron [26] (Table 1). Refractive index of the compressed argon was takenfrom [27,28].

One-side laser heating of the iron foil was performed in the CW mode with a near-IR (1.06 μm)Ytterbium fiber laser (200 W, IPG Photonics). A detailed description of the laser heating systemis provided in [29,30]. Laser power was measured at the position of the DAC and corrected forthe effects of reflections and absorption. A standard formula for normal reflectance was appliedto diamond–air and diamond–argon interfaces, using refractive indices of diamond [31] andargon [27,28]. For evaluating the laser power absorption by the sample, room temperature high-pressure data on the infrared reflectivity of iron [32] have been applied. Overall, 39(2)% and70(3)% of the laser power reaching DAC was absorbed by iron at 38 and 70 GPa, respectively.The focal spot size and power distribution, used as input parameters in the numerical model, weremeasured with a laser beam profiler (Thorlabs) in two perpendicular directions. The intensityprofiles, described by a Gaussian function exp[−1/2(r2/R2)], signified a small ellipticity of thebeam (semi axes 7.8 and 8.8 μm).

At each pressure, a series of temperature measurements at an increasing laser power wascarried out. For each laser power, two temperatures were measured simultaneously using twoCCD detectors from the opposing points on the two sides of the foil. Measured tempera-ture difference thus represents the axial temperature gradient across the foil. A wavelength-independent emissivity approximation was used when calculating temperatures using Planck’slaw [33]. Recent study by Seagle et al. [32] on the high-pressure optical properties of ironshows that the hcp phase of iron exhibits only a small wavelength dependence of emissivity.Moreover, a small residual effect of this approximation on the calculated thermal conduc-tivity largely cancels out when using temperature gradients as a primary input parameter inthe model.

3. Numerical modeling

The heat conduction in the 3D pressure chamber (Figure 1) was modeled by the finite-elementmethod using a software package Comsol Multiphysics. In case of metals, the laser power isabsorbed in a very thin surface layer proportional to the skin depths of metallic materials (δ =1/

√πf μσ , where μ is the permeability, σ the electrical conductivity and f the frequency of the

wave), which is usually of the order of tens of nanometers. This requires a different approach fromthat applied in the case of (semi)transparent materials exhibiting a bulk absorption [34]. Thermal



Figure 1. Three-dimensional geometry of the pressure chamber arrangement for modeling the temperature distributionin the sample and pressure medium.

transport in iron and argon obeys the steady-state heat equation

∇ · (−k(T )∇T ) = 0. (1)

The laser-heating source, Qlaser was represented by the heat flux at the heated Fe/Ar interface(boundary condition), partitioned between iron and argon according to the equation

−�nFe(−kFe∇TFe) − �nAr(−kAr∇TAr) = Qlaser. (2)

Measurements using the laser beam profiler yielded a slightly elliptical distribution of the laserpower (semi-axes a and b) in the form Q exp(− 1

2 (x2/a2 + y2/b2)). Q is typically of severalGW/m2.

For a given pressure, thermal conductivity coefficient of iron, kFe, was modeled as a simple,temperature-dependent scalar. The interpretation of results, in regard to this assumption, is dis-cussed below. Thermal conductivity of solid argon crystallizing at low pressures with the cubicsymmetry (fcc) is intrinsically isotropic. Its values at high pressures (up to 50 GPa) and tem-peratures (up to 2000 K) have been calculated in a molecular dynamics study [35]. Errandoneaet al. [36] found a sluggish fcc-to-hcp structural transformation in argon after stress relaxationby laser heating above 50 GPa. Although this is relevant for the uppermost pressure–temperatureconditions of this study, only about 10% of the hcp polymorph of argon is expected to be presentin the fcc/hcp assembly, as follows from the reported data on the progress of the transition. More-over, a close structural similarity between the two phases (different stacking polytypes) lends asupport to the assumption of similar values of their thermal conductivities. Within the accuracyof the current approach and limits imposed by the absence of textural information, a single crys-tal data of Tretiakov and Scandolo [35] for the thermal conductivity of fcc argon is used in thenumerical model (e.g. 3.6 W/m K at 38 GPa, 1500 K and 2.9 W/m K at 70 GPa, 2500 K). Maxi-mum temperatures reached in the experiment remained at least 700 K below the melting point ofargon at the studied pressures [37].

The model calculations were performed in a system which included the iron–argon assemblage,forming the pressure chamber, bounded by diamonds and gasket. In laser-heating experiments,temperature of the diamond culet is difficult to measure, and an assumption of close-to-roomtemperature value is commonly made [34]. In order to constrain better the temperature on theargon–diamond interface, a 2D axisymmetric model of the entire DAC has been run initially. Forthe thermal conductivities of the individual cell parts, 40 W/m K was used for the boron carbideseat, and 25 W/m K for the cell’s body machined from Vascomax steel. The conductivity ofdiamond anvil can vary between 500 and 2000 W/m K, depending on the amount of impurities [38]and the stress state. For the average thermal conductivity of our type I diamonds (stressed culet+ pavilion + crown), two values were used, 500 and 1000 W/m K. The conductivity of rheniumgasket is 48 W/m K. Figure 2 illustrates the calculated temperature distribution in the pressurechamber and its immediate vicinity, for the outer surface temperature of 320 K, typical for a



Figure 2. Two-dimensional axisymmetric model of the DAC including the diamonds and cell body parts (figure iszoomed to the pressure chamber and parts of the surrounding diamonds and gasket). T 0 (310, 320 K) is the temperatureof the cell’s body. The lower figure shows the temperature profiles along argon–anvil interface for diamond conductivitiesof 500 and 1000 W/m K.

passively air-cooled laser-heated diamond anvil cell in our setup. The temperature profiles on thediamond-medium interface, shown in Figure 2 for several combinations of parameters, reveal thatdiamond anvils can sustain a non-negligible temperature gradient under conditions of localizedlaser heating and thin insulating layers. In the modeling of the sample chamber, the temperatureboundary condition on the medium-diamond interface was held as a parameter, investigated inrelation to the conductivity of argon and absorbed laser power.

4. Results

Laser heating was carried out at two pressures, 38 and 70 GPa. The experimental points inFigures 3(a)–(d) present the laser-power dependence of the central peak temperatures on theopposing sides (heated/unheated) of the foil. The peak temperatures were derived in a self-consistent manner from the experimentally determined average temperatures of the central hotspot of diameter 8 μm, defined by the spatial resolution of the imaging system. Using thecalculated temperature profiles in a two-step loop, the upward average-to-peak temperature cor-rections were found to span the range 25–120 K for temperatures 1000–2000 K. Errors of ±50 K



Figure 3. Laser-power dependence of axial temperatures across the iron foil. Dots, experimental values; lines, modelcalculations. Heating cycle was performed through (a) the standard-size diamond anvil at 38 GPa, (b) the beveled diamondanvil at 38 GPa, (c) the standard-size diamond anvil at 70 GPa; (d) the beveled diamond anvil at 70 GPa.

shown in the plot reflect the reproducibility of temperature measurement. The overall accuracy isabout ±100 K.

Thermal conductivity of iron, kFe, has been found in several iterative steps. Initially, a sweep-ing search for its temperature independent value, best matching the experimentally measuredaxial gradients, has been carried out for the nominal values of input parameters (absorbed power(ALP), boundary temperature (Tb), conductivity of argon (kAr) and laser beam radius (LBR)). Inthe following iteration, a linear temperature dependence of the conductivity, k(T ) = A + B/T ,has been considered. At high temperatures, the dependence became constrained by tempera-tures measured in this study, while the low temperature dependence is anchored by the highpressure room temperature data on electrical resistivity of iron [17] through the application of theWiedemann–Franz law. Finally, the so far fixed nominal values of the input parameters have beenrelaxed toward the best agreement between the model and the experiment.

The study by Reichlin [17], carried out up to 33 GPa, shows a factor-of-two increase in theelectrical resistance of iron in the range 15–25 GPa, marking the bcc–hcp phase transition. Apply-ing the Wiedemann–Franz law to Reichlin’s data [17] yields value k (30 GPa, 300 K) of about43 W/m K for the thermal conductivity of the hcp phase of iron. Using the linearized equationof state k(P, 300 K) = k (30 GPa, 300 K)[1 + (K ′

30/K30)P ] [39], conductivity values of 69 and91 W/m K were obtained at 38 and 70 GPa, respectively. At 30 GPa, bulk modulus, K30, and itspressure derivative, K ′

30, are 313 GPa and 4.94, respectively, as calculated from the equation ofstate of hcp-Fe [40].

Temperature dependencies of the thermal conductivity of hcp iron obtained in this study atpressures 38 and 70 GPa are shown in Figure 4.



Figure 4. Temperature dependence of the thermal conductivity of hcp. Solid lines, temperature range of this study. Blackdots, values calculated from the room temperature electrical resistivity measurements by [17] using the Wiedemann–Franzlaw. The vertical bar indicates the estimated uncertainty of the conductivity. The temperature dependence is expressed ask(T ) = A + B/T : A = 1.2 W/m K, B = 20352 W/m (38 GPa) and A = 21.8 W/m K, B = 20754 W/m (70 GPa).

5. Discussion

Within the empirical approach used here, a simple, inversely proportional to temperature, decreasein thermal conductivity fits the data. Figure 4 shows that the temperature dependence of thermalconductivity of hcp iron at 70 GPa is not much different from that of the bcc phase at ambientpressure. This results mainly from the dramatic decrease in the conductivity of iron across thebcc-to-hcp phase transformation [16–19]. At the upper pressure–temperature conditions of thisstudy, pressure increases the thermal conductivity at the rate of about 0.6–0.7 Wm−1 K−1 GPa−1.

The effects of uncertainties in sample thickness and position, absorbed laser power andlaser beam diameter have been assessed by the method of variation of these parameters in themodel. Treating these as independent random errors (±4 W/m K: 0.2 μm in sample thickness;±3 W/m K: sample misplaced by 0.5 μm from the central position, ±2 W/m K: 5% in ALP;±5 W/m K: 0.6 μm in LBR), the overall uncertainty in the thermal conductivity at high pressureand temperature is estimated to be around ±7 W/m K at 70 GPa and 1800 K (Figure 4). Thermalconductivity of argon and the temperature boundary condition do not contribute to the iron con-ductivity error as they do not affect the temperature gradient. However, they affect the amount ofabsorbed laser power needed to bring Fe to a certain temperature, which in turn affect the soughtconductivity of the sample. Further possible contribution to a systematic bias includes the effectof temperature-induced change in the laser absorption. At high temperatures, reflectance of metalsusually decreases because the apparent electron–lattice collision time shortens [41]. Correctingfor this effect would increase the values of conductivity.

The interpretation of the acquired values of thermal conductivities is texture-dependent. Somedegree of a preferred orientation of crystal grains in the iron foil at high pressure is likely.As shown,for example, in [42–44], hexagonally closed packed iron subjected to uniaxial stress in a diamondanvil cell displays strong preferred orientation, with the c-axis parallel to the axis of the cell.The degree of alignment depends on a number of factors, including sample deformation historyand elastic strength of pressure medium. Errandonea et al. [36] applied laser heating on iron atconditions close to this study (same pressure medium, similar pressure and temperature, 49.6 GPa



and 2250 K, respectively) and show a rather intense, mutually comparable (100), (002) and (101)lattice reflections of hcp iron in the axial diffraction pattern [36, Figure 4]. This signifies thatneither of the corresponding basal-, prismatic- and pyramidal-slip-related preferential alignmentshave progressed to a high degree. Argon, being an elastically softer pressure medium, appearsto act as a weaker promoter to the preferred orientation as compared, for example, to MgO.For our study, this observation may indicate that the assumption of an isotropic aggregate isnot strongly violated and the obtained conductivity could be interpreted as an effective (average)thermal conductivity of polycrystalline hcp-iron, representing a value between the upper and lowerlimits on the average conductivity, given by kav = (kc + 2ka)/3 and 1/kav = (1/kc + 2/ka)/3,respectively [45]. kc and ka are thermal conductivities along the c-axis and in the basal plane,respectively. Single crystal studies or textural analysis using X-rays [42,43] can be used to addressthis issue in future experiments.

Acknowledgements

The Swedish Research Council (Vetenskapsrådet) is gratefully acknowledged for the finan-cial support via grant 621-2005-4857. A.F.G. and V.V.S. acknowledge support from NSF EAR0711358, Carnegie Institution of Washington, DOE/BES and DOE/NNSA (CDAC). We thankN. Subramanian for help with the experiment.

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Thermal conductivity of hcp iron at high pressure and temperature

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