Lattice Thermal Conductivity of Lower Mantle Minerals and Heat Flux from Earth’S Core

Lattice thermal conductivity of lower mantle minerals and heat flux from Earth’s core

Geeth M. Manthilake1, Nico de Koker, Dan J. Frost, and Catherine A. McCammon

Bayerisches Geoinstitut, Universität Bayreuth, D-95440 Bayreuth, Germany

Edited by Mark H. Thiemens, University of California, La Jolla, CA, and approved September 9, 2011 (received for review June 30, 2011)

The amount of heat flowing from Earth’s core critically determines extrapolation to the conditions of the lowermost mantle (11, 18). the thermo-chemical evolution of both the core and the lower Recent first-principles computations of lattice k in MgO periclase mantle. Consisting primarily of a polycrystalline aggregate of has shown that Debye theory (Eq. 2, see below) gives an excellent silicate perovskite and ferropericlase, the thermal boundary layer representation of the density dependence of k in a simple oxide at the very base of Earth’s lower mantle regulates the heat flow (19–21), while a recent set of experimental measurements (22) from the core, so that the thermal conductivity (k) of these mineral indicates that the density dependence of k in more complex phases controls the amount of heat entering the lowermost solids can be represented by removing the assumption that the mantle. Here we report measurements of the lattice thermal con- phonon group velocities are proportional to the Debye tempera- 3 ductivity of pure, Al-, and Fe-bearing MgSiO3 perovskite at 26 GPa ture (13; Eq. , see below). up to 1,073 K, and of ferropericlase containing 0, 5, and 20% Fe, at In this study we seek to constrain the lattice thermal conduc- 8 and 14 GPa up to 1,273 K. We find the incorporation of these tivity of a realistic lower mantle mineral assemblage. We per- elements in silicate perovskite and ferropericlase to result in a formed high pressure and temperature measurements of k for the ∼50% decrease of lattice thermal conductivity relative to the end following systems: Pure MgSiO3 (Pv), MgSiO3 þ 3 mol% FeSiO3 þ 2 % ð Þ member compositions. A model of thermal conductivity con- (PvFe03), and MgSiO3 mol Al2O3 PvAl02 at 26 GPa up to þ 5 % ð Þ strained from our results indicates that a peridotitic mantle would 1,073 K; pure MgO (Pe), MgO mol FeO PeFe05 , and þ 20 % ð Þ have k¼9.11.2W∕m K at the top of the thermal boundary layer MgO mol FeO PeFe20 at 8 and 14 GPa, up to 1,273 K. and k¼8.41.2W∕m K at its base. These values translate into a We then apply these results to reassess the value of k for the heat flux of 11.01.4 terawatts (TW) from Earth’s core, a range of lowermost mantle and propose a refined value for core-mantle values consistent with a variety of geophysical estimates. boundary heat flux. Results D” ∣ core-mantle boundary ∣ high pressure We find that addition of even a small amount of minor elements to the mineral composition strongly reduces the lattice conduc- he 46 3 terawatts (TW) of heat flowing from the solid Earth tivity (Fig. 1). This reduction is of similar magnitude for larger (1, 2) is one of the prime geophysical constraints on the dy- T concentrations of Fe in ferropericlase, as well as for substitution namic state of the planet. As an integrated quantity, this global of Al rather than Fe in perovskite. The measured k for MgSiO3 value does not distinguish individual contributions from the man- perovskite is notably higher than the values Osako and Ito mea- tle and the core. Knowledge of the proportion of surface heat sured at ambient pressure and supercooled conditions (14), while flow originating from the core is of enormous importance to our – our measurements for MgO periclase are somewhat lower than understanding of mantle dynamics (3 5), the generation of the the previous experimental determination of Katsura (12). magnetic field by flow in the outer core (6, 7), and the distribution To construct a set of models for the pressure-temperature of radiogenic isotopes between mantle and core (8). Current es- dependence of lattice conductivity in lower mantle minerals, we timates place core-mantle boundary (CMB) heat flux at around SI Appendix S6–S12 – combine our thermodynamic models ( , Eqs. ) 10 30% of the surface value (4), an uncertainty which signifi- with the relation cantly hampers constraints on the thermal evolution of the core. This large spread of values further implies notable uncertainty in a ρ g ¼ Tref [1] the extent of heat generated in the lower mantle, because heat k kref ρ ; not originating from the core must necessarily come from the T ref overlying mantle. The principle factor controlling the heat flux from the core is in which a allows for deviation from the theoretically expected 1∕T relation (23, 24), and g ¼ð∂ ln k∕∂ ln ρÞ , with the ability of the thermal boundary layer overlying the core-man- T GEOPHYSICS tle boundary to conduct heat. Thermal conductivity of the low- ¼ 3γ þ 2 − 1∕3 [2] ermost mantle is therefore a key physical property to constrain g q ; if we are to narrow the bounds of uncertainty on core-mantle boundary heat flux. The polycrystalline nature of the lower man- for the oxide phases (Debye theory; 19, 20), and tle aggregate, combined with the presence of minor elements, g ¼ γ þ 2q þ K0 − 4∕3; [3] most notably Fe, will likely suppress a strong radiative contribu- T tion to thermal conduction in the lowermost mantle (9, 10), mak-

ing heat transport via lattice vibrations the dominant factor to Author contributions: G.M.M., N.d.K., and D.J.F. designed research; G.M.M. performed consider. research; G.M.M., N.d.K., and C.A.M. analyzed data; and G.M.M., N.d.K., D.J.F., and Existing estimates of lattice thermal conductivity for the low- C.A.M. wrote the paper. ermost mantle range between k ¼ 4–16 W∕m K, based on extra- The authors declare no conflict of interest. polation of measurements for pure end member phases (MgO, This article is a PNAS Direct Submission. – MgSiO3, SiO2) at upper mantle pressures (11 16). While minor Freely available online through the PNAS open access option. elements have been shown to have a large effect on k in upper 1To whom correspondence should be addressed. E-mail: Geeth.Manthilake@ mantle minerals (17), their effect at high pressure in lower mantle uni-bayreuth.de. minerals has not been rigorously considered. A nontrivial source This article contains supporting information online at www.pnas.org/lookup/suppl/ of uncertainty in k of the lowermost mantle is the means of doi:10.1073/pnas.1110594108/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1110594108 PNAS ∣ November 1, 2011 ∣ vol. 108 ∣ no. 44 ∣ 17901–17904 Downloaded by guest on September 29, 2021 60 18 26 GPa Katsura 1997 0.1 GPa Kanamori 1968 Touloukian 1972 12 Pv 40 k (W/m K) 6 PvFe03 k (W/m K) 20

PvAl02 1/T 0 0 GPa extrapolation 0 45 14 GPa 500 750 1000 1250 Pe T (Kelvin)

30 Fig. 2. Thermal conductivity of MgO periclase at ambient pressure, ex- trapolated from our high pressure measurements using Eq. 2 and our thermodynamic model. Values compare well at high temperatures with the

k (W/m K) Pe 15 Fe05 measurements of Touloukian, et al. (25) for polycrystalline MgO, though it falls somewhat below those of Kanamori, et al. (26) and Katsura (12). The model underestimates the lowest T values, as does a simple 1∕T relation. PeFe20 0 compositions considered is consistent with the use of Eq. 2 for the oxide systems. Ambient pressure thermal conductivity for 45 8 GPa MgO determined from our model agrees well with experiment, although it fails to capture the very steep increase in k seen at Pe room temperature (Fig. 2). The agreement is very encouraging 30 for the larger extrapolation to high P and T at which we perform below. k (W/m K) 15 PeFe05 Discussion Representing the lower mantle as a randomly oriented polycrys- Pe 0 Fe20 talline aggregate of 20% ferropericlase and 80% perovskite, and 500 750 1000 1250 assuming Fe to partition preferentially into ferropericlase (27), Kelvin we construct a kðP;TÞ model for the lowermost mantle by combining our models for Pe 20 and Pv 03 given in Table 1. Fig. 3 Fig. 1. Measured lattice thermal conductivities for the respective mineral Fe Fe compositions considered (see SI Appendix for more detail). We use the values shows the subsequent Hashin-Shtrikman averaged mantle k determined from the amplitude ratio (Θ), in order to minimize contributions from direct radiative transfer. Solid lines indicate the fits of Eq. 1, using the 24 parameters reported in Table 1. 0.8 Pv + 0.2 Pe

0 for the perovskite phases (modified Debye theory; 13, 22). KT is 18 the pressure derivative of the isothermal bulk modulus, γ the 1000 ¼ −ð∂ γ∕∂ ρÞ Grüneisen parameter, and q ln ln T . This approach 12 4000 further allows us to correct for the fact that our measurements k (W/m K) are at constant pressure while theoretical descriptions of kðTÞ are strictly valid at constant volume. With ρ and g constrained 6000 6 from the respective thermodynamic models, we determine the 1 optimal values of kref and a in Eq. for each set of measurements (Table 1). 12 0.8 Pv We find that the T-dependence for MgO is somewhat more Fe03 + 0.2 Pe 1000 gradual than 1∕T, while k flattens out for the more complex struc- Fe20 9 tured MgSiO3 perovskite as well as for samples containing Fe or Al. This behavior is consistent with larger values to the lower 4000 limits of the phonon mean free path lengths in the more complex 6 k (W/m K) structured and/or disordered crystal lattices, in which case it is 6000 theoretically expected that a ¼ 1∕2 (23). We also note that the 3 density dependence of the kref values in all three ferropericlase

660 1200 1800 2400 2895 kð Þ Table 1. Parameters for P,T in the respective compositions, depth (km) modeled through Eq. 1 Fig. 3. Thermal conductivity models computed from our results using Pe PeFe05 PeFe20 Pv PvFe03 PvAl02 Hashin-Shtrikman averaging (28) for aggregates of 20% Pe þ 80% Pv (top) ρ 3∕ and 20% Pe 20 þ 80% Pv 03 (bottom). The mantle geotherm is constructed ref (cm mol) 3.71 3.84 4.21 4.45 4.49 4.45 Fe Fe k ∕ by combining a 1,600 K pyrolite adiabat (29) with a superadiabatic tempera- ref (W m K) 21.0 8.94 6.02 12.8 4.46 3.69 a 0.76 0.24 0.24 0.43 0.20 0.22 ture increase in the bottom 200 km through the thermal boundary layer to reach the range of temperature estimates for the core-mantle boundary T ¼ 700 P ¼ 8 In all cases ref K, with ref GPa for periclase and 26 GPa for (30, 31). Temperatures in the thermal boundary layer thus increase from perovskite. þ200 2;600 200 K at its top to 4;100−800 K at the CMB.

17902 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1110594108 Manthilake et al. Downloaded by guest on September 29, 2021 model thus obtained, computed along a model geotherm (29). Expected value Our subsequent prediction of the lattice thermal conductivity in the lowermost mantle is 9.1 1.2 W∕m K at the top of the 24 thermal boundary layer, and 8.4 1.2 W∕m K at its base, notably lower than the corresponding values of 18.9 1.6 and 15.4 Estimates 1.4 W∕m K we predict for a pure MgO-MgSiO3 aggregate. from plumes In the light of the relatively small differences seen between 16 150 km Fe- and Al-bearing perovskite, and between different Fe-concentrations in ferropericlase, it is unlikely that the small change in 200 km ionic radius resulting from a transition in the spin state of Fe will affect the lattice conductivity significantly. Indeed, at mantle tem- 8 300 km peratures the transition will be very gradual and to an intermedi- CMB heat flow (TW) ate rather than low spin state (32, 33). In contrast to metals, dielectric materials do not exhibit a simple direct relation between thermal conductivity and electrical resistivity (34), so that 0 room temperature changes in the latter associated with the spin 500 1000 1500 2000 transition (35, 36) do not imply a corresponding change in k in ∆T (Kelvin) the mantle. To account for possible variations in our models for k due to these factors, we increase the uncertainty on the kref value Fig. 4. CMB heat flux as a function of the temperature contrast across the from 5% to 10%. D’’, for different thermal boundary layer thicknesses, computed from our The estimated heat flux computed from our thermal conduc- model values for the lower-most mantle by iteratively adjusting the thermal 20% þ 80% tivity model is 11.0 1.4 TW (Fig. 4), which is within the range of profile for steady state. Solid envelopes represent a PeFe20 PvFe03 assemblage, accounting for uncertainty on the k values. Dashed lines indicate values based on geophysical arguments (2, 4), and also consistent the positions of these envelopes if the silicate component of the thermal with the minimum bounds of heat flux (3-4 TW) required by geo- boundary layer was in the postperovskite (PPv) phase for T < 3;300 K, assum- k ¼ 2 × k dynamic models to sustain the geomagnetic field (2). Due to the ing PPv Pv (37). Independent core-mantle boundary heat flux values steep thermal gradient in the basal thermal boundary layer, post- based on mantle plume buoyancy flux estimates are shown for comparison perovskite will occur as lenses surrounded by perovskite (29, 38). (see ref. 2). Using a simplified scaling based on measurements in analogue materials (37), we estimate that a layer of postperovskite would tivity, we use thermal diffusivity values determined from the increase CMB heat flow by about 40%. amplitude ratio of temperature signals between the two thermo- Our results indicate that lateral compositional heterogeneities SI Appendix S2 of Fe and Al in major lower mantle mineral assemblages will couples ( , Eq. ) to minimize spurious effects due to direct radiative transfer between the thermocouples. In Fe- not result in large variations of thermal conductivity on its own. þ3 However, to the extent that lateral variations in temperature and containing samples, Fe concentrations are measured using chemistry bring about changes in mineral abundances, either by Mössbauer spectroscopy (refer to SI Appendix, section S1). To changing the perovskite-ferropericlase ratio, or by substituting represent the thermodynamic properties of the various phases as perovskite for postperovskite, lateral variations in heat flux from a function of pressure and temperature, we use a self-consistent the core on the order of 50% can be expected. Mie-Grüneisen formulation of the Helmholtz free energy (40, 41; SI Appendix Experimental Methods refer to , section S3). Thermal conductivity measurements were performed on pre- ACKNOWLEDGMENTS. We thank S. Linhardt for technical support and two synthesized polycrystalline samples using the Ångström method anonymous reviewers for their comments and suggestions. This project (e.g., ref. 39) in a 5,000 tonne multianvil apparatus at the Bayer- was financially supported by the Bayerisches Geoinstitut Visitor’sProgram isches Geoinstitut (refer to SI Appendix, section S2). To accu- for G.M.M., and the European Union (MRTN-CT-2006-035957) and German rately represent the temperature dependence of lattice conduc- Science Foundation (DFG) (KO 3958/1-1) to N.d.K and D.J.F.

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