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Lattice thermal conductivity of MgO at conditions of Earth’s interior

Edited by Russell J. Hemley, Carnegie Institution of Washington, Washington, DC, and approved January 11, 2010 (received for review June 26, 2009)
Abstract
Thermal conductivity of the Earth’s lower mantle greatly impacts the mantle convection style and affects the heat conduction from the core to the mantle. Direct laboratory measurement of thermal conductivity of mantle minerals remains a technical challenge at the pressuretemperature (PT) conditions relevant to the lower mantle, and previously estimated values are extrapolated from low PT data based on simple empirical thermal transport models. By using a numerical technique that combines firstprinciples electronic structure theory and Peierls–Boltzmann transport theory, we predict the lattice thermal conductivity of MgO, previously used to estimate the thermal conductivity in the Earth, at conditions from ambient to the coremantle boundary (CMB). We show that our firstprinciples technique provides a realistic model for the PT dependence of lattice thermal conductivity of MgO at conditions from ambient to the CMB, and we propose thermal conductivity profiles of MgO in the lower mantle based on geotherm models. The calculated conductivity increases from 15 –20 W/Km at the 670 km seismic discontinuity to 40 –50 W/Km at the CMB. This large depth variation in calculated thermal conductivity should be included in models of mantle convection, which has been traditionally studied based on the assumption of constant conductivity.
Thermal conductivity (κ) is one of the most important mineral properties in determining the heat budget of the Earth. Heat in the Earth’s interior is transferred by convection in the mantle and core and regulated by conduction at thermal boundary layers. As defined by Fourier’s law of heat conduction J _{Q} = κ·∇T, determines the conducting heat flow density (J _{Q}) in the presence of a temperature gradient ∇T. κ also appears in the Rayleigh number, which measures the convective vigor of a system. Thus, the thermal conductivity of the lower mantle affects the structure, thickness, and dynamics of the CMB (1, 2), the style and structure of mantle convection (3 –5), and the amount of heat conducted from the core to the mantle (6) that in turn influences the generation of the Earth’s magnetic field (7).
Despite its importance, thermal conductivity remains as one of the least constrained physical properties of minerals, especially at lowermantle pressures (P) and temperatures (T) (23–135 GPa (8) and approximately 1,900–4,000 K (9 –12). Experimental data at deep mantle conditions are scarce due to the technical difficulty of measuring thermal conductivity at these extremes. Thermal conductivity of lowermantle minerals is often estimated either by extrapolating data from lower PT conditions and/or employing theoretical models with parameters fitted with lower PT data (1, 13). However, direct extrapolation to deep mantle conditions can be unreliable beyond the PT range of the measurements, and empirical models are often based on untested assumptions. For example, the sound velocities are used to approximate phonon velocities, and the pressure dependence of phonon lifetime is assumed to be given by equilibrium thermodynamic properties, such as lattice thermal expansion and/or Grüneisen parameters.
MgO, the endmember of the secondmost abundant mineral in the lower mantle, has historically served as a model system for evaluating the thermal conductivity of the deep mantle (14). Its thermal conductivity is also an order of magnitude larger than that of Mg endmember of the most abundant mantle mineral—silicate perovskite (15). Hence, studying its thermal conductivity provides a useful approach to constraining the thermal conductivity of the lower mantle. Though efforts have been devoted to measuring the thermal conductivity of MgO (κ _{MgO}) at both high pressures and temperatures (16 –24), most of those measurements have been limited to below 7 GPa. First, nonempirical calculations of lattice thermal conductivity (25) of MgO at high pressure were performed based on molecular dynamics simulations and Green–Kubo theory (26). An order of magnitude underestimation of κ _{MgO} in that study is maybe due to the ionic potential model adopted in the simulation that does not adequately account for lattice anharmonicity.
In this paper, we report a firstprinciples study of the lattice thermal conductivity of MgO in the pressure and temperature ranges of 0–150 GPa and 300–4,000 K, respectively. Our method combines firstprinciples lattice vibration calculations, quantum phonon scattering theory, and Peierls–Boltzmann transport equation within the singlemode excitation approximation. We first investigate the microscopic heat transport characteristics of all the acoustic and optic phonons and examine the density and temperature dependence of κ _{MgO}. We then determine κ _{MgO} at lowermantle pressures and temperatures with the thermal equation of state predicted from our firstprinciples calculations. Finally, we propose a depth profile of κ _{MgO} in the lower mantle based on previous estimates of cold and hot geotherms.
Results
To predict the lattice thermal conductivity of MgO crystals at conditions from ambient to those of the Earth’s CMB, we first computed the harmonic force constant matrices of lattice vibration and the thirdorder lattice anharmonicity tensors at seven densities, ranging from 3.35 to 5.15 g/cm^{3} (27). At each density, we explicitly evaluated phonon scattering rates of all the irreducible phonon modes at nine temperature points from 300–4,000 K. Finally, we derived microscopic phonon mode conductivity using the calculated heat capacity c _{V}, group velocity v _{g}, and scattering rate/lifetime τ of each phonon mode on a 16 × 16 × 16 point Brillouin zone grids for all the 63 densitytemperature configurations.
Fig. 1 shows our calculated average mode conductivity κ _{mode} for phonons within different phonon frequency (ω) ranges at ρ = 3.70 g/cm^{3} and T = 300 K. While all phonons transfer heat, the lower frequency acoustic modes are clearly much more efficient in heat conduction than the higher frequency optic modes. This is consistent with the fact that the acoustic modes, especially those near the Brillouin zone center, have much larger group velocities than optical modes. For modes near zone center, we find that the ratio between acoustic and optic group velocities can be as large as 12. Furthermore, our calculations reveal that the average phonon lifetime of acoustic phonons is approximately a factor of threelonger than those of optic phonons. MgO crystalline with only two atoms per unit cell, contain equal numbers of acoustic branches and optic branches. Yet, the overall contribution of acoustic phonons accounts for nearly 85% of the total thermal conductivity at 300 K (Fig. 1 Inset). This ratio deceases slightly with increasing temperature as more optic phonons are thermally excited, and it approaches 80% above the Debye temperature. The high temperature limit of κ _{acoustic}/κ _{total} increases mildly under large compression (for example, about 87% at ρ = 5.15 g/cm^{3}).
Fig. 2 presents our theoretical prediction of the temperature dependence of lattice thermal conductivity of MgO at selected densities. The inverse of the calculated lattice thermal conductivity scales consistently as a linear function (A + BT) over the entire calculated temperature range (from 300–4000 K) when density is held constant. This predicted linear temperature dependence of 1/κ can’t be extended to low temperature as our calculations do not take into account the scattering mechanisms (such as point defects, dislocations, or grain boundary scattering) that are much less important at temperature conditions relevant to the Earth’s hot interior. For instance, Gibert et a.l (28) has studied experimentally the effect of grain boundary scattering on the thermal diffusivity. By comparing the measured thermal diffusivity of both single crystal and polycrystalline olivine (with grain sizes varying from 0.01– 2 mm), they concluded that the grain boundary scattering has a negligible effect on thermal diffusivity at ambient conditions. To extend this finding to elevated temperatures and pressures (larger density), we adopt a simple empirical model of Callaway (29) and find that grain boundaries of polycrystalline MgO start to play a role at ambient condition only if the characteristic length l is in the order of μm or smaller. Taking l = 1 μm for example, the estimated reduction in κ _{MgO} due to grain boundary scattering is 13% and 26% at room temperature for density of 3.70 g/cm^{3} and 5.15 g/cm^{3}, resp., and they are significantly lower at 2,000 K and become only 2% and 2.6%, resp. A similar trend is also found in our modeling of defect scattering. Hence, we conclude that it is valid to neglect the effect of defects and boundary scattering in studying the thermal conductivity of Earth’s hot interior, and phonon–phonon interaction is the major scattering process that hinders the heat transport at high temperatures.
In this study, the temperatureindependent A term mainly represents the contribution associated with the mass disorder (i.e., isotope) induced phonon scattering (30). As shown in the Inset of Fig. 2, the neglection of isotopeinduced phonon scattering leads to a noticeable overestimation of thermal conductivity around ambient temperature. At 300 K, this can lead to an overestimation as large as 46% for ρ = 3.70 g/cm^{3}. However, the contribution of the isotope effect diminishes at high temperature—falling to 4% at 4000 K at the same density. This reveals that it is necessary to include the isotope effect when one compares the κ(T)/κ(T = 300 K) ratio between measurements and calculations. The temperaturedependent BT term can be primarily attributed to the anharmonicityinduced 3phonon scattering mechanism that dominates at high temperature, and lattice thermal conductivity approaches to the wellknown relation at the highT limit (31). As both A and B terms are nonnegatively defined, we fitted the log functions of A and B terms of the seven studied densities with second order polynomial functions of 1/ρ: [1]
The fitting parameters are listed in Table 1.
Discussion
We first compare our results with available experiments, which are performed under either ambient pressure or room temperature conditions. Based on the proposed densitytemperature model (Eq. 1 and Table 1), κ _{MgO} at isobaric conditions can be readily obtained with the thermal equation of state ρ(T,P). Fig. 3 shows the calculated κ _{MgO}(T,P) at ambient pressure based on our calculated (Solid Line) and measured (Dashed Line) equation of state (32), in comparison with four sets of experimental data (16 –18, 23, 33) in symbols. Our predicted κ _{MgO} at 300 K is about 66 W/Km, which is in agreement with the reported experimental data scattering from 36 to 70 W/Km. A detailed review of experimental data for κ _{MgO}, including systematic errors in various techniques and qualities of measured samples, can be found in ref. 13. The adopted firstprinciples local density approximation (LDA) theory is known to overestimate density, and we find adopting the experimental equation of state (32) lowers our calculated κ _{MgO} by about 10%. Nevertheless, our calculations in general slightly overestimate κ, and this is consistent with the fact that all experimental samples are of small sizes and contain intrinsic imperfections, whereas calculations assume perfect crystallinality. While decreasing with temperature, lattice thermal conductivity increases under compression. The Inset of the Fig. 3 shows the calculated κ(P)/κ(P = 0) ratio at 300 K (Solid Line) that falls in the scattered experimental high pressure measurements (19 –22). The LDAcalculated isothermal pressure coefficient (d ln(κ _{MgO})/dP) is about 3.90% GPa^{1} at T = 300 K near ambient pressure, which is in good agreement with the measured value of 4% GPa^{1} (20, 33) and 4.9% GPa^{1} (21).
Due to the lack of direct measurements at deep mantle conditions, various empirical models (1, 33, 34) have been proposed to describe the relative change in thermal conductivity upon compression to extrapolate experimental data to the relevant pressure conditions of the Earth’s interior. Most of these models are based on the simple empirical expression of proposed by Dugdale and McDonald (35), where a is the interatomic distance, v is the averaged phonon velocity, B _{T} is the isothermal bulk modulus, and γ is the Grüneisen parameter. Additional approximations are often adopted to describe the density dependency of these relevant thermal properties. For example, Poirier (34) derived d ln(κ)/dP = (2γ + 5/3)/B _{T}. Hofmeister (36) proposed a damped harmonic oscillator (DHO) model that originates directly from the microscopic phonon transport theory (Eq. 2 ). However, because of the insufficient experimental data on individual phonon modes (such as their group velocities and lifetimes), further simplifications are inevitable for studying real minerals. The simplified DHO model predicts the upper limit of the d ln(κ)/dP coefficient as (4γ + 1/3)/B _{T} (23).
To compare our firstprinciples results with empirical models, we first calculated d ln(κ _{MgO})/dP based on the above two empirical models by using our LDAcalculated γ and B _{T} (Fig. 4 A and B), and then derived their κ(P)/κ _{o} ratio by integration (Fig. 4 C and D). The simplified DHOmodelpredicted d ln(κ _{MgO})/dP at 300 K and ambient pressure is in agreement with our firstprinciples calculation. However, our calculated d ln(κ _{MgO})/dP decays faster under compression than that predicted by the simplified DHO model, and incidentally approaches the prediction of the Poirier model at higher pressures (Fig. 4 A). Consequently, the κ(P)/κ _{o} ratio predicted by the Poirier model is closer to that calculated with our firstprinciples method, whereas the predictions by the simplified DHO model are more than 20% larger at pressures higher than 80 GPa (Fig. 4 C). The incidental agreement between the Poirier model and our firstprinciples calculations does not hold for all temperatures. For example, at 3,000 K our calculated d ln(κ _{MgO})/dP is significantly higher than that predicted by empirical models at ambient pressure (Fig. 4 B), and the difference in the prediction of κ(P)/κ _{o} between empirical models and current calculations becomes more significant at higher temperatures and pressures (Fig. 4 D). Our results suggest that such empirical models are inadequate even for structurally simple minerals like MgO, and they are more likely to yield larger uncertainties for complex minerals such as silicate perovskite. Furthermore, we find that temperature has a strong effect on the pressure derivative d ln(κ _{MgO})/dP. It increases rapidly with the increase of temperature at pressures lower than 50 GPa, and becomes almost temperature independent above 80 GPa. At CMB conditions (P = 135 GPa and T = 3,000 K), our firstprinciples calculations predict κ _{MgO} and d ln(κ _{MgO})/dP to be around 43 W/Km and 0.36% GPa^{1}, resp.
The geotherm of the Earth’s lower mantle depends on the thermal conductivity of the lower mantle. In the current study, we adopt a hot and cold geotherm based on experimental constraints (37) to give a direct estimation of κ _{MgO} at the lowermantle conditions (Fig. 5). The cold geotherm corresponds to wholemantle convection, whereas the hot geotherm to partiallylayered convection. Large differences between the hot and cold geotherms are due to the uncertainties in the temperature structure as well as the lateral inhomogeneity. Nevertheless, these two geotherms provide a reasonable temperature bound for the lower mantle. At the same depth, κ _{MgO} can be 5–10 W/Km lower along the hot geotherm than along the cold geotherm. However, it holds true that κ _{MgO} changes greatly with depth, from 15–20 W/Km at the 670 + km transition zone to 40–50 W/Km on the mantle side of the CMB. The calculated depth dependence of κ _{MgO} along the hot and cold geotherms above the CMB are and resp., where z (in the unit of km) is the depth in the lower mantle relative to the 670 km seismic discontinuity. An 800 Ktemperature difference across the CMB layer can reduce κ _{MgO} from 43 W/Km to 34 W/Km, indicating depth dependence of thermal conductivity within the layer should not be neglected when one considers the heat conduction at the boundary layer; e.g., it affects the thickness of the thermal boundary layer δD through for a heat flux from the core of q.
Our current firstprinciples calculations represent an improvement in modeling the lattice thermal conductivity of this model oxide material. Several key issues need to be resolved to realistically constrain the thermal conductivity of the lower mantle, which is mainly controlled by the compositeaveraged thermal conductivity of (Mg,Fe)O and (Mg,Fe)SiO_{3}. First, the effect of iron in mineral solid solutions is important yet poorly understood. The iron content not only modifies the density, interatomic forces, and lattice anharmonicity but also adds microscopic disorder. Our firstprinciples technique can be expanded to include this effect, but this is beyond the scope of this study. A preliminary analysis reveals that the key effect is a significant reduction of phonon group velocities with increasing iron contents. Effects associated with the FeMg mass disorder, although significant at 300 K, diminish with increasing temperature. A quantitative evaluation of iron effects requires more comprehensive theoretical treatment.
Second, accurate modeling of compositeaveraged thermal conductivity requires knowledge of the thermal conductivity of both the individual constituents and the structures of the composite. The perovskite endmember (MgSiO_{3}) is known to have a much lower thermal conductivity than the magnesiowüstite endmember (MgO). At ambient conditions, the thermal conductivity of MgSiO_{3} perovskite is 5.1 W/Km (38), which is less than 10% of that of MgO. If MgSiO_{3} perovskite behaves similarly to MgO upon heating and compression, we estimate that the upper bound of the thermal conductivity at the top of the CMB will be around 11 ∼ 12 W/Km, assuming that MgO and MgSiO_{3} are layered sidebyside along the direction of heat flow (39). More realistic composite structures should be included in future studies, along with better constraints on the thermal conductivity of silicate perovskite and the effects of iron content in solid solution at lowermantle conditions.
Finally, in the Earth’s hot interior, an additional effective thermal conductivity due to the intergrain thermal radiation can be considered. Radiative thermal conductivity increases rapidly with increasing temperature, and becomes significant at high temperatures (40, 41). On the other hand, the effect of radiative heat transfer is diminished with the reduction of grain size; it is also controlled by the iron concentration of the minerals (42). Optical absorption measurements at high pressure have been used to infer the radiative thermal conductivity of lowermantle minerals (43 –46). Large discrepancies were found among the estimates of radiative thermal conductivity from these measurements that might be due to the differences in sample grain size, iron concentration, or different experimental setups. Further experimental investigation is needed to constrain the average radiative thermal conductivity at conditions near the CMB, including the contribution due to the highspin/lowspin transition.
Method
Many microscopic processes contribute to the overall heat conduction in a solid. For insulating mantle minerals, heat conducts mainly via lattice vibrations (31). The Peierls–Boltzmann transport equation expresses the lattice thermal conductivity as: [2]where V _{o} is the volume of the unit cell, N _{a} is the number of atoms in the unit cell, and the integration is over the first Brillouin zone in the reciprocal space. Each phonon mode in the space is labeled with its crystal momentum and polarization index i (from 1 to 3N _{a}), and c _{V}, v _{g}, τ, and d = v _{g} τ are its heat capacity, phonon group velocity, phonon lifetime, and phonon mean free path, resp.
The firstprinciples local density approximation techniques are now routinely adopted to accurately predict harmonic lattice phonon spectra that can be readily used to derive phonon heat capacity and group velocity in Eq. 2 . To evaluate phonon lifetime τ, we consider two types of perturbation to the nearly independent phonon model—the thirdorder lattice anharmonicity and the isotope mass disorder. The phonon scattering rate ( ) that is the inverse of , can be calculated from Fermi’s Golden rule: , where is the transition rate from the initial state Φ_{i} to the final state Φ_{f} of a manyphonon system under perturbation δH. In the present study, instead of directly solving for the phonon distributions at the presence of temperature gradients, we further adopt the singlemode excitation approximation to evaluate the relaxation rate of a phonon mode when only this phonon mode is perturbed out of its thermal equilibrium (47). Each type of phonon scattering is treated individually, and the overall transition rate is approximated as the sum of all the transition rates from different scattering mechanisms [Matthiessen’s rule (48)]. More details of the phonon lifetime calculations are provided in SI Text .
Conclusions
The firstprinciplesderived model has been developed to advance our understanding of lattice thermal conductivity of minerals at deep mantle conditions. By combining the microscopic transport theory and firstprinciples lattice dynamics calculations, we have predicted the value for MgO over a broad range of PT conditions of the lower mantle without empirical extrapolation. The good agreement with low PT measurements suggests that the firstprinciplesbased implementation of Peierls–Boltzmann transport theory within singlemode excitation approximation can be used to predict the thermal conductivity of insulating mantle minerals at high PT conditions. Our study indicates that the calculated values for MgO vary significantly with depth in the lower mantle, increasing by a factor of 2–3 from the 670 km discontinuity (15–20 W/Km) to the mantle side of the CMB (40–50 W/Km). This finding starkly contrasts with the assumption of constant thermal conductivity that is widely adopted in many geodynamics simulation studies of the lower mantle. Our firstprinciples technique could be readily adapted to study lattice thermal conductivity of ironbearing lowermantle minerals. In light of further improved experimental data of lattice thermal conductivity at lower pressure (49) and radiative thermal conductivity at the condition of the CMB, our study should serve as a useful stepping stone to realistically constrain the total thermal conductivity of the lower mantle.
Acknowledgments
We thank A. Kavner, A.M. Hofmeister, D.A. Drabold, and G. Poirier for helpful discussions and R.J. Hemley and anonymous reviewers for comments and suggestions. This work is supported by National Science Foundation Grants EAR0757847 and EAR0510914.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: xtang{at}igpp.ucla.edu.

Author contributions: X.T. and J.D. designed research, performed research tools, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0907194107/DCSupplemental.
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