Thermal conductivity of CaGeO3 perovskite at high pressure



[1] The pressure dependence of thermal conductivity (equation image) in Earth's lower mantle is poorly understood, resulting in very uncertain equation image values in the D″. Using orthorhombic CaGeO3 perovskite as an analogue for MgSiO3 perovskite, we measured equation image in a multi-anvil apparatus using the Ångström method at pressures of 8, 10, 14 and 18 GPa, and temperatures of 573–1073 K. We find the pressure dependence of thermal conductivity for orthorhombic perovskite shows a somewhat higher slope than that predicted by Debye theory, and account for this by relating phonon velocities to the bulk modulus, rather than to the Debye temperature. From this relation, we estimate the thermal conductivity of MgSiO3 perovskite at the top and bottom of the D″ to be 3.3 W/m K and 2.2 W/m K, corresponding to values of 6.4 W/m K and 4.5 W/m K for the mantle.

1. Introduction

[2] The amount of heat flowing across the core-mantle boundary (CMB) critically determines the thermal evolution of both the mantle and the core. By determining the fraction of the total heat budget not produced in the mantle, it sets key constraints for the required budget of heat producing elements in the mantle, and is likely the main source of heat for the formation of mantle plumes. It constrains the amount of energy required to drive the geodynamo, sets the age of the inner core, and therefore also the required budget of radiogenic isotopes in the core. By limiting the rate at which heat is conducted across the thermal boundary layer at the very base of the mantle, often associated with the seismic D″, the thermal conductivity (equation image) of the lower-most mantle is the key physical property in constraining geodynamical models of CMB heat flux from geophysical observations.

[3] Estimates for equation image in the lowermost mantle are very poorly constrained, ranging between 4–16 W/m K [Goncharov et al., 2009; Lay et al., 2008; Hofmeister, 1999; Brown and McQueen, 1986]. equation image for MgSiO3 perovskite is not well known (see below), while its pressure and compositional dependence has yet to be investigated. Extrapolation to the extreme pressure conditions at the base of the mantle (136 GPa) is therefore a very uncertain exercise. The importance of heat transport by radiation remains an additional source of debate. Though some studies have suggested that MgSiO3 perovskite will have a high radiative conductivity in the lower mantle [Keppler et al., 2008], this is unlikely to be the case for a polycrystalline aggregate of (Mg, Fe)O periclase and (Mg, Fe)SiO3 perovskite [Goncharov et al., 2008]. Another factor contributing to uncertainty is the post-perovskite phase, which is likely present in the colder parts of the D″ [e.g., Catalli et al., 2009; Tsuchiya et al., 2004]. Of course, even if its conductive properties differ strongly from that of perovskite, this will be of secondary importance to the heat flux from the core, since the mantle temperature directly adjacent to the core is likely outside the post-perovskite stability field.

[4] When considering the large degrees of compression relevant to Earth's mantle, the change of lattice equation image with compression is usually expressed relative to some reference value equation image at ρref and Tref through a power law relation

equation image

with different relations for the exponent g = ∂ ln equation image/∂ ln ρ, all of which are based on approximations to the expression for a kinetic gas of phonons

equation image

where v is the sound speed and τ the mean phonon lifetime (Table S1 of Text S1 in the auxiliary material gives a glossary of mathematical symbols used). The most frequently used relation, often referred to as Debye theory, follows by applying the Debye approximation to the theoretically predicted form of τ [de Koker, 2010; Ross et al., 1984; Roufosse and Klemens, 1973], to get

equation image

where γ is the Grüneisen parameter and q = −∂ ln γ/∂ ln ρ. If one further relates v to equation image rather than to the Debye frequency [Manga and Jeanloz, 1997], one can show that

equation image

If the density dependence of τ is neglected, g can also be represented as [Hofmeister, 2007]

equation image

[5] With only sparse high pressure data available with which to test how best to perform extrapolations, there is very little consensus on which level of approximation is sufficient. Using first-principles molecular dynamics calculations, de Koker [2010] showed that g(γ) (equation (3)) works extremely well for MgO periclase, finding that equation image is strongly dominated by acoustic phonons, indicating that the relation should perform reasonably well for more complex solids as well.

[6] Direct measurement of equation image in MgSiO3 perovskite is very challenging, and has only been performed in a supercooled sample at ambient pressures [Osako and Ito, 1991]. Obtaining experimental constraints on the pressure dependence of equation image in this mineral is therefore not currently possible. Analogue materials have proven invaluable in elucidating the trends to be expected in the physical properties of mantle phases. As lattice conductivity is primarily controlled by the nature of the crystal lattice [e.g., Slack, 1979], analogue materials should give a good indication of the pressure dependence of equation image in deep mantle phases. CaGeO3 perovskite is an ideal analogue phase for MgSiO3 perovskite, because they have the same space group (Pbnm), similar ratios among unit cell parameters, and similar radius ratios between the cations. CaGeO3 is stable above 6 GPa, making it an ideal analogue to study the pressure dependence of thermal conductivity in mantle perovskite.

2. Experimental Method

[7] Polycrystalline CaGeO3 perovskite was pre-synthesized by keeping CaGeO3 wollastonite powder at 8 GPa and 1373 K for 2 hours [Liu et al., 1991]. X-ray powder diffraction analysis of the samples before and after conductivity measurements confirmed the presence of single phase CaGeO3 perovskite with orthorhombic symmetry.

[8] To measure thermal diffusivity we used the Ångström method [e.g., Fujisawa et al., 1968], in which a sinusoidally varying temperature with angular frequency ω = 2πν is applied to the circumference of a cylindrical sample, and the phase lag (ϕ) and amplitude ratio (χ) in the temperature signal is measured at radii r1 and r2 in the sample (Figures S1 and S2 in Text S1 of the auxiliary material). Thermal diffusivity D is determined by solving

equation image
equation image
equation image

where ber(u) and bei(u) are the Kelvin functions [e.g., Zhang and Jin, 1996]. Thermal conductivity is then obtained as equation image = ρCPD, with a thermodynamic model (see below) used to compute ρ and CP at P and T, and to adjust r2r1 to account for compression.

[9] Measurements were performed using the 5000 tonne press at the Bayerisches Geoinstitut [Frost et al., 2004], with a 25/15 configuration for 8, 10 and 14 GPa measurements, and 18/11 for 18 GPa. The experimental procedure was described in detail by Xu et al. [2004]. After compressing samples to the desired pressure, temperature signals at the two thermocouples are collected at 100 K intervals along a heating and cooling cycle to 1673 K. At each temperature, the sample was allowed to reach steady state before measurements were made at ν = 0.1, 0.4, 0.7, 1.0 and 1.3 Hz. Amplitude and phase at each modulation frequency were obtained by fitting 10 sinusoidal cycles to the fitting equation

equation image

Amplitude ratio and phase shift were calculated by dividing inner amplitude by outer amplitude and subtracting outer phase from inner phase, respectively. After each experiment, a cross section normal to the axis of the sample cylinder was made in order to determine the distance between the two thermocouples. For calculation of thermodynamic properties we use the average temperature of the thermocouple readings.

3. Thermodynamic Treatment

[10] The thermodynamic model we employ for CaGeO3 perovskite, uses a Mie-Grüneisen formulation for the Helmholtz free energy F, which allows all thermodynamic information to be derived self-consistently by Legendre transformations and differentiation [Ita and Stixrude, 1992; Callen, 1985]. The relation is

equation image

where Fc describes isothermal compression

equation image
equation image

and Fth describes isochoric heating in the quasiharmonic approximation

equation image

The Debye temperature at volume V is obtained as

equation image

with γ(V) expressed through the finite strain based form of Stixrude and Lithgow-Bertelloni [2005]

equation image

γ0′ and γ1′ being constants with

equation image

This expression allows q to vary with density; first principles computations have shown it to better represent γ(ρ) compared to the more commonly used constant-q approximation [de Koker and Stixrude, 2009; Stixrude and Lithgow-Bertelloni, 2005; Oganov and Dorogokupets, 2003; Karki et al., 2000; Anderson, 1974]. Model parameters for CaGeO3 perovskite are reported in Table 1.

Table 1. Parameters for the Mie-Grüneisen expression of the Helmholtz free energy, at ambient pressure and T0 = 300 K
V0 (cm3/mol)K0 (GPa)K0equation image0 (K)γ0q0

4. Results

[11] Uncertainties on individual measurements are estimated to be 4% [Xu et al., 2004]. Due to greater noise in determining χ than for ϕ, values determined from the former show more scatter as a function of frequency (Figures S3S6), resulting in larger total uncertainty of the mean Dχ than for Dϕ. The assumptions on which equations (6)(8) are based are only valid when Dχ = Dϕ, so that we consider in our analysis only those measurements for which diffusivities determined from phase lag and amplitude ratio agree to within experimental uncertainty. Measurements that satisfy these criteria lie between 573 and 1073 K. Dχ and Dϕ diverge above ∼1100 K, likely due to direct radiative heat transfer, which increases with temperature and primarily affects ϕ, and also below 573 K where χ measurements contain severe scatter.

[12] Mean thermal conductivities thus determined show an increase with pressure and an asymptotic decrease with temperature (Figure 1). We determine a reference lattice conductivity for each pressure by fitting results at each pressure to the relation

equation image

with Tref = 700 K. The heat capacity ratio accounts for the temperature dependence of CP. Measurements show good agreement with the 1/T dependence relevant for phonon-phonon scattering.

Figure 1.

Thermal conductivity as a function of temperature, determined from amplitude ratio measurements. Lines indicate inverse temperature fits (equation (17)) to each isobar.

[13] We can now assess the extent to which equations (3)(5) are able to represent equation image(ρ). Best fitting values for equation image (equation (1)), with γ, q and K0′ evaluated from equation (10) were determined; the resulting relations are shown in Figure 2. g(γ, K′) (equation (4)) is found to be notably superior to g(γ) (equation (3)) and g(K′) (equation (5)), neither of which are able to account for the values at 8 GPa and 18 GPa.

Figure 2.

equation image at Tref = 700 K (symbols). Lines show best fit relations for equations (3)(5). Inset: g = ∂ ln equation image/∂ ln ρ as a function of density from equations (3)(5), computed using equation (10).

5. Discussion and Conclusions

[14] The improved agreement of g(γ, K′) with equation image(ρ) compared to g(γ) reflects the ambiguity inherent in the definition of θ in materials with many atoms in their unit cells (orthorhombic perovskite has 20). Because these materials have many optic modes, θ values determined as a moment of the vibrational density of states [e.g., Wallace, 2002], or from heat capacity data as we have done here, will be very different from what one would determine using the sound velocity, with the result that a relation which treats separately the density dependence of θ (through γ), and v (through K′), should be more successful in such cases.

[15] First-principles simulations have shown g(γ) to represent very well the conductivity of MgO periclase [de Koker, 2010], a material for which the vibrational density of states remains Debye-like to very high pressures. These simulations further indicate that τ varies strongly with pressure, so that g(K′) is unlikely give robust extrapolations of equation image. The close agreement between g(γ) and g(K′) is likely fortuitous, and is not present when K0′ is lower, as is the case in many mantle phases including MgSiO3 perovskite [Xu et al., 2008].

[16] Viewing the behavior of ∂ ln equation image/∂ ln ρ in CaGeO3 as an analogue for MgSiO3, we may use our results to obtain a revised estimate of the thermal conductivity of perovskite in the lowermost mantle. Using g(γ), de Koker [2010] extrapolated the ambient pressure measurement of Osako and Ito [1991] to the base of the mantle along a model adiabat to obtain 2.6(1) W/m K at the top of the thermal boundary layer and 1.6(1) W/m K at its base, which combined with first-principles computations for MgO periclase corresponds to values for a 80:20 perovskite:periclase aggregate of 5.9(6) W/m K and 4.0(5) W/m K, respectively. Using the same extrapolation model with g(γ, K′), the conductivity of MgSiO3 perovskite at the top of the thermal boundary layer becomes 3.3(1) W/m K, and 2.2(1) W/m K at the base. The revised Hashin-Shtrikman averages translate into values for the polycrystalline aggregate of 6.4(6) W/m K and 4.5(5) W/m K.

[17] To apply these estimates to the mantle it is assumed that the decreasing effect of adding other major elements to MgO and MgSiO3, notably Fe and Al, would be approximately cancelled by an increase due to radiative conductivity. Of course, it should be kept in mind that the validity of this assumption is yet to be investigated rigorously. Assuming a thermal gradient of 7.0 ± 1.5 K/km in the D″ [Lay et al., 2008; Buffett, 2002], our revised estimate for mantle conductivity in this region yields a heat flux from the core of 5.5 ± 1.1 TW, which falls at the lower end of the spectrum of heat flux values consistent with geophysical observations [e.g., Lay et al., 2008]. Although this leaves some room for additional contributions to the conductivity by radiation, our results indicate that the geophysical requirements can be satisfied by simple lattice conduction of heat; large increases in conductivity in the lower mantle are not required.


[18] This research was supported by the European Union under contract MRTN-CT-2006-035957, and the German Science Foundation under contract KO 3958/1-1 to NdK.

[19] The editor thanks Bernhard Steinberger and an anonymous reviewer for their assistance in evaluating this paper.