Measurements and models of thermal transport properties

1. Measurements and models of thermal transport properties by Anne Hofmeister Many thanks to Joy Branlund, Maik Pertermann, Alan Whittington, and Dave Yuen

2. Thermal conductivity largely governs mantle convection vs. vs. viscous damping buoyancy heat diffusion

3. Microscopic mechanisms of heat transport: Partially transparent insulators (silicates, MgO) Metals (Fe, Ni) Opaque insulators (FeO, FeS) Material type: Electron scattering Phonon scattering klat Photon diffusion (krad,dif) Mechanisms inside Earth: Ballistic photons Unwanted mechanisms only in experiments:

4. Phonon scattering (the lattice component) • With few exceptions, contact measurements were used in geoscience, despite known problems with interface resistance and radiative transfer • Problematic measurements and the historical focus on k and acoustic modes has obfuscated the basics • Thermal diffusivity is simpler: k = rCPD = Heat = Light Macedonio Melloni (1843)

5. PZ LO 2TO PY PX z EX Problems with existing methods: Spurious direct radiative transfer: Light crosses the entire sample over the transparent frequencies, warming the thermocouple without participation of the sample source sink Polarization mixing because LO modes indirectly couple with EM waves Thermal losses at contacts Electron-phonon coupling provides an additional relaxation process for the PTGS method sample metal Few LO Many LO

6. The laser-flash technique lacks these problems and isolates Dlat(T) furnace near-IR detector Sample under cap furnace support tube laser cabinet

7. How a laser-flash apparatus works SrTiO3 at 900o C IR detector pulse Signal t half sample emissions hot furnace Suspended sample Time For adiabatic cooling (Cowan et al. 1965): laser pulse IR laser

8. How a laser-flash apparatus works SrTiO3 at 900o C IR detector pulse Signal t half sample emissions hot furnace Suspended sample Time For adiabatic cooling (Cowan et al. 1965): laser pulse IR laser More complex cooling requires modeling the signal

9. Advantages of Laser Flash Analysis: emissions End cap Sample holder Laser pulse emissions No physical contacts with thermocouples Au Thin plate geometry avoids polarization mixing sample c u graphite Au/Pt coatings suppress direct radiative transfer laser pulse Mehling et al’s 1998 model accounts for the remaining direct radiative transfer, which is easy to recognize olivine Bad fits are seen and data are not used

10. Laser-Flash analysis gives Absolute values of D (and k), verified by measuring standard reference materials We find: Higher thermal conductivity at room temperature because contact is avoided Lower k at high temperature because spurious radiation transfer is avoided Pertermann and Hofmeister (2006) Am. Min.

11. Contact resistance causes underestimation of k and D On average, D at 298 K is reduced by 10% per thermal contact Hofmeister 2006 Pertermann and Hofmeister 2006 Branlund and Hofmeister 2007 Hofmeister 2007ab Pertermann et al. in review Hofmeister and Pertermann in review

12. LFA data accurately records D(T) A consistent picture is emerging regarding relationships of D and k with chemistry and structure D of clinopyroxenes: Hofmeister and Pertermann, in review

13. LFA data do not support different scattering mechanisms existing at low and high temperature (umklapp vs normal) Instead the “hump” in k results from the shape of the heat capacity curve contrasting with 1/D = a +bT+cT2…. Hofmeister 2007 Am Min.

14. Pressure data is almost entirely from conventional methods, which have contact and radiative problems: Can the pressure derivatives be trusted? 2006

15. At low pressures, dD/dP is inordinately high and seems affected by rearrangement of grains, deformation or changes in interface resistance The slopes are ~100 x larger than expected for compressing the phonon gas. The high slopes correlate with stiffness of the solid and suggest deformation is the problem. Derivatives at high P are most trustworthy but are approximate Hofmeister in review

16. Heat transfer via vibrations (phonons) + phonon gas analogy of Debye damped harmonic oscillator model of Lorentz gives D =<u>2/(3ZG) or (Hofmeister, 2001, 2004, 2006) where Gequals the full width at half maximum of the dielectric peaks obtained from analysis of IR reflectivity data

17. IR Data is consistent with general behavior of D with T, X, and P • FWHM(T) is rarely measured and not terribly inaccurate, but increases with temperature. • Flat trends at high T are consistent with phonon saturation (like the Dulong-Petit law of heat capacity) arising from continuum behavior of phonons at high n • FWHM(X) has a maximum in the middle of compositional joins, leading to a minimum in D (and in k) All of the above is anharmonic behavior FWHM is independent of pressure (quasi-harmonic behavior), allowing calculation of dk/dP from thermodynamic properties:

18. Pressure derivatives are predicted by the DHO model with accuracy comparable to measurements Hard minerals cluster

19. Conclusions: Phonon Transport • Laser flash analysis provides absolute values of thermal diffusivity (and thermal conductivity) which are higher at low temperature and lower at high temperature than previous measurements which systematically err from contact resistance and radiative transfer • Contact resistance and deformation affect pressure derivatives of phonon scattering – data are rough, but reasonable approximations. • Pressure derivatives are described by several theories because these are quasi-harmonic. The damped harmonic oscillator model further describes the anharmonic behavior (temperature and composition).

20. Earth 990 K ~1 km 1000 K Diffusive Radiative Transfer is largely misunderstood because: • We are familiar with direct radiative transfer • Diffusive radiative transfer is NOT really a bulk physical property as scattering and grain-size are important • In calculating (approximating) diffusive radiative transfer from spectroscopy, simplifying approximations are needed but many in use are inappropriate for planetary interiors Diffusive: the medium is the message Direct: the medium does not participate Space

21. d Modeling Diffusive Radiative Transfer Earth’s mantle is internally heated and consists of grains which emit, scatter, and partially absorb light. • Light emitted from each grain = its emissivity x the blackbody spectrum • Emissivity = absorptivity (Kirchhoff, ca. 1869) which we measure with a spectrometer. • The mean free path is determined by grain-size, d, and absorption coefficient, A. (Hofmeister 2004, 2005); Hofmeister et al. (2007)

22. P2 P1 A n The pressure dependence of Diffusive Radiative Transfer comes from that of A, not from that of the peak position Positive for n<nmax, negative for n>nmax Over the integral, these contributions roughly cancel And d krad/ dP is small (Hofmeister 2004, 2005)

23. A By assuming A is constant (over n and T) and ignoring d, Clark (1957) obtained kradT3/AObviously, there is no P dependence with no peaks n Dependence of A on n and on T and opaque spectral regions in the IR and UV make the temperature dependence weaker than T3 (Shankland et al. 1979) Accounting for grain-size and grain-boundary reflections is essential and adds more complexity (Hofmeister 2004; 2005; Hofmeister and Yuen 2007)

24. Emissivity (), a material property, is needed, as confirmed with a thought experiment: Removing one single grain from the mantle leaves a cavity with radius r. The flux inside the cavity is sT4, where s is the Stefan-Boltzmann constant (e.g. Halliday & Resnick 1966). From Carslaw & Jaeger (1960). Irrespective of the particular temperature gradient in the cavity, Eq. 2 shows that krad is proportional to the product s. Dimensional analysis provides an approximate solution: krad ~ sT3r. The result is essentially emissivity multiplied by Clark’s result [krad = (16/3) sT3L], because the mean free path L is ~r for the cavity.

25. Conclusions: Diffusive Radiative Transfer • Not considering grain-size, back reflections, and emissivity and/or assuming constant A (krad ~T3, i.e., using a Rosseland mean extinction coeffiecient) provides incorrect behavior for terrestrial and gas-giant planets. • High-quality spectroscopic data are needed at simultaneously high P and T to better constrain thermodynamic and transport properties and to understand this mesoscopic and length-scale dependent behavior of diffusive radiative transfer