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

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

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

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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 = C P D Heat = Light Macedonio Melloni (1843 ) =

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

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

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How a laser-flash apparatus works IR detector hot furnace Suspended sample IR laser laser pulse sample emissions SrTiO 3 at 900 o C Time Signal t half For adiabatic cooling (Cowan et al. 1965): pulse

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How a laser-flash apparatus works IR detector hot furnace Suspended sample IR laser laser pulse sample emissions SrTiO 3 at 900 o C Time Signal t half For adiabatic cooling (Cowan et al. 1965): pulse More complex cooling requires modeling the signal

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Sample holder End cap Laser pulse emissions laser pulse emissions sample graphite cu Advantages of Laser Flash Analysis: Thin plate geometry avoids polarization mixing Au/Pt coatings suppress direct radiative transfer 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 Au No physical contacts with thermocouples

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

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

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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

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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+cT 2 …. Hofmeister 2007 Am Min.

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

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At low pressures, dD/dP is inordinately high and seems affected by rearrangement of grains, deformation or changes in interface resistance Hofmeister in review 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

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

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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 FWHM(X) has a maximum in the middle of compositional joins, leading to a minimum in D (and in k) FWHM is independent of pressure (quasi-harmonic behavior), allowing calculation of dk/dP from thermodynamic properties: All of the above is anharmonic behavior

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

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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).

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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 Radiative Transfer is largely misunderstood because: Space Diffusive: the medium is the message Earth 990 K ~1 km 1000 K Direct: the medium does not participate

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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. Modeling Diffusive Radiative Transfer (Hofmeister 2004, 2005); Hofmeister et al. (2007) d

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The pressure dependence of Diffusive Radiative Transfer comes from that of A, not from that of the peak position (Hofmeister 2004, 2005) Positive for max Over the integral, these contributions roughly cancel And d k rad / dP is small A P1 P2

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By assuming A is constant (over and T) and ignoring d, Clark (1957) obtained k rad T 3 /A Obviously, there is no P dependence with no peaks Dependence of A on and on T and opaque spectral regions in the IR and UV make the temperature dependence weaker than T 3 (Shankland et al. 1979) A Accounting for grain-size and grain-boundary reflections is essential and adds more complexity (Hofmeister 2004; 2005; Hofmeister and Yuen 2007)

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Removing one single grain from the mantle leaves a cavity with radius r. The flux inside the cavity is T 4, where 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 k rad is proportional to the product . Dimensional analysis provides an approximate solution: k rad ~ T 3 r. The result is essentially emissivity multiplied by Clark’s result [k rad = (16/3) T 3 ], because the mean free path is ~r for the cavity. Emissivity ( ), a material property, is needed, as confirmed with a thought experiment:

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Conclusions: Diffusive Radiative Transfer Not considering grain-size, back reflections, and emissivity and/or assuming constant A (k rad ~T 3, 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

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MECHANISMS OF HEAT TRANSFER

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