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
Photon diffusion (krad,dif)
Phonon scattering klat
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)
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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
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The laser-flash technique lacks these problems
and isolates Dlat(T)
furnace
near-IR detector
Sample under cap
furnace
support tube
laser cabinet
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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
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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
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Advantages of Laser Flash Analysis
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 als 1998 model accounts for the
remaining direct radiative transfer, which is
easy to recognize
olivine
Bad fits are seen and data are not used
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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.
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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
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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
bTcT2.
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
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
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Heat transfer via vibrations (phonons)

damped harmonic oscillator model of Lorentz
phonon gas analogy of Debye
gives
D ltugt2/(3ZG)
or
(Hofmeister, 2001, 2004, 2006)
where G equals the full width at half maximum of
the dielectric peaks obtained from analysis of IR
reflectivity data
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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 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
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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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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
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Modeling Diffusive Radiative Transfer

• Earths 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)
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The pressure dependence of Diffusive Radiative
Transfer comes from that of A, not from that of
the peak position
Positive for nltnmax, negative for ngtnmax Over the
integral, these contributions roughly cancel And
d krad/ dP is small
(Hofmeister 2004, 2005)
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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)
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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 Clarks result krad (16/3) sT3L, because
  the mean free path L is r for the cavity.

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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