Measurements and models of thermal transport properties

by Anne Hofmeister

Many thanks to Joy Branlund, Maik Pertermann, Alan Whittington, and Dave Yuen

Thermal conductivity largely governs mantle convection

buoyancy

vs.

heat diffusion

vs.

viscous damping

Microscopic mechanisms of heat transport:

Metals(Fe, Ni)

Opaque insulators (FeO, FeS)

Partially transparent insulators(silicates, MgO)

Electron scattering

Photon diffusion(krad,dif)

Ballistic photons

Material type:

Mechanisms inside Earth:

Unwanted mechanisms only in experiments:

Phonon scattering klat

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 = CPD

Heat = Light

Macedonio Melloni (1843)

=

Problems with existing methods:

PX

PY

PZLO

2TO

EX

z

sample

metal

0

2

4

6

8

10

12

14

16

0

2

4

6

8

10

12

14

2000 3000 4000 5000 6000 7000

A, cm-1

IBB

,

W/mm2/mm

Wavenumber, cm-1

OlivineE||c

IBB

1000 C

800 C

600 C25 C

300 C

1000 C

800 C

300 C

25 C

QuartzE⊥c

579 C

25 C

A=0

300 C

500 C

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 thePTGS method

Few LO Many LO

The laser-flash technique lacks these problems and isolates Dlat(T)

lasercabinet

near-IR detector

supporttube

Sampleundercap

furnace

furnace

How a laser-flash apparatus works

half

2

139.0t

LD =

IR detector

hot furnace

Suspendedsample

IR laser

laser pulse

sample emissions

-2000 -1000 0 1000 2000 3000 4000 5000 6000

Time /ms

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Signal/V

SrTiO3 at 900o C

Time

Signal t half

For adiabatic cooling (Cowan et al. 1965):

pulse

How a laser-flash apparatus works

half

2

139.0t

LD =

IR detector

hot furnace

Suspendedsample

IR laser

laser pulse

sample emissions

-2000 -1000 0 1000 2000 3000 4000 5000 6000

Time /ms

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Signal/V

SrTiO3 at 900o C

Time

Signal t half

For adiabatic cooling (Cowan et al. 1965):

pulse

More complex cooling requires modeling the signal

Sampleholder

End cap

Laser pulse

emissions

0

1

2

0 200 400 600 800Time, mst

half

room temp., phonons only

1300 C,photons + phonons

u

u

c

b

laser pulse

emissions

sample

graphite

c u

0

0.5

1

1.5

2

2.5

-500 0 500 1000 1500 2000

Time, ms

gadolinium gallium garnetT = 1570 K

direct radiative contribution

phononcontribution

thalf

surfacecoolingby radiation

baseline

laserpulse

data

model

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

Laser-Flash analysis gives

2

3

4

5

6

7

8

400 600 800 1000 1200 1400

ktot

,

W/m-K

Temperature, K

Kanamori et al.[001]

Kobayashi 13 [001]

K 8 [001]

K 8 [100]

K 8 [010]

Gibert et al[100]

Gibert et al.[001]

Scharmeli [010]

[001]

olivine single-crystals

Chai et al.

Gibert et al.[010]

Scharmeli [100]

S&S 14[010]

Schatz & Simmons 8[010]

Pertermann & Hofmeister[010]

P&H [001]

P&H [100]

Beck et al.

2.5 GPa

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.

Hofmeister 2006Pertermann and Hofmeister 2006Branlund and Hofmeister 2007Hofmeister 2007abPertermann et al. in reviewHofmeister and Pertermann in review

1

3

5

7

0 1 2

D

mm2/s

Number of contacts

quartz

NaCl

Fo90

Fo90

(Chai et al. 1996)

spinel

diopside

orthoclase

cryogenicstudy

NaCl (Pangilinan et al. 2000)

Directionally averaged values

almandine

On average, D at 298 K is reduced by 10% per thermal contact

Contact resistance causes underestimation of k and D

LFA data accurately records D(T)

0.5

1

1.5

2

2.5

3

3.5

4

400 600 800 1000 1200 1400 1600

Temperature, K

CaMgSi2O

6

Na ~(001)

Na (001)

558.49/T0.88859

358.54/T0.89472

Na (010)Na (100)

Fe (110)

Fe (100)

Fe (001)

Fe ~(001)

A consistent picture is emerging regarding relationshipsof D and k with chemistry and structure

D of clinopyroxenes: Hofmeister and Pertermann, in review

1.5

2

2.5

3

3.5

4

2 2.05 2.1 2.15 2.2 2.25 2.3 2.35

Average of M2 and M1 bond lengths, A

solid solutions [001]

end-members [001]

solid solutions ||c

end-members ||c

o

LFA data do not support different scattering mechanisms existing at low and high temperature

(umklapp vs normal)

0

20

40

60

80

100

120

140

0 200 400 600 800 1000 1200 1400

klat

W/m-K

Temperature,K

R42

R54

ceramicdisordered

ordered

MgAl2O

4

spinelSlack 1962contact

power law fit to LFA data

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.

1

10

100

1000

104

105

106

107

0 100 200 300 400 500 600 700

D

mm2/s

Temperature, K

LFApolynomial fit to 1/D

contact measurementsSlack 1962

Spinelordered

LFApower law fit to D

Data

Pressure data is almost entirely from conventional methods, which have

contact and radiative problems:

0.5

1

1.5

2

2.5

3

400 600 800 1000 1200 1400 1600

D,

mm2/s

Temperature, K

[100][001]+[100]

[001]

[010]10 GPa

1 atm single-crystal Fo93

LFAPertermann and Hofmeister

(in review)

7 GPa4 GPa

1 atmextrap.

Fo90

ceramic at P

Xu et al. (2004)D(10 GPa) = 0.266+492/ TD(7 GPa) =0.256+459/ TD(4 GPa) = 0.315+341/ T

single-crystal Fo93

at 8.3 GPa

Osako et al. (2004)

Can the pressure derivatives be trusted?

2006

At low pressures, dD/dP is inordinately high and seems affected by rearrangement of grains,

deformation or changes in interface resistance

0

5

10

15

20

25

30

0 2 4 6 8 10

d(ln D)/dP %/GPa

Pmax

, GPa

NaCl

Fo90

MgO

Pangilinan et al. 2000

Chai et al.1996

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

Heat transfer via vibrations (phonons)

damped harmonic oscillator modelof 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

=

1

3

2

0 uCMZ

k V

ρD =<u>2/(3Z)or

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

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

0

10

20

30

0 10 20 300

10

20

30

Models of d(ln k)/dP %/GPa

Measurements of d(ln k)/dP, %/GPa

(3K'/2+2q-11/6)/KT

where q = 1K'/K

T

(4γth+1/3)/K

T

NaCl

DHO model

acoustic model

Hardmineralscluster

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).

• 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

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

í)],í( [

í )1(

)e1(

3

4)(

02

2

, dT

TI

dA

dnTk BB

dA

difrad ∂∂

+−

= ∫∞ −

(Hofmeister 2004, 2005); Hofmeister et al. (2007)

d

The pressure dependence of Diffusive Radiative Transfer comes from that of A, not from that of the peak position

í)],í( [

í )1(

)e1(

3

4)(

02

2

, dT

TI

dA

dnTk BB

dA

difrad ∂∂

+−

= ∫∞ −

(Hofmeister 2004, 2005)

Positive for <max, negative for >max

Over the integral, these contributions roughly cancelAnd d krad/ dP is small

dP

d

d

dA

dP

dA

=A

P1 P2

By assuming A is constant (over and T) and ignoring d, Clark (1957) obtained kradT3/AObviously, there is no P dependence with no peaks

í)],í( [

í )1(

)e1(

3

4)(

02

2

, dT

TI

dA

dnTk BB

dA

difrad ∂∂

+−

= ∫∞ −

Dependence of A on and on T and opaque spectral regions in the IR and UV make the temperature dependence weaker than T3 (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)

Removing one single grain from the mantle leaves a cavity with radius r. The flux inside the cavity is T4, 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 krad is proportional to the product .

Dimensional analysis provides an approximate solution:

krad ~ T3r.

The result is essentially emissivity multiplied by Clark’s result [krad = (16/3) T3], because the mean free path is ~r for the cavity.

Emissivity (), a material property, is needed, as confirmed with a thought experiment:

flux 4rad T

r

Tk ==

∂∂

−

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