Transcript Thermal Transport in the Transition Zone

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
vs.
buoyancy
vs.
heat diffusion viscous damping
Microscopic mechanisms of heat
transport:
Material type:
Mechanisms inside Earth:
Unwanted mechanisms
only in experiments:
Metals
(Fe, Ni)
Opaque
insulators
(FeO, FeS)
Electron
scattering
Phonon
scattering
klat
Ballistic photons
Partially
transparent
insulators
(silicates, MgO)
Photon
diffusion
(krad,dif)
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)
Problems with existing methods:
14
16
Spurious direct radiative transfer:
Light crosses the entire sample over 14
the transparent frequencies,
12
warming the thermocouple without
10
-1
participation of the sample
A, cm
579 C
I
BB
1000 C
1000 C
Quartz
Ec
25 C
12
10
8
800 C
IBB,
800 C
A=0
8
Olivine
E||c
300 C
W/mm2/mm
6
25 C
6
4
4
300 C
600 C
25 C
2
2
500 C
300 C
0
source
2000
sink
3000
4000
6000
0
7000
-1
Wavenumber, cm
Thermal losses at Polarization mixing because
LO modes indirectly couple
contacts
with EM waves
sample
PZ LO
z
metal
5000
PY
EX
Few LO
2TO
PX
Many LO
Electron-phonon
coupling provides
an additional
relaxation process
for the
PTGS method
The laser-flash technique lacks these
problems and isolates Dlat(T)
furnace
near-IR
detector
furnace
laser
cabinet
Sample
under
cap
support
tube
How a laser-flash apparatus works
SrTiO3 at 900o C
1.8
1.6
IR detector
1.4
pulse
sample emissions
Signal
Signal/V
1.2
1.0
t half
0.8
0.6
hot furnace
0.4
0.2
Suspended
sample
laser pulse
IR laser
0.0
-0.2
-2000
-1000
0
1000
2000
3000
4000
5000
6000
Time /ms
Time
For adiabatic cooling
D  0.139
(Cowan et al. 1965):
t
L2
half
How a laser-flash apparatus works
SrTiO3 at 900o C
1.8
1.6
IR detector
1.4
pulse
sample emissions
Signal
Signal/V
1.2
1.0
t half
0.8
0.6
hot furnace
0.4
0.2
Suspended
sample
laser pulse
IR laser
0.0
-0.2
-2000
-1000
0
1000
2000
3000
4000
5000
6000
Time /ms
Time
For adiabatic cooling
D  0.139
(Cowan et al. 1965):
t
More complex cooling
requires modeling the
signal
L2
half
Advantages of Laser Flash Analysis:
emissions
No physical contacts with thermocouples
emissions
Au
Thin plate geometry avoids
polarization mixing
End cap
sample
c
u
graphite
Sample
holder
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
b
olivine
2.5
surface
cooling
by radiation
laser
pulse
2
u
1
phonon
contribution
data
1
direct radiative
contribution
0.5
c
baseline
u
0
0
200
thalf
room temp.,
phonons only
400
600
Time, ms
800
model
thalf
1.5
2
1300 C,
photons + phonons
laser pulse
gadolinium gallium garnet
T = 1570 K
0
-500
Bad fits are seen and
data are not used
0
500
1000
Time, ms
1500
2000
Laser-Flash analysis gives
olivine single-crystals
8
Beck
et al.
Gibert et al
[100]
7
k ,
W/m-K
We find:
Schatz & Simmons 8
[010]
tot
Absolute values of D (and k),
verified by measuring
standard reference materials
6
Higher thermal conductivity at
room temperature because
contact is avoided
Kanamori et al.
[001]
Scharmeli
[010]
5
K8
[100]
Gibert et al.
[001]
4
Scharmeli
2.5 GPa
[100]
[001]
P&H [100]
K 8 [001]
3
Chai
et al.
Gibert et al.
[010]
S&S 14
[010]
P&H [001]
Kobayashi 13 [001]
2
Pertermann &
Hofmeister
[010]
K 8 [010]
400
600
800
1000
Temperature, K
1200
1400
Lower k at high temperature
because spurious radiation
transfer is avoided
Pertermann and Hofmeister
(2006) Am. Min.
Contact resistance causes
underestimation of k and D
Directionally averaged values
spinel
cryogenic
study
7
On average,
D at 298 K
is reduced by
10% per thermal
contact
D
mm2/s
quartz
5
NaCl
NaCl (Pangilinan et al. 2000)
3
Fo 90
Fo
90
1
(Chai et al. 1996)
diopside
almandine
orthoclase
0
1
Number of contacts
2
Hofmeister 2006
Pertermann and Hofmeister 2006
Branlund and Hofmeister 2007
Hofmeister 2007ab
Pertermann et al. in review
Hofmeister and Pertermann in review
LFA data accurately records D(T)
A consistent picture is emerging regarding relationships
of D and k with chemistry and structure
4
CaMgSi O
Na (001)
3.5
2
4
6
end-members [001]
3.5
3
Fe ~(001)
3
2.5
558.49/T
358.54/T 0.89472
2
1.5
1
2.5
Fe
(110)
Fe (001)
0.5
400
600
Na (010)
Fe (100)
800
1000
1200
end-members ||c
2
Na ~(001)
Na (100)
solid solutions [001]
0.88859
1400
1600
1.5
solid solutions ||c
2
2.05
2.1
2.15
2.2
2.25
2.3
o
Average of M2 and M1 bond lengths, A
Temperature, K
D of clinopyroxenes: Hofmeister and Pertermann, in review
2.35
LFA data do not support different scattering
mechanisms existing at low and high temperature
(umklapp vs normal)
140
10
2
10
4
spinel
Slack 1962
contact
100
klat
MgAl O
R42
120
10
D
80
2
mm /s
R54
W/m-K
10
7
Spinel
ordered
6
5
4
contact measurements
Slack 1962
1000
60
power law fit
to LFA data
40
20
ceramic
100
10
ordered
200
1
400
600
800
1000
Temperature,K
LFA
polynomial fit to 1/D
disordered
0
0
LFA
power law fit to D
1200 1400
0
100
200
300
Data
400
500
600
Temperature, K
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.
700
Pressure data is almost entirely from
conventional methods, which have
contact and radiative problems:
single-crystal Fo
93
at 8.3 GPa
Osako et al. (2004)
3
1 atm single-crystal Fo
93
LFA
Pertermann and Hofmeister
(in review)
2006
2.5
D,
2
2
Fo
mm /s
ceramic at P
Xu et al. (2004)
D(10 GPa) = 0.266+492/ T
D(7 GPa) =0.256+459/ T
D(4 GPa) = 0.315+341/ T
1.5
1
90
[100]
1 atm
extrap.
[001]
[010]
4 GPa
0.5
400
[001]+[100]
600
800
1000
1200
Temperature, K
10 GPa
7 GPa
1400
1600
Can the pressure
derivatives be
trusted?
At low pressures, dD/dP is inordinately high and
seems affected by rearrangement of grains,
deformation or changes in interface resistance
d(ln D)/dP
%/GPa
The slopes are ~100 x larger
than expected for
compressing the phonon gas.
30
25
NaCl
20
The high slopes correlate
with stiffness of the solid and
suggest deformation is the
problem.
Pangilinan
et al. 2000
15
10
Derivatives at high P
are most trustworthy
but are approximate
Fo 90
5
MgO
0
0
2
Chai et al.
1996
4
6
Pmax , GPa
8
10
Hofmeister in review
Heat transfer via vibrations (phonons)
+
phonon gas analogy
of Debye
damped harmonic oscillator model
of Lorentz
gives
k0 
r
3MZ
CV u
2
1
G
or
D =<u>2/(3ZG)
(Hofmeister, 2001, 2004, 2006)
where G equals the full width at half maximum of the
dielectric peaks obtained from analysis of IR reflectivity data
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:
Pressure derivatives are predicted by
the DHO model with accuracy
comparable to measurements
30
30
NaCl
acoustic model
(3K'/2+2q-11/6)/KT
20
20
where q = 1
Models of
d(ln k)/dP
%/GPa
K'/KT
10
10
Hard
minerals
cluster
(4th+1/3)/KT
DHO model
0
0
10
20
Measurements of d(ln k)/dP, %/GPa
30
0
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).
Diffusive Radiative Transfer is largely
misunderstood because:
• We are familiar with direct radiative transfer
Direct: the medium
does not participate
Space
Diffusive: the medium
is the message
Earth
990 K
~1 km
1000 K
• 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
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.
d
• The mean free path is determined by
grain-size, d, and absorption coefficient, A.
2 
4dn
krad ,dif (T ) 
3
 dA
(1  e ) [ I BB ( ν, T )]
0 (1  dA) ν 2 T dν
(Hofmeister 2004, 2005); Hofmeister et al. (2007)
2 
4dn
krad ,dif (T ) 
3
(1  e  dA ) [ I BB ( ν, T )]
0 (1  dA) ν 2 T dν
The pressure dependence of Diffusive
Radiative Transfer comes from that of A,
not from that of the peak position
dA dA dn

dP dn dP
P1
A
P2
n
Positive for n<nmax, negative for n>nmax
Over the integral, these contributions roughly cancel
And d krad/ dP is small
(Hofmeister 2004, 2005)
2 
4dn
krad ,dif (T ) 
3
(1  e  dA ) [ I BB ( ν, T )]
0 (1  dA) ν 2 T dν
By assuming A is constant (over n and T) and ignoring
d, Clark (1957) obtained kradT3/A
A
Obviously, 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)
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).
T
 k rad
 flux  s T 4
r
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.
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