JOURNAL OF APPLIED PHYSICS 111, 112609 (2012)
Thermal conductivity of argon at high pressures and high temperatures
Alexander F. Goncharov,1 Michael Wong,1,2 D. Allen Dalton,1 J. G. O. Ojwang,1
Viktor V. Struzhkin,1 Zuzana Konôpková,3,4 and Peter Lazor3
1
Geophysical Laboratory, Carnegie Institution of Washington, Washington, D.C. 20015, USA
Department of Earth & Planetary Science, University of California, Berkeley, Berkeley,
California 94720, USA
3
Department of Earth Sciences, Uppsala University, SE-752 36 Uppsala, Sweden
4
Deutsches Elektronen Synchrotron (DESY), 22607 Hamburg, Germany
2
(Received 15 April 2011; accepted 13 November 2011; published online 15 June 2012)
Knowledge of the thermal conductivity of Ar under conditions of high pressures and temperatures
(P-T) is important for model calculations of heat transfer in the laser heated diamond anvil cell
(DAC) as it is commonly used as a pressure transmitting medium and for thermal insulation. We
used a modified transient heating technique utilizing microsecond laser pulses in a symmetric DAC
to determine the P-T dependent thermal conductivity of solid Ar up to 50 GPa and 2500 K. The
temperature dependent thermal conductivity of Ar was obtained by fitting the results of finite
element calculations to the experimentally determined time dependent temperature of a thin Ir foil
surrounded by Ar. Our data for the thermal conductivity of Ar are larger than that theoretically
calculated using the Green-Kubo formalism, but they agree well with those based on kinetic
theory. These results are important for ongoing studies of the thermal transport properties of
C 2012 American Institute of
minerals at pressures and temperatures native to the mantle and core. V
Physics. [http://dx.doi.org/10.1063/1.4726207]
INTRODUCTION
Knowledge of the thermal conductivity of materials
under extreme conditions of high pressure and high temperature is important for a number of applications, including geoand planetary sciences, technology, and industry. At ambient
pressures and variable temperatures, there are well established experimental techniques that provide very accurate
data on the thermal transport properties using both contact
and noncontact methods.1–5 Remarkably, thermal properties
of very thin (nm scale) films can be also measured using
ultrafast laser pump-probe techniques.6,7 Measurements of
the thermal conductivity at high P-T conditions remain a
challenging task because of limitations in space and sample
access imposed by high-pressure devices. Moreover, the
thermochemical properties of all materials change at high
P-T conditions, which cause additional complications in
determining thermal transport properties. Currently, there
are well established techniques for measuring thermal conductivity at moderate P (<20 GPa) and high-T in large
presses (e.g., Ref. 8). The Ångstrom technique uses the
standard cylindrical sample geometry and determines the
thermal conductivity analytically, based on the phase and
amplitude shift of the temperature wave created at the sample axis. At higher pressures, where diamond anvil cells
(DACs) are commonly used, this technique is difficult to use
due to the necessity of very local temperature probes. For
this reason, extrapolations based on scaling relations4,9 and
theoretical calculations9–12 are mainly used under P-T conditions relevant to the planetary interiors. Measurements of the
thermal conductivity (more accurately, thermal diffusivity
D ¼ K/(q Cp), where K is the thermal conductivity, q is the
density, and Cp is the specific heat capacity) in the DAC are
rare.13,14 Model calculations of the heat fluxes through the
0021-8979/2012/111(11)/112609/6/$30.00
sample cavity15–17 are the necessary part of these measurements due to the complex sample geometry and limited
capabilities of temperature measurements in the sample
cavity of the DAC.
Measurements of thermal conductivity in the DAC are
based on the determination of thermal gradients across a sample of known thickness or on the heat transfer rate from the
heated material spot. In the former case, a continuous heating
method is appropriate, and the thermal conductivity can be
deduced from simultaneous radiative temperature measurements from both sample sides when the sample is heated from
one side.14 The results depend moderately on the thermal conductivity of the medium, so better knowledge of this key parameter would increase the accuracy of the measurements. In
the latter case, time-dependent radiative temperature measurements are needed. For materials transparent to laser heating,
the laser radiation should be absorbed by a material with a
short skin depth (coupler). The absorptive and thermochemical properties of the coupler can also affect the measurements,
so these in principle need to be known under conditions of
high P-T to make the measurements more accurate. In the
transient heating technique (THT),13 a 10 ns long laser pulses
was employed, which was assumed to thermalize (make temperature even) the coupler much faster than the pulse duration
thus making the thermal diffusivity of the coupler material
unimportant for determination of the temperature decay rate.
In these measurements and the following finite element (FE)
model calculations, the energy of the laser pulse has not been
used to constrain the temperature rise of the coupler; instead,
it was used as a free parameter in the fitting of the measured
temperature-time dependencies. In subsequent work using the
THT on other materials,18 it was determined that the use of
energetic ns pulses (which are needed to heat the sample to
high temperatures, which can be measured radiometrically)
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112609-2
Goncharov et al.
creates non-equilibrium conditions on the coupler-sample
(medium) interface. These conditions cannot be described by
FE calculation using classical heat transfer equations, thus
diminishing the accuracy of the results. It is interesting that
classic heat transfer formalism works very well for much
shorter (100 fs) laser pulses (albeit of much lower energy),
which are used for the time-domain-thermoreflection (TDTR)
measurements of the thermal conductivity, including those
under high pressures.19,20 Those measurements also need an
input for the thermal conductivity of the medium and the coating material (usually metal).
Here, we address the issues mentioned above and the
current needs to improve the accuracy of the thermal conductivity measurements in the DAC. We utilized a modified
transient heating technique by using pulses of microsecond
duration (which have much lower peak power) to determine
the thermal conductivity of Ar surrounding the metallic coupler. We have chosen Ar because it is a common pressure
transmission medium for the laser experiments (e.g.,
Ref. 21), and because its thermal conductivity and other thermochemical properties have been recently calculated using
modern molecular dynamics simulation techniques in a wide
P-T range,22 thus making an easy comparison available. Previous experimental works were all performed at ambient
pressure.23–25 We determined the thermal conductivity of
solid Ar up to 50 GPa and 2500 K by fitting the measured
time-dependent temperature of the coupler, which in this
case followed the classic heat transfer equations. Unlike the
previous THT works,13,18 the value of the laser pulse energy
served as an important constraint in this determination.
EXPERIMENTAL PROCEDURE
We used a symmetric DAC with diamond anvils possessing flat culets of 300 lm diameter up to 50 GPa and
2500 K. Rhenium foil of 300 lm initial thickness preindented to 20 GPa served as a gasket. A thin Ir foil (1 lm
thick) of approximately 80 lm diameter was prepared by
compressing a small amount of material between two diamond anvils to the desired thickness controlled by spectral
measurements of the interference fringes. The foil was positioned in a recessed gasket hole (Fig. 1) as parallel as possible to the diamond tips and approximately axially centered
between them to insure that there is no direct contact
between the foil and diamond anvils and to reduce the axial
temperature gradients. Pressure was determined by the ruby
J. Appl. Phys. 111, 112609 (2012)
fluorescence technique at room temperature and no correction for the thermal pressure has been made. The sample cavity was filled with Ar (loaded at room temperature at
0.2 GPa), and the iridium foil served as a laser absorber (coupler) to pump thermal energy into the sample (surrounding
Ar).
To determine the temperature history of the sample, we
measured the iridium coupler’s time resolved thermal emission spectra as a function of time. We used a custom optical
microscope system similar to that described in Ref. 13,
except that achromatic glass optics were used. The fiber laser
power was finely controlled using a k/2 waveplate and polarizing cube beamsplitter. The laser power at the sample position was calibrated in a separate experiment using a power
meter positioned at the sample location. The laser power
reaching the sample within the DAC was corrected for transmission of the DAC using the Fresnel formula. The laser
beam profile in the focal spot was determined by a laser
beam profiler. The determined beam spot profile was
approximated to a 2D round Gaussian function of 22 lm full
width (at 1/e height).
We used 6 ls width pulses from electronically modulated Yb-based fiber laser21 operating at 10 kHz repetition
rate. This allowed the coupler temperature to return to essentially room temperature between laser pulses. This is verified
by the FE calculations, which reproduces the temperature
history of the coupler in the available measurement range.
The temporal pulse profile (measured by a photodiode and
oscilloscope) was kept constant for the whole set of measurements and used in FE calculations (approximated by a sum
of the Gaussians). The thermal emission spectra were accumulated using a 300 mm focal length spectrograph with
300 gr/mm grating equipped with a gated CCD detector.13
The detector gate was synchronized with the laser pulses.
The spectra were measured with a 500 ns gate time by averaging the laser heating events for a total accumulation time
between 0.1 to 120 s. The emission spectra measured within
the spectral window of 550-710 nm were energy calibrated
using a standard lamp with the NIST calibrated spectral irradiance. An automation program controlling the delay generator electronically adjusted the temporal delay of the detector
gate relative to the heating laser pulse. The temperature was
determined synchronously with the variation of the time
delay using a Wien’s fitting procedure (Fig. 2).26 By performing this fit as a function of electronic delay of the spectrometer gate, we are able to collect a full history of the
thermal evolution of the sample. Our system allows measurement of the radiative temperatures as low as 1300 K. The
temperature determination uncertainty varies with temperature: It is approximately 50 K above 1900 K and up to 150 K
approaching the low temperature limit of the measurements.
This has been estimated based on the scatter between different consecutive measurements.
COMPUTATIONAL METHODS
FIG. 1. The DAC sample schematic. Ar sample fills the volume above and
below the Ir foil (coupler). Since the coupler has a rectangular shape, Ar can
freely move between the upper and the lower pockets.
We used a commercial FE solver code (FlexPDE6 3D
professional, PDE Solutions, Inc.) for modeling the timedependent heat fluxes in the DAC cavity (e.g., Ref. 27). The
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112609-3
Goncharov et al.
J. Appl. Phys. 111, 112609 (2012)
FIG. 3. The distances between the Ir foil and diamond culets measured
using the spectral distance between the interference fringes. The fits applied
to the data are used for the sample cavity dimensions in the FE calculations.
FIG. 2. Representative radiometric time-resolved temperature measurements. The temperature is determined as an inverse slope of the linear lines
fitted to the data. The data represent the thermal emission spectra (Ik) transformed as shown and plotted as a function of an energetic variable. C1 and
C2 are first and second radiation constant with values of C1 ¼ 119.1044
(W nm2), C2 ¼ 1.4388 107 (nm K), respectively.
based on the high melting temperature at ambient pressure
(2683 K) and the results of our experiments (e.g., Ref. 31).
Using FE calculation methods, we simulated the heat
flux transfer in the DAC cavity using the experimentally
geometrical parameters used for calculations were determined experimentally; the thermochemical parameters were
chosen as described below. The thermal conductivity of Ar
was approximated as suggested in the theoretical calculations22 and then fit to the experimental data (2-parameters
fit) by forward modeling. The results of calculations obtained
using PDE solver were confirmed using the software package
COMSOL Multiphysics,14 which gave nearly identical
results.
RESULTS AND DISCUSSION
Two experimental runs were performed ranging from 10
to 50 GPa. At each pressure point, we determined the sample
geometry by measuring the cavity thickness and the distances
between the Ir foil and diamond culets using the optical interferometry technique (Fig. 3). The pressure dependences of
these parameters have been fit and thus averaged values have
been used in the FE calculations (see below). The data show
some scatter related to the foil deformation under pressure,
resulting in spatially variable foil–diamond culet distances.
We neglect these effects in our simplified FE model calculations. The refractive index of Ar, which is required for calculation, was taken from the Brillouin data of Ref. 28 and was
extrapolated to higher pressure as described in Ref. 29.
The laser heating experiments were performed to variable maximum temperatures depending on the nominal pressure. In the experimental runs used for comparison with FE
calculations (Fig. 4), we did not want to exceed the melting
temperature of Ar30 to avoid probing the fluid state, which
can have substantially different thermochemical parameters.
This would make our model calculations less reliable. We
assumed that Ir has a higher melting temperature than Ar
FIG. 4. Temperature history of the DAC for pulse laser heating at 43 GPa.
Many such plots were constructed for pressures up to 50 GPa. Radiometric
data from the Wien’s fits (the error bars are the temperature determination
uncertainties) illustrate an increase in the sample temperature corresponding
to the front edge of the laser pulse and then plateaus before decaying below
the detection limit. The thick solid line is the results of the FE calculations,
which represent the best fit to these data yielding the following parameters
for the temperature dependent thermal conductivity of Ar: K300 ¼ 72 W/(K m),
m ¼ 1.35—see Table I for the description of parameters. The best fit to the
data calculated in the assumption that the emissivity of Ir decreased by 10%
at 43 GPa essentially coincides with this curve (not shown); the parameters
yielded are K300 ¼ 79 W/(K m), m ¼ 1.7. The thermal history, calculated
taking into account the isobaric changes in density of Ar and Ir and the isobaric change in the thermal heat capacity of Ir with temperature, is shown by
a thin dashed line. The temporal profile (a.u. of intensity) of the incident
laser pulse (measured via a photodiode) is also included.
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112609-4
Goncharov et al.
J. Appl. Phys. 111, 112609 (2012)
determined geometric and laser heating parameters (Fig. 3).
This also includes the laser power needed to reach the measured temperature-time profiles (Fig. 4). The thermochemical
parameters and optical emittance data of Ir at ambient pressure and high temperature are available in the literature.32
The thermal conductivity of Ir at high pressures was determined by scaling these data to high pressures using the available equation of state31 and scaling relations.33 We assumed
that the temperature dependence of the thermal conductivity
of Ir (taken from Ref. 32) changes congruently with pressure
(Table I). As there are no data, we assumed that the temperature dependent optical properties of Ir32 are pressure independent. A possible departure from this assumption would
affect the final results for the thermal conductivity of Ar as
the absorbed laser pulse energy used in our calculations is
directly proportional to the optical emissivity of Ir. The
uncertainty in the value of the thermal conductivity, which
results from the unknown emissivity of Ir at high pressures,
has been estimated by performing calculations with the emissivity varied by 10% of the nominal value (maximum estimated pressure induced change) (Fig. 4); the results obtained
are within the error bars shown in the Figs. 5 and 6.
To evaluate whether the thermal expansion of Ar and Ir
would affect our results intrinsically through the change in
density with temperature, we performed finite element calculations using the thermal equations of state, which were
determined experimentally for Ir31 and theoretically for
Ar.22 We found that the calculated changes in time profiles
are small (within the experimental uncertainties in measured
temperatures) (Fig. 4), so they can be neglected. We did not
attempt calculating the extrinsic effects related to the change
in the interface distances as such calculations are beyond our
current capabilities. We also neglected the temperature
dependences of the specific heat capacities (Table I) as these
are expected to be small for Ar in the temperature range
studied,22 while for Ir the use of the temperature dependent
specific heat capacity32 affects the calculated time series
very little (Fig. 4). Finally, we neglected the stress and temperature dependences of the thermochemical parameters of
diamond (it remains at almost 300 K in all calculations).
An initial estimate for the Ar thermal conductivity was
applied using the theoretical work.22 The temperature dependent thermal conductivity of Ar was determined by fitting
the results of FE calculations to the experimentally determined time dependent coupler temperature for each experimental pressure point (Figure 4). The uncertainty in the
FIG. 5. Thermal conductivity as a function of temperature. Dashed lines
show the uncertainty interval. The points indicate thermal conductivity
results from MD simulations from Ref. 22 and the corresponding line is the
fit to these results.
thermal conductivity was determined by finding the range of
parameters for which the FE model and the experimental
curves agree within the experimental uncertainty.
In comparing the dependence with temperature of our
thermal conductivity data to that of Tretiakov and Scandolo,22 we find that (on examples of the results for 10 and
50 GPa—Fig. 5) the thermal conductivity is higher than the
results of the Green-Kubo (GK) MD simulations at all temperatures. However, our data agree well with the theoretical
calculations obtained using kinetic theory22 (Fig. 6). From
the log(K)-log(T) plots, simulations give a slope of 1.3
while our data give a slope of 1.4. With respect to the pressure dependence of thermal conductivity for Ar at 300 K
(Figure 6), our data have a similar dependence on a log(K)log(P) scale (theoretical value 1.29 vs. our data 1.24).
The pressure effect on the thermal conductivity can be estimated using the Leibfried-Schlömann (LS) equation
K¼A
V 1=3 xD 3
;
c2 T
(1)
TABLE I. Thermochemical parameters of materials used in model FE calculations.
Property/material
Diamond
Density, q0 (kg/m3)
Bulk modulus, B0 (GPa)
BM derivative, B00
Thermal conductivity (W/(m K))
Specific heat capacity (J/(kg K))
Emissivity
3500
2000
509
0
Sample/medium
(Ar)
1809
5.513
5.20
K300 (300/T)m
568.7
0
Coupler (Ir)
22 650
390.5
3.26
(q/q0)4 (147 þ 3.2 106 (T 300) 7 1010 (T 300)2)
130
0.551 (5.661 106 K1) T þ (8.586 109 K2) T2
(5.784 1012 K3) T3
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112609-5
Goncharov et al.
FIG. 6. Thermal conductivity as a function of pressure at 300 K. The data
points (closed circles) are those determined in this work at 300 K with the
corresponding errors. The separate solid and dashed lines show the results of
the MD calculations using the Green-Kubo method and kinetic theory,
respectively.22 The squares and the thin dashed line show the LeibfriedSchlömann equation (1) fit to our data.
where V is the volume, xD is the Debye frequency, c is the
Grüneisen parameter, T is the temperature, and A is a constant that is independent of pressure.34 We estimated the values on the right side of Eq. (1) at 300 K based on the
available experimental data29,35 and by fitting the constant A
to our 300 K data. The results are shown in Fig. 6. The slope
of the thermal conductivity dependence on pressure obtained
from Eq. (1) agrees well with our experimental data. Moreover, the slope agrees well with the scaling expression
obtained from kinetic theory (e.g., Refs. 15 and 22).
It is remarkable that our data for the temperature dependence agree reasonably well with the results of theoretical calculations22 and ambient pressure experimental data.23
The departure of the temperature dependence of the thermal
conductivity from a common T1 form has been found at
ambient pressure23–25 and attributed to various mechanisms
such as, for example, the effects of thermal expansion on the
lattice vibrational frequencies.23 Our measurements show
that this departure holds at high pressures (see also Ref. 22)
and possibly even increases with pressure. Our high-pressure
experiments cover a much wider temperature range than that
in the ambient pressure studies,23–25 mainly probing the classical high-temperature region T HD (HD is the Debye
temperature; HD ¼ 85 K at 0 GPa, increasing slightly above
300 K at 50 GPa36), thus making difficult the direct
comparison.
CONCLUSIONS
Our flash laser heating measurements in the DAC with
ls pulses determined the thermal conductivity of Ar based
on good agreement between experimentally observed temperature history and the results of finite element calculations
with realistic geometric and thermochemical parameters.
The experiment has been designed to minimize the effect of
J. Appl. Phys. 111, 112609 (2012)
the unknown high-pressure optical emissivity and thermal
conductivity of the Ir coupler, while other thermochemical
parameters could be constrained rather accurately. Using the
theoretically predicted power dependencies of the thermal
conductivity of Ar on temperature, we find good agreement
between experimental and FE models calculated temperature
histories. Knowledge of the laser power and laser focal spot
shape allowed constraining the Ar thermal conductivity values. We find that larger low-temperature values and slightly
faster temperature dependencies of thermal conductivity better match our observations compared to the GK theoretical
predictions, while our data nearly coincide with the theoretical results based on kinetic theory. The effect of thermal
pressure increase at high temperature was found to be negligible for the current determination of thermal conductivity.
The measurements presented have several sources of
uncertainty, which could affect the results. We believe that
the major one is related to inhomogeneity of the optical
properties of Ir coupler arising from micro-defects created
during previous heating cycles or because of coupler deformation. Larger than nominal Ir emissivity values would
result in increased values of the thermal conductivity.
Accurate thermal conductivity data of Ar at high pressures and high temperatures are essential to future measurements of material thermal conductivity under extreme P-T
conditions using the modern ultrafast pump-probe TDTR
technique.19,20 Ar is a commonly used and convenient pressure medium; recent measurements of the thermal conductivity of MgO show that it can be used at high pressures up to
60 GPa.37 Measurements of the thermal conductivity of Fe
using the static laser heating technique14 would also benefit
from these new data. Our measurements open the possibility
of using Ar under extreme P-T conditions with higher confidence using the experimentally determined values of the
thermal conductivity.
ACKNOWLEDGMENT
We thank R. S. McWilliams for important comments
and suggestions on the manuscript. We acknowledge support
from NSF EAR 0711358 and EAR 1015239, Carnegie Institution of Washington, Army Research Office, STINT
IG2010-2 062, and DOE/NNSA (CDAC).
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