Atomistic Green-Kubo simulations are performed to evaluate the lattice thermal conductivity for single crystals of the ultra high temperature ceramics ZrB2 and HfB2 for a range of temperatures. Recently developed interatomic potentials are used for these simulations. Heat current correlation functions show rapid oscillations which can be identified with mixed metal-Boron optical phonon modes. Agreement with available experimental data is good

Bauschlicher, Charles W., Jr.

Lawson, John W.

Murray, Daw S.

NASA Technical Reports Server

Lattice thermal conductivity of ultra high temperature ceramics ZrB2 and
HfB2 from atomistic simulations
John W. Lawson,1, a) Murray S. Daw,2 and Charles W. Bauschlicher, Jr.3
1)Thermal Protection Materials Branch, Mail Stop 234-1,
NASA Ames Research Center, Moffett Field, CA 94035
2)Department of Physics and Astronomy, Clemson University, Clemson,
SC 29631
3)Entry Systems and Technology Division, Mail Stop 230-3,
NASA Ames Research Center, Moffett Field, CA 94035
Atomistic Green-Kubo simulations are performed to evaluate the lattice ther-
mal conductivity for single crystals of the ultra high temperature ceramics
ZrB2 and HfB2 for a range of temperatures. Recently developed interatomic
potentials are used for these simulations. Heat current correlation functions
show rapid oscillations which can be identified with mixed metal-Boron optical
phonon modes. Agreement with available experimental data is good.
a)Electronic mail: John.W.Lawson@nasa.gov
1
https://ntrs.nasa.gov/search.jsp?R=20110015597 2019-08-30T17:16:47+00:00Z
The class of materials referred to as ultra high temperature ceramics (UHTC),
including ZrB2 and HfB2, have high melting points, good chemical stability and
reasonable mechanical properties. They are candidate materials for applications in-
volving high temperature environments such as sharp leading edges for hypersonic ve-
hicles, propulsion systems and refractory crucibles1–3. They have also been discussed
as buffer materials for microelectronics4. Unlike most ceramics, these materials are
distinguished by their high thermal conductivity. High thermal conductivity offers a
number of advantages for high temperature applications, including improved thermal
shock resistance and enhanced thermal radiation by efficient distribution of thermal
energy over available surfaces.
ZrB2 and HfB2 are broadly similar materials, but have nontrivial differences.
Zr and Hf have comparable atomic radii and valence electronic structure, making
their diboride analogues electronically almost indistinguishable. However, the large
mass difference between Zr and Hf atoms lead to significant deviations especially
in the vibrational spectra of the diborides. These differences are expected to affect
their lattice thermal conductivity as well. The crystal structure of ZrB2 and HfB2
is the AlB2-type C32 with space group P6/mmm. These materials are layered with
alternating planes of closed packed hexagonal Zr/Hf and open hexagonal, graphitic
B. Metal atoms are situated directly above/below 6-membered rings in the adjacent
Boron planes. A detailed ab initio study of the structure and properties of these
materials was recently presented6.
High thermal conductivity ceramics such as ZrB2 and HfB2 have significant con-
tributions from both electronic and phononic thermal carriers to the total thermal
conductivity κtot
5. The electronic component κe can be estimated from the electri-
cal conductivity σ using the Weidemann-Franz (WF) empirical relation, κe = κ0σT ,
where κ0 is the Lorentz constant (2.45× 10
−8W ·Ω ·K−2) and T is the temperature.
The phonon part κph cannot be measured directly and is usually inferred by subtract-
ing κe from κtot. The lattice thermal conductivity can be accessed directly however
from molecular dynamics (MD) simulations.
Thermal conductivity measurements for single crystal ZrB2 at room temperature
have been reported as 140W/(m ·K) in the basal direction and 100W/(m ·K) along
2
the c-axis11. These single sample results included neither a characterization of defects
which will reduce κtot nor individual estimates for κe and κph. Thermal conductivity
data has not been reported for single crystal HfB2 although such samples have been
fabricated12.
Thermal conductivity of polycrystalline ZrB2 and HfB2 has been more thor-
oughly studied8–10. Room temperature measurements give 60W/(m · K) for ZrB2
and 104W/(m ·K) for HfB2. The reduction of κtot relative to single crystals comes
primarily from the thermal resistance of grain boundaries. Variations of κtot with grain
size can be substantial8,10. For ZrB2 at 300K, κe was estimated to be 32W/(m ·K)
using the WF relation, giving 28W/(m · K) for κph. As a function of temperature,
κtot for polycrystalline ZrB2 rises gradually up to at least 1800K primarily due to a
monotonically increasing κe. Conversely, κph decreases as 1/T , saturating to a value
of 18W/(m·K) above 1000K which is well above the Debye temperature (TD = 750K)
for this material. ForHfB2, κtot is generally higher than for ZrB2 although a detailed
comparison controlling for defect distributions and microstructure has not been done.
Separate estimates for κph in HfB2 have not been made although behavior similar
to ZrB2 is expected. Based on these results, we estimate the lattice contribution
to κtot to be on the order of 40%. This suggests for single crystals that the lattice
component will be at least 56W/(m ·K) within the planes and 40W/(m ·K) in the
normal direction.
Recently, we developed Tersoff style interatomic potentials7 for ZrB2 and HfB2.
In this paper, we report results of atomistic Green-Kubo simulations of the lattice
thermal conductivity κph for single crystal ZrB2 and HfB2. Direct measurement of
κph is difficult in materials with high κe, therefore, simulations of the lattice compo-
nent can be useful to isolate that particular thermal conduction mechanism. These
computations represent the first atomistic simulations for ZrB2 and HfB2 and also
the first nontrivial test of these new potentials.
3
I. METHOD
The Green-Kubo theorem relates the lattice thermal conductivity tensor κij to the
integral of the ensemble average of the time correlation function of the heat current
Ji(t)
κij =
V
kBT 2
∫ ∞
0
〈Ji(0)Jj(t)〉dt (1)
where i, j = x, y, z, V is the volume, kB is Boltzmann’s constant and T is the tem-
perature. The instantaneous heat current can be evaluated from the phase space
configuration of the system at time t as
~J(t) =
1
V
[
∑
i
ei~vi +
∑
i<j
(~fij · ~vj)~xij ] (2)
where ei is the per atom energy of atom i, ~fij is the force on atom i due to neighbor
j, ~vi is the velocity of atom i and ~xij = ~xi−~xj where ~xi is the position vector of atom
i.
MD simulations were performed using the LAMMPS package14. Systems were
prepared first with a NPT run to allow the volume to expand, followed by a NV E
run to equilibrate the system at the desired T . Timsteps ranging from 0.1fs to 1fs were
used as required to maintain energy conservation. Larger timesteps were generally
used at low T and smaller timesteps at higher T . After equilibration, time series
data for the heat current was collected under NV E. To reduce statistical noise for κ,
long NV E runs were required. For lower temperatures, T = 300K for example, runs
of 10ns with a 1fs timestep were performed. For higher temperatures with smaller
timesteps, runs with a comparable or larger number of steps were used. The largest
number of steps considered was 4 × 107 steps. Eight independent simulations were
performed for each system and at each temperature to control for the randomized
initial velocities.
Correlation functions were evaluated from the heat current time series. To increase
the statistics, correlations were considered in a sliding window through the data set.
This approach gives errors that increase with the correlation time. The integral of
the heat current correlation function (HCCF ) Cij(t) = 〈Ji(0)Jj(t)〉 was evaluated
using the trapezoid rule. Time correlations up to 200ps gave well-converged values
4
for the thermal conductivity.
Recently derived Tersoff style interatomic potentials7 were used for both ZrB2 and
HfB2. For ZrB2, two potentials were developed that were denoted “Pot 1” and “Pot
2”. “Pot 2” however gives lattice thermal conductivity values for κzz that are higher
than the in-plane component which is not consistent with experiment. “Pot 1” does
give the correct ordering, and therefore we focus our investigations using it.
Unit cells contain 12 atoms with 4 Zr/Hf atoms and 8 Boron atoms. Simulation
cells of size 8×8×16 were utilized giving 12,288 atoms. Thermal conductivity values
are well converged with respect to system size at this level.
II. RESULTS
Lattice thermal conductivity was obtained for both ZrB2 and HfB2 at tempera-
tures of T = 300K, 600K, 800K, 1000K. In Fig 1, the normalized HCCF Cxx(t) is
presented for three different systems: ZrB2 at T = 300K, HfB2 at T = 300K and
HfB2 at T = 1000K. Analogous results can be obtained for Cyy and Czz. A dis-
tinctive feature of these curves is their rapid and regular oscillation. The oscillation
frequency and decay for ZrB2 and HfB2 at 300K is similar, reflecting the broad
commonalities of these materials. However, for HfB2 at 1000K, there is a signifi-
cantly stronger damping. These features will be directly reflected in the values for
κ. This oscillatory behavior is different from many HCCF reported in the literature.
The HCCF for Silicon, for example, decays monotonically without any oscillations15.
Oscillatory HCCF have been observed previously, usually in systems with nontrivial
mass differences16.
The HCCF oscillations can be analyzed in more detail by examining the Fourier
transform. In Fig 2, the power spectra for six different HCCF are displayed: the
Cxx and Czz components for ZrB2 at T = 300K and the same components for
HfB2 at T = 300K and at T = 1000K. Interestingly, the spectra are peaked
around single frequencies which can be identified with specific optical phonon modes.
Phonon spectra for these interatomic potentials have been calculated previously7. For
example, for ZrB2 at T = 300K, the power spectra for Cxx spikes at ω = 364cm
−1.
5
This frequency corresponds precisely to the in-plane Zr − B optical phonon mode.
Similarly, the Czz spectra spikes at ω = 510cm
−1 which matches the out-of-plane
Zr − B optical phonon mode. Analogous results are shown for HfB2 at T = 300K.
For HfB2 at 1000K, we see a broadening of the peaks due to anharmonic effects.
Numerical integration of the HCCF is shown in Fig 3 for ZrB2 and HfB2 at
T = 300K as a function of the upper integration limit τ . For each material, two
independent components of the thermal conductivity tensor are shown: κ|| is the
average of the symmetric in-plane directions κxx and κyy, and κzz is normal to the
planes. As can be seen, the curves plateau above 100ps indicating largely converged
integrals. Values for the ZrB2 lattice thermal conductivity when integrated out to
200ps are 60W/(m ·K) for the in-plane component and 41W/(m ·K) in the normal
direction. For HfB2, the values are 74W/(m · K) and 63W/(m · K), respectively.
For both of these materials, κzz is less than κ|| which agrees with the experimental
ordering. Values for HfB2 are also higher than for ZrB2.
Results for theHCCF are relatively free of statistical noise, which is a consequence
of the very long runs we performed. Other methods have been reported in the liter-
ature to obtain reliable numbers from shorter runs. For example, fitting the HCCF
to sums of exponentials has been proposed. However, the correct analytic form is
not known at this time. Therefore, this approach could introduce errors. Modern
computer power appears to be adequate to obtain sufficient statistical convergence
without fitting.
Next we consider the temperature dependence of the lattice thermal conductiv-
ity. In Fig 4, we plot both the in-plane κ|| and the normal κzz conductivities at
four different temperatures T = 300K, 600K, 800K, 1000K for ZrB2 and HfB2. As
expected, there is a rapid decrease as a function of T . In this temperature regime,
anharmonic phonon scattering dominates with the phonon mean free path decreasing
as 1/T . Fits of the different data sets to 1/T are shown in the graph. All four curves
converge to similar values for κ at high T , which at T = 1000K is about 5W/(m ·K).
Compared to experiment however the high T values obtained from our simulations
for κ are probably too low. Lattice thermal conductivity for polycrystalline ZrB2 for
example saturates around 18W/(m ·K) which can be considered a lower bound for
6
our single crystal results. The low κ values at high T most likely reflect a limitation
in the description of the anharmonic part of the interatomic potentials.
III. DISCUSSION
Further insight can be obtained by considering the approximate relation κ¯ph =
1
3
ρCvsλph where ρ is the density, C is the specific heat, vs is the velocity of sound, λph
is the phonon mean free path and κ¯ph is the isotropic lattice thermal conductivity.
Both experimental and computed values for these quantities are given in Table I. The
specific heat of these materials is comparable with a value of ∼ 48J/(mol ·K). The
velocity of sound can be evaluated using vs = 0.87
√
E/ρ where E is the bulk modulus,
giving 5.16km/s for ZrB2 and 3.78km/s for HfB2 based on the experimental values
while giving 5.45km/s for ZrB2 and 5.29km/s for HfB2 based on values from the
interatomic potentials. Since vs ∼
√
1/M , ZrB2 is expected to have a higher vs
than HfB2 which is reflected in the vs derived from experimental numbers. However,
the HfB2 interatomic potential gives a modulus too high by a factor of 2X, thus
the theoretical values of vs for the two materials are comparable. Estimates for the
phonon mean free path can be obtained using experimental and simulation results
for κ¯ph. For ZrB2 at 300K, an experimental estimate of 51W/(m ·K) for κ¯ph gives
λph = 11.3nm while the simulation value of 54W/(m ·K) for κ¯ph gives λph = 11.4nm.
These results are in very good agreement, although an accurate experimental estimate
for κ¯ph requires further study. For HfB2 at 300K, experimental data is not available.
The simulation value of 70W/(m ·K) for κ¯ph gives λph = 14.3nm for that material.
In this paper, atomistic Green-Kubo simulations for the lattice thermal conduc-
tivity of ZrB2 and HfB2 were performed for a range of temperatures. Interatomic
potentials for these materials were recently developed and this paper represents the
first non-trivial test of those potentials. Oscillations in the heat current correlation
functions can be identified with mixed Zr/Hf−B optical phonon modes arising from
the large metal/Boron mass difference. Agreement with available experimental results
is very good at room temperature, but is probably too low at higher temperatures
e.g. T = 1000K.
7
IV. ACKNOWLEDGMENTS
J.W.L. and C.W.B. are civil servants in the Entry Systems and Technology Divi-
sion. M.S.D was supported under a NASA prime contract to ELORET Corporation.
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9
ρ(g/cm3) Tmelt(K) TD(K) C(J/(mol ·K)) E(GPa) vs(km/s) λph(nm) κ¯ph(W/m ·K)
ZrB2 Exp. 6.119 3518 750 48.2 215 5.16 11.3 51
MD 240 5.45 11.4 54
HfB2 Exp. 11.212 3653 565 49.5 212 3.78 N/A N/A
MD 415 5.29 14.3 70
TABLE I. Room temperature properties for single crystal ZrB2 and HfB2 where ρ is the
density, Tmelt is the melting temperature, TD is the Debye temperature, vs is the velocity
of sound calculated from vs = 0.87
√
E/ρ where E is the bulk modulus, λph is the phonon
mean free path calculated from κ¯ph =
1
3ρCvsλph and κ¯ph is the isotropic lattice thermal
conductivity.
10
-1
-0.5
 0
 0.5
 1
 0  5  10  15  20
<
J(τ
)J(
0)>
/<J
(0)
J(0
)>
τ (ps)
ZrB2 300K
HfB2 300K
HfB2 1000K
-1
 0
 1
 0  0.25  0.5
FIG. 1. Normalized heat current correlation function (HCCF ) for Cxx(τ) as a function
of time. Unlike monoatomic systems, the HCCF for ZrB2 and HfB2 have rapid, regular
oscillations.
11
 300  350  400  450  500  550  600
Po
w
er
 (a
rb.
 un
its
)
ω (cm-1)
Cxx(ω) Czz(ω)
ZrB2 300K
HfB2 300K
HfB2 1000K
FIG. 2. Power spectra of six heat current correlation functions obtained from Fourier
transformation. Low temperature spectra are peaked around a single frequency identified
with mixed metal-Boron optical phonon modes.
12
 0
 20
 40
 60
 80
 100
 120
 0  50  100  150  200
κ
 
(W
/(m
.K
))
τ (ps)
HfB2 κ||
ZrB2 κ||
HfB2 κzz
ZrB2 κzz
FIG. 3. Integrated HCCF as a function of the upper integration limit τ giving both in-
plane (x-y plane) κ|| and out-of-plane (z-direction) κzz conductivities at 300K for ZrB2
and HfB2.
13
 0
 20
 40
 60
 80
 100
 400  600  800  1000  1200  1400
κ
 
(W
/(m
.K
))
T (K)
HfB2 κ||
ZrB2 κ||
HfB2 κzz
ZrB2 κzz
FIG. 4. In-plane κ|| and out-of-plane κzz lattice thermal conductivity as a function of
temperature for ZrB2 and HfB2. Curves are fits of the data to 1/T .
14


Lattice Thermal Conductivity of Ultra High Temperature Ceramics ZrB2 and HfB2 from Atomistic Simulations

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