Electrical Resistivity of Substitutional Disordered hcp Fe

46th Lunar and Planetary Science Conference (2015)
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Electrical Resistivity of Substitutional Disordered hcp Fe-Si and Fe-Ni Alloys at High Pressure: Implications
for Core Energy Balance. H. Gomi1, K. Hirose2, H. Akai3, Y. Fei1, 1Geophysical Laboratory, Carnegie Institution
of Washington, 5251 Broad Branch Road, N.W., Washington, DC 20015 ([email protected]), 2Earth-Life
Science Institute, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan, 3The Institute for Solid State
Physics, University of Tokyo, Kashiwanoha 5-1-5, Kashiwa, Chiba 277-8581, Japan.
Introduction: There is a delicate energy balance
between the energy requirement for driving a dynamo
process in a metallic fluid core and the heat transfer by
convection or conduction. A better constrained value
of the thermal conductivity of a planetary core is critically important to model the generation of planet magnetic field through self-sustained dynamo and evolution
of the core through its history. It also controls the
thermal evolution of the planets. The thermal conductivity of the metallic core can be estimated from the
electrical resistivity via the Wiedemann-Franz law, k =
LT/ρ, where k is the thermal conductivity, L = 2.45 ×
10-8 WΩ/K2 is the Lorenz number, T is the temperature
and ρ is the electrical resistivity. Gomi et al. [1]
demonstrated that the resistivity saturation phenomenon is dominant to determine the resistivity at high
temperature and high concentration of impurity elements, pertinent to the core conditions.
The physics of the resistivity saturation is largely
investigated by theoretical studies [2]. The basic idea is
that the mean free path of conduction electrons should
not become shorter than the interatomic distance,
which is so-called “Ioffe-Regel condition” [3]. Butler
[4] developed the first-principles calculation method of
the resistivity of disordered alloys combined with the
coherent potential approximation (CPA), which is applied to chemically, thermally and spin disordered metals and alloys [5].
In this study, we focused on chemically disordered
iron alloys and demonstrated that the resistivity of
planetary cores is largely constrainted by the resistivity
saturation.
Methods: We have conducted experiments to
measure the electrical resistivity at high pressure in a
diamond anvil cell (DAC) up to 90 GPa. A rhenium
foil with Al2O3 insulating layer was used as gasket material. The sample resistance was obtained by the fourterminal method under a constant DC current of 10 mA
with a digital multi-meter (ADCMT 6581). Five separate runs were conducted using iron-silicon alloys with
different Si contents. The samples were foils of Fe alloyed with 1wt.%Si (1.97 at.%Si), 2wt.%Si (3.90
at.%Si), 4 wt. %Si (7.65 at.%Si), 6.5 wt.% Si (12.14
at.%Si) and 9 wt.%Si (16.43 at.%Si) (Rare Metallic)
with initial thickness of ~10 μm.
We have also performed first-principles calculations by using the AkaiKKR (machikaneyama) package,
which employed the Korringa-Kohn-Rostoker (KKR)
method with the atomic sphere approximation (ASA)
within a framework of the local density approximation
(LDA) for exchange-correlation potential. The relativistic effects are taken into account within the scalar
relativistic approximation. The wave functions are calculated up to l = 3, where l is the angular momentum
quantum number. The coherent potential approximation (CPA) is adopted to treat the substitutional disorder effect on the electronic band structure [6]. The
electrical resistivity is calculated from the KuboGreenwood formula [4].
Results: Figure 1 shows the measured electrical
resistivity of iron-silicon alloys as a function of pressure at 300 K with different Si contents. A sharp resistivity enhancement was observed across the bcc-hcp
phase boundary, which is consistent with previous
studies [1, 7]. The resistivities of iron-silicon alloys are
very close to that of Gomi et al. [1] and Seagle et al.
[7] at around 20 GPa, but slightly lower than Gomi et
al. [1] at higher pressure. The resistivity increases linearly with increasing silicon impurity concentration as
predicted by the Matthiessen’s rule.
Fig.1 Electrical resistivity of iron-silicon alloys as a
function of pressure. Yellow (1 wt.% Si), light green (2
wt.% Si), deep green (4 wt. % Si), blue (6.5 wt.% Si)
and purple (9 wt.% Si) circles and diamonds are present measurement. Previous measurements conducted
by Gomi et al. [1] (Red and orange circle for pure Fe
and gray and white square for 2 wt.% Si alloy) and
Seagle et al. [7] (purple triangle for 9 wt.% Si alloy)
were also plotted for comparison.
46th Lunar and Planetary Science Conference (2015)
The electrical resistivities of iron-silicon and iron
nickel alloys are calculated by using the KuboGreenwood formula [4] at several volumes, equivalent
to experimentally determined room-temperature volumes of pure hcp iron at 0, 40, 80 and 120 GPa, respectively [8]. Figure 2 shows the electrical resistivities
of iron-silicon and iron-nickel alloys. The electrical
resistivity increases with increasing impurity content up
to about 50 at%, then the resistivity decreases to zero.
To simplify, we considered the average as the resistivity of bulk polycrystalline alloys: (2ρx + ρz)/3 where ρx
and ρz are the resistivity calculated perpendicular and
parallel to the c-axis, respectively.
Fig.2 Present (circile) and previous (square) electrical resistivity of iron-silicon (blue) and iron-nickel
(green) alloys as a function of silicon and nickel concentration. The open symbols indicate experimental
results and the filled symbols represent values from
first-principles calculations.
Discussion: Figure 2(a) summarized the electrical
resistivities of iron-silicon and iron-nickel alloys at
ambient pressure. Blue open square symbols indicate
the measured resistivity of single-phase iron-silicon
alloys at ambient temperature [9 and references therein]. The resistivity increases with increasing silicon
concentration within the bcc stability field, and then the
resistivity decreases because of formation of Fe3Si
component with DO3 structure. A local minimum is
observed at the stoichiometric composition. They also
found unexpected high resistivity at χSi ≈ 35 at.%,
which may be related to the DO3 structure. Literature
values of iron nickel solid solutions at 300 K are plotted as green broken line [10]. Impurity resistivity of
nickel in iron is smaller than that of silicon. The resistivity is increases with increasing nickel impurity concentration within the bcc stability field. The discontinu-
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ity in resistivity at χNi ≈ 30 at.% is attribute to the phase
transition from bcc to fcc. Then, the resistivity decreases with increasing the nickel content. First-principles
results on hcp iron-nickel solid solutions are plotted as
solid green circles for comparison. Figures 2 (b, c, d)
show comparison of our experimental results with firstprinciples calculations and previous shock compression
experiments [11]. Our experimental and first-principles
results of iron-silicon alloys show excellent agreement
with each other. Furthermore, previous shock wave
data [11] are in general agreement with our results,
except for the value at 34.2 at.% Si, although these data
are taken along Hugoniot. This is consistent with the
prediction from the saturation theory that temperature
dependence should become small at high resistivity [1].
In addition, the first-principles result of iron-silicon
alloys clealy shows a kink at χSi ≈ 15 at.%, which is
considered to be the result of the resistivity saturation.
Because both thermally and chemically disordered alloys follow the resistivity saturation, electrical resistivity should become about 1 × 10-6 Ωm at strongly disordered condition.
Terrestrial planetary cores consists of iron-nickel
alloys with different amounts of impurity such as Si
depending on the accretion consitions and size of the
planets. This study provides systematic data and theoretical predictions that reveal the effect of Ni and Si
concentractions on the electrical resistivity over a wide
pressure range. The derived thermal conductivity values via the Wiedemann-Franz law can be directly used
to model conductive energy losses from the core to
power the dynamo. Understanding the effect of Si on
the thermal conductivity is particularly important for
Mercury because Si could be a major light element in
the core [12,13]. Sigificant increase in resistivity with
Si impurity would substantially low the capacity for the
core to transport heat by conduction.
References: [1] Gomi et al. (2013) PEPI. [2]
Gunnarsson et al. (2003) Rev. Mod. Phys. [3] Gurvitch
(1981) PRB. [4] Butler (1985) PRB. [5] Glasbrenner et
al. (2014) PRB. [6] Akai (1989) J. Phys.: Cond. Mat.
[7] Seagle et al. (2013) GRL [8] Dewaele et al. (2006)
PRL [9] Varga et al (2002) J. Phys.: Cond. Mat. [10]
Ho et al. (1983) J. Phys. Chem. Ref. [11] Matassov
(1977) PhD thesis. [12] Fei et al. (2011) 42nd LPSC
Abstract #194. [13] Malavergne V. et al. (2010) Icarus,
206, 199-209.