Review of Electrical Resistivity Measurements and Calculations of Fe and Fe-Alloys Relating to Planetary Cores

At a Glance

Metadata	Details
Publication Date	2021-09-10
Journal	Frontiers in Earth Science
Authors	Meryem Berrada, Richard A. Secco
Institutions	Western University
Citations	25
Analysis	Full AI Review Included

Executive Summary

Core Research Focus: This review synthesizes theoretical and experimental data on the electrical resistivity (ρ) and thermal conductivity (k) of pure Fe and Fe-alloys (Ni, O, Si/S) under the extreme high-pressure (P) and high-temperature (T) conditions found in terrestrial planetary cores.
Critical Constraint: Accurate ρ values are essential for calculating core thermal conductivity (k) via the Wiedemann-Franz Law (WFL), which dictates the adiabatic heat flow (q_ad) required to sustain planetary dynamos and model thermal evolution.
Earth Core Findings: Average ρ values for Fe at Earth’s Core-Mantle Boundary (CMB) and Inner-Core Boundary (ICB) are centered around 1.07 µΩm and 0.97 µΩm, respectively. The inclusion of light elements (Ni, O, Si/S) results in ρ values that remain largely within the uncertainty range of pure Fe measurements.
Methodological Discrepancies: Significant variations in reported ρ values exist (ranging from 0.25 µΩm to 3.7 µΩm for pure Fe at Earth’s CMB), often attributed to experimental artifacts (e.g., laser misalignment in DAC, shunting effects in shock compression) or large extrapolations from low P-T data.
Planetary Core Variability: The effect of light element impurity scattering is highly dependent on the planetary body. The effect of Si and S is negligible for Mars’ core, but Fe-S compositions at Ganymede’s CMB show large deviations in ρ (up to 8.01 µΩm) compared to pure Fe.
Modeling Utility: Contour maps of ρ(P, T) generated from the compiled literature provide a generalized tool for estimating electrical resistivity across the P-T regimes of Earth, Moon, Mercury, Mars, and Ganymede cores.

Technical Specifications

Parameter	Value	Unit	Context
Earth CMB Pressure (P)	136	GPa	Reference condition for Earth’s outer core top
Earth ICB Pressure (P)	330	GPa	Reference condition for Earth’s inner core boundary
Earth CMB Temperature (T)	4,000	K	Reference condition for Earth’s outer core top
Earth ICB Temperature (T)	5,000	K	Reference condition for Earth’s inner core boundary
Average ρ (Pure Fe, Earth CMB)	1.07	µΩm	Mean resistivity value reported in literature
Average ρ (Pure Fe, Earth ICB)	0.97	µΩm	Mean resistivity value reported in literature
Lorenz Number (Sommerfeld, L₀)	2.44 x 10^-8	WΩK^-2	Theoretical value used for WFL conversion (k_e = L₀T/ρ)
ρ (Fe-Ni, Earth CMB Range)	0.58 - 2.77	µΩm	Range for Fe alloys containing 5-51.6 wt% Ni
ρ (Fe-Si/S, Earth CMB Range)	0.60 - 4.6	µΩm	Range for Fe alloys containing 1.8-35.5 wt% Si or S
ρ (Fe36.5S, Ganymede CMB)	4.12 ± 0.07	µΩm	Measured value at 1,411 K, 5 GPa (Littleton et al., 2021)
ρ (Fe, Mercury CMB)	0.88	µΩm	Average resistivity at 1,900 K, 5 GPa
ρ (Fe, Mars CMB)	0.95	µΩm	Average resistivity at 1,770 K, 23 GPa
Maximum Shock Compression P	208	GPa	Achieved in shock wave experiments on Fe

Key Methodologies

The determination of electrical resistivity (ρ) and thermal conductivity (k) at planetary core conditions relies on a combination of static compression, dynamic compression, and advanced theoretical modeling:

Experimental Techniques (High P-T Generation)

Diamond-Anvil Cell (DAC):
- Used for static compression up to 170 GPa and laser heating up to 3,000 K.
- Resistivity (ρ) is typically measured using the four-probe method or the Van der Pauw method.
- Direct thermal conductivity (k) measurements are achieved by monitoring heat pulse propagation across Fe foils.
Multi-Anvil Press (MAP):
- Used for static compression up to 24 GPa and high T (up to 2,200 K).
- Employs a four-wire measurement technique for ρ, often used to constrain liquid Fe and Fe-alloy behavior near the melting boundary at lower pressures.
Shock Compression:
- Dynamic method using metallic flyer plates to generate simultaneous high P (up to 208 GPa) and high T.
- Conductivity (σ) is measured via contact or contactless methods, relating to electromotive forces generated by the shock wave.

Theoretical and Computational Methods

First-Principles Calculations (FPC):
- Utilizes Density-Functional Theory (DFT) and Dynamical Mean-Field Theory (DMFT) to model electronic band structure and electron scattering mechanisms (electron-phonon, electron-magnon, electron-electron).
Kubo-Greenwood Formula:
- A standard FPC approach used to calculate frequency-dependent electron conductivity (σ), yielding the linear contribution to current response and resistivity (ρ).
Boltzmann Equation:
- Used in FPC to describe the diffusion of electrons, considering electronic band structure, phonon dispersion, and electron-phonon interactions to calculate ρ.
Wiedemann-Franz Law (WFL):
- Used to convert measured or calculated electrical resistivity (ρ) into electronic thermal conductivity (k_e) using the relationship k_e = L*T/ρ, where L is the Lorenz number (L₀ is the Sommerfeld value).

Commercial Applications

The fundamental research on transport properties of Fe and Fe-alloys under extreme P-T conditions has direct relevance to several engineering and scientific domains:

Geophysical Modeling and Resource Exploration:
- Core Heat Flux Determination: Accurate ρ and k values are essential inputs for thermal evolution models, determining the heat flow (q_ad) from the core, which influences mantle convection and the viability of planetary magnetic fields (dynamos).
- Planetary Interior Structure: Data on Fe-alloy behavior under pressure helps constrain the composition and phase stability of cores in terrestrial bodies (e.g., determining the light element content of Earth’s inner core).
High-Pressure/High-Temperature Material Synthesis:
- Extreme Environment Materials: The methodologies (DAC, MAP, Shock Compression) and resulting data inform the synthesis and characterization of materials designed to operate under extreme stress, pressure, and temperature, relevant to industrial processes or specialized equipment (e.g., high-pressure vessels, deep-earth drilling components).
Computational Physics and Simulation:
- Validation of Transport Codes: Experimental ρ and k data provide crucial validation points for advanced computational models (DFT, MD) used to predict the behavior of metallic alloys and high-energy density matter where direct physical testing is impractical or impossible.
Shock Physics and Defense:
- Equation of State (EOS) Development: Shock compression data on Fe alloys are used to refine the EOS models for metals, critical for simulations involving high-velocity impacts, blast dynamics, and material response under rapid compression.

View Original Abstract

There is a considerable amount of literature on the electrical resistivity of iron at Earth’s core conditions, while only few studies have considered iron and iron-alloys at other planetary core conditions. Much of the total work has been carried out in the past decade and a review to collect data is timely. High pressures and temperatures can be achieved with direct measurements using a diamond-anvil cell, a multi-anvil press or shock compression methods. The results of direct measurements can be used in combination with first-principle calculations to extrapolate from laboratory temperature and pressure to the relevant planetary conditions. This review points out some discrepancies in the electrical resistivity values between theoretical and experimental studies, while highlighting the negligible differences arising from the selection of pressure and temperature values at planetary core conditions. Also, conversions of the reported electrical resistivity values to thermal conductivity via the Wiedemann-Franz law do not seem to vary significantly even when the Sommerfeld value of the Lorenz number is used in the conversion. A comparison of the rich literature of electrical resistivity values of pure Fe at Earth’s core-mantle boundary and inner-core boundary conditions with alloys of Fe and light elements (Si, S, O) does not reveal dramatic differences. The scarce literature on the electrical resistivity at the lunar core suggests the effect of P on a wt% basis is negligible when compared to that of Si and S. On the contrary, studies at Mercury’s core conditions suggest two distinct groups of electrical resistivity values but only a few studies apply to the inner-core boundary. The electrical resistivity values at the Martian core-mantle boundary conditions suggest a negligible contribution of Si, S and O. In contrast, Fe-S compositions at Ganymede’s core-mantle boundary conditions result in large deviations in electrical resistivity values compared to pure Fe. Contour maps of the reported values illustrate ρ( P , T ) for pure Fe and its alloys with Ni, O and Si/S and allow for estimates of electrical resistivity at the core-mantle boundary and inner-core boundary conditions for the cores of terrestrial-like planetary bodies.

Tech Support

Original Source

DOI: https://doi.org/10.3389/feart.2021.732289

References

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