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Comprehensive calculations of Fe-Si system via deep learning force field: the core structure of Mercury

This study employs a deep learning force field framework to comprehensively calculate the physical properties of the Fe-Si system, revealing that Mercury's core likely consists of a solid inner core overlaid by a thermally stratified layer.

Original authors: Wei-Jie Li, Zi Li, Weijie Wang, Wen-shuo Liu, Cong Wang, Ping Zhang

Published 2026-08-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Wei-Jie Li, Zi Li, Weijie Wang, Wen-shuo Liu, Cong Wang, Ping Zhang

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Cosmic Detective Story: Peering into Mercury's Heart

Imagine trying to figure out what's inside a giant, glowing marble that's hurtling through space, but you can't touch it, you can't drill into it, and you can't even see inside it. That's the challenge scientists face when studying the cores of planets like Mercury. To understand these hidden worlds, researchers rely on a special branch of physics called materials science, which asks: "How do atoms behave when they are squeezed by immense pressure and heated to temperatures hotter than the surface of the Sun?"

The key to solving this puzzle lies in understanding how different materials, specifically mixtures of iron and silicon, act under these extreme conditions. Think of it like trying to predict how a pot of soup will behave if you crank the stove to maximum and then squeeze the pot with a hydraulic press. Does the soup freeze? Does it boil? Does it get thicker or thinner? In the real world, testing this is incredibly hard and expensive because you need massive machines to create that kind of pressure. However, scientists have developed a new "superpower" called deep learning. This is a type of artificial intelligence that learns from the laws of physics to predict how atoms move without needing to run the most expensive, slow simulations every single time. By combining this AI with the laws of quantum mechanics, researchers can now build a virtual model of a planet's core and watch it evolve, helping us understand why planets have magnetic fields, how they cool down, and what their insides are actually made of.

Cracking the Code of Mercury's Core

In this study, a team of researchers used a clever mix of artificial intelligence and high-tech physics simulations to build a detailed map of Mercury's core. They focused on a specific mixture of iron and silicon (with up to 25% silicon), which is believed to be the main ingredient of the planet's heart. Instead of just guessing, they trained a "deep learning force field"—think of it as a super-smart robot that learned the rules of how iron and silicon atoms dance together by studying millions of examples from a computer program that knows the laws of physics perfectly. Once the robot learned the rules, it could run simulations of the entire core much faster than before, allowing the scientists to calculate a whole bunch of properties all at once, from how the material squishes under pressure to how it conducts heat.

The results paint a surprisingly specific picture of Mercury's interior. The simulations suggest that Mercury likely has a solid inner core, much like Earth does, but it didn't form the way some scientists previously thought. For a while, there was a popular idea called "iron snow," where solid iron would form at the very top of the core and rain down like snowflakes. However, the data in this paper rules that out. The calculations show that the temperature and pressure curves inside Mercury don't line up in a way that would let snow form at the top. Instead, the solid core likely formed from the bottom up, growing inward over billions of years.

Another major finding concerns how heat moves through this core. The team calculated that the total heat conductivity of Mercury's core is about 48.0 W/m/K. This number is high enough to suggest that the very top layer of the core might be a "thermal stratification" zone. Imagine a layer of warm soup sitting on top of a cooler layer that refuses to mix; this is what might be happening at the top of Mercury's core, creating a stable layer that doesn't churn around. This could explain why Mercury's magnetic field might have behaved differently in the past compared to today.

The researchers also looked at how "thick" or "runny" the liquid metal is. They found that at high temperatures, the liquid flows easily with a viscosity in the order of mPa s and atoms move around with a diffusion coefficient of 10⁻⁹ m² s⁻¹. But, as the temperature drops to around 1600 K, the liquid gets much thicker (viscosity jumps to 45 mPa s) and the atoms stop moving as freely. This change confirms that the material is indeed turning solid at lower temperatures, supporting the idea of a solid inner core.

So, what does this all mean for the story of Mercury? The paper suggests a timeline where Mercury started as a completely liquid ball of iron and silicon. As it cooled, a solid core began to grow from the bottom up, not the top down. Today, we likely have a solid inner core surrounded by a liquid outer layer, with a possible calm, stratified layer sitting right at the top of the core. While the scientists are very confident in these numbers based on their simulations, they note that this is a theoretical picture. They didn't find any "iron snow" or a completely liquid core, but they did find that the core's structure is complex and depends heavily on how much silicon is mixed in. It's a vivid, simulated glimpse into a world we can't visit, showing us that Mercury's heart is a solid, growing sphere wrapped in a hot, flowing, and perhaps slightly layered shell.

Technical Summary: Comprehensive Calculations of Fe-Si System via Deep Learning Force Field

Problem Statement
The internal structure and thermal evolution of Mercury's core remain poorly understood due to a lack of observational constraints and the limitations of traditional high-pressure experiments, which often provide only isolated physical quantities. Existing models frequently treat thermoelastic properties as fixed values, disregarding their variations with temperature, pressure, and composition. Furthermore, theoretical approaches face a trade-off: ab initio molecular dynamics (AIMD) offers high accuracy but is computationally prohibitive for large systems and long timescales, while classical molecular dynamics (MD) lacks the necessary transferability and accuracy for complex chemical environments. There is a critical need for a unified framework that can systematically extract the variation patterns of all relevant physical quantities (equation of state, thermoelastic properties, melting curves, and transport properties) as functions of state variables to accurately model Mercury's core.

Methodology
The authors propose a hybrid framework combining ab initio calculations with deep neural networks to achieve ab initio accuracy at a reduced computational cost.

Deep Learning Force Field (DLFF): A deep potential energy surface was constructed using the DeePMD-kit software. The model was trained on data generated from AIMD simulations (VASP code) using the PBE functional, PAW potentials, and a Hubbard U model (U ≈ 4.3 eV) to account for magnetic effects in Fe-Si liquids. The training dataset included 15,000 configurations from 47 isothermal-isobaric AIMD trajectories, with a root mean square error (RMSE) of ~1 meV for energy and ~50 meV/Å for forces.
Molecular Dynamics Simulations: Large-scale MD simulations (up to 2,000 atoms) were performed using the trained DLFF within the LAMMPS package. These simulations covered Fe-Si systems with Si concentrations ranging from 5 to 25 at.% under pressures of 6–40 GPa and temperatures of 1600–3000 K.
Property Calculation:
- Equation of State (EOS): Multivariate polynomial fitting was applied to P-ρ-T and E-ρ-T data to derive thermoelastic properties (bulk modulus, thermal expansion, Grüneisen parameter, sound velocity).
- Melting Curves: Thermodynamic integration (TI) methods were employed to calculate the Gibbs free energy difference between liquid and solid (bcc) phases, identifying melting points where $G_{liquid} = G_{solid}$ .
- Transport Properties: Electronic thermal conductivity was calculated via the Kubo-Greenwood formula using AIMD snapshots. Lattice thermal conductivity, viscosity, and diffusion coefficients were derived from MD simulations using Green-Kubo formulas and Einstein relations.
- Core Modeling: A 1D spherically averaged thermodynamic model was used to solve for adiabatic temperature and pressure profiles, incorporating the calculated EOS and boundary conditions at the core-mantle boundary (CMB).

Key Contributions and Results

Unified Physical Property Coverage: The study successfully generated a comprehensive dataset for the Fe-Si system (<25 at.% Si) covering the equation of state, thermoelastic properties, adiabatic curves, melting curves, and transport properties with ab initio accuracy.
Validation of DLFF: The deep learning force field successfully reproduced AIMD-level accuracy for pressure and energy, with discrepancies falling within the statistical error of AIMD (1%). This validates the use of DLFF for calculating complex properties like viscosity and diffusion in large systems.
Melting and Adiabatic Curves: The calculated melting curves for Fe-10 at.% Si and Fe-25 at.% Si show that a significant portion of Mercury's core is likely solid. The adiabatic curves are relatively flat compared to the melting curves.
Transport Properties:
- Viscosity and Diffusion: At low temperatures (1600 K), the system exhibits high viscosity (~~45 mPa s) and low diffusion coefficients (~~0.71 × 10⁻⁹ m² s⁻¹), indicating a solid phase. At higher temperatures (2000 K), viscosity drops to ~6.4 mPa s and diffusion increases to ~4.0 × 10⁻⁹ m² s⁻¹, consistent with liquid behavior.
- Thermal Conductivity: The total thermal conductivity is estimated at 48.0 W/m/K (electronic: ~~46.6 W/m/K; lattice: ~1.4 W/m/K). This value is significantly higher than that of FeS (~~4 W/m/K) but lower than non-magnetic Fe liquid in Earth's core.
Mercury's Core Structure:
- Solid Inner Core: The intersection of the adiabatic and melting curves suggests the existence of a solid inner core, comprising approximately half to two-thirds of the core's radius. The Si concentration in the solid inner core is inferred to decrease with radius.
- Rejection of Iron Snow: The slopes of the calculated adiabatic and melting curves ( $dP_{ad}/dT_{ad} > dP_{melt}/dT_{melt}$ ) do not support top-down crystallization (iron snow) mechanisms.
- Thermal Stratification: The high thermal conductivity (48.0 W/m/K) results in an adiabatic heat flux (~20 mW m⁻²) that exceeds estimates for the core-mantle boundary heat flux (10–15 mW m⁻²). This condition ( $Q_{ad} > Q_{cmb}$ ) suggests the possibility of thermal stratification on top of the core.

Significance and Claims
The paper claims to provide a "most possible physical picture" of Mercury's core structure and thermal evolution based on the calculated properties. The authors conclude that Mercury's core likely contains a solid inner core overlain by a thermally stratified layer, while explicitly ruling out the "iron snow" mechanism (top-down crystallization) under the current modeled conditions.

The study emphasizes that these conclusions are derived specifically for Fe-Si systems with Si concentrations below 25 at.%. The authors modestly note that their melting curve calculations focus on the liquid-to-bcc transition and do not account for the kinetics of complex solid-phase transitions or other potential elements (e.g., S) that could lead to immiscibility or different thermal conductivities. The work serves as a foundational dataset for future energy and entropy budget modeling of Mercury's thermal evolution.

The Cosmic Detective Story: Peering into Mercury's Heart

Cracking the Code of Mercury's Core

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