[Paper Review] Ab initio melting temperatures of bcc and hcp iron under the Earth's inner core condition

Yang Sun, Mikhail I. Mendelev+6 more|arXiv (Cornell University)|May 4, 2022

High-pressure geophysics and materials4 citations

TL;DR

This study computes the ab initio melting temperatures of bcc and hcp iron under Earth's inner core conditions using a hybrid thermodynamic integration (TI) approach that combines classical molecular dynamics with ab initio calculations. By minimizing finite-size effects and uncertainty through a smooth coupling path, the method confirms that hcp iron is the thermodynamically stable phase, with a free energy difference of only ~10s meV/atom between hcp and bcc near melting, indicating near-degeneracy of the two phases at high pressure and temperature.

ABSTRACT

There has been a long debate on the stable phase of iron under the Earth's inner core conditions. Because of the solid-liquid coexistence at the inner core boundary, the thermodynamic stability of solid phases directly relates to their melting temperatures, which remains considerable uncertainty. In the present study, we utilized a semi-empirical potential fitted to high-temperature ab initio data to perform a thermodynamic integration from classical systems described by this potential to ab initio systems. This method provides a smooth path for thermodynamic integration and significantly reduces the uncertainty caused by the finite-size effect. Our results suggest the hcp phase is the stable phase of pure iron under the inner core conditions, while the free energy difference between the hcp and bcc phases is tiny, on the order of 10s meV/atom near the melting temperature.

Motivation & Objective

• To resolve the long-standing uncertainty in the melting temperatures of bcc and hcp iron under extreme Earth's inner core conditions.
• To address the significant uncertainty in experimental and simulation-based melting temperature measurements, which currently range from 4,850 K to 7,600 K.
• To develop and apply a thermodynamically consistent method that reduces finite-size effects and improves accuracy in ab initio melting point calculations.
• To determine the thermodynamic stability of hcp versus bcc iron by computing the free energy difference near the melting temperature.

Proposed method

• The study employs thermodynamic integration (TI) from a classical system (with a semi-empirical potential) to an ab initio system (using density functional theory) to compute the free energy difference.
• The TI path is constructed via a coupling parameter λ that interpolates between the classical potential and ab initio forces, enabling smooth free energy evaluation.
• The method uses the Gibbs-Helmholtz equation to compute the classical melting temperature and free energy difference, which are then corrected via TI and PV work terms.
• The free energy perturbation (FEP) method is applied to improve accuracy by re-evaluating ab initio energies from high-efficiency DFT trajectories using higher-accuracy DFT settings.
• Equations (S3)–(S11) formalize the decomposition of the ab initio free energy difference into classical melting contribution, TI correction, and pressure-volume work terms.
• The method accounts for volume and pressure differences between classical and ab initio systems, ensuring consistency in thermodynamic integration.

Experimental results

Research questions

• RQ1 Which phase—bcc or hcp—has the higher melting temperature for pure iron under Earth's inner core pressure and temperature?
• RQ2 How accurately can ab initio melting temperatures be computed when finite-size effects and potential inaccuracies are minimized?
• RQ3 What is the free energy difference between hcp and bcc iron near the melting temperature under inner core conditions?
• RQ4 Can the thermodynamic stability of iron phases be reliably assessed when the melting temperature difference between phases is small?
• RQ5 How does the combination of classical and ab initio simulations reduce uncertainty in melting point predictions?

Key findings

• The hcp phase is the thermodynamically stable phase of pure iron under Earth's inner core conditions.
• The free energy difference between hcp and bcc iron is on the order of 10s meV/atom near the melting temperature, indicating near-degeneracy of the two phases.
• The melting temperature of hcp iron is computed with significantly reduced uncertainty compared to prior ab initio studies, which had uncertainties of ~100–500 K.
• The method successfully reduces finite-size effects by using a smooth coupling path between classical and ab initio systems.
• The free energy perturbation (FEP) method confirms the reliability of the ab initio energy differences, with fluctuations of 3–8 meV/atom, much smaller than kBT (~500 meV/atom) at 6000 K.
• The study provides the first ab initio melting temperature for bcc iron under inner core conditions via thermodynamic integration, resolving a key gap in prior work.