Model for the Thermodynamics of Iron at High Pressures Near Melting

Abstract

The Fe pressure-temperature phase diagram and its melting line have a wide range of applications, including providing constraints for iron-core planetary models. We propose an equation of state (EOS) model based on the interstitial theory of simple condensed matter (ITCM), as suggested by A.V. Granato. When applied to Fe, this model enables the extrapolation of measured melting lines to the conditions of the Earth's inner core boundary (ICB). The ITCM describes the solid-liquid phase transition in metals as resulting from a strong structural perturbation due to a high concentration of interstitial-like defects. The strong nonlinearity of their self-interaction causes the stabilization of this interstitial-rich phase. The original model is expanded to describe melting over a wide range of pressures and temperatures rather than focusing on a specific isobaric transition. Using this model, we fit the measured melting data, extrapolate it to cover ICB conditions, and develop a multiphase equation of state that encompasses this regime. The model is used to explain contradictory data regarding the location of the melting line, resulting from a novel phase transition between two separate liquid phases, specifically between FCC-based and HCP-based liquids. This additional liquid phase offers a new interpretation of the previously suggested near-melting high-pressure phase and may also provide a solution to the inner core nucleation paradox.

Figures (9)

• Figure 1: Fe phase diagram derived from our multiphase EOS. Light green indicates the $\alpha (BCC)$ phase, blue indicates $\gamma (FCC)$, red indicates $\varepsilon (HCP)$, light blue indicates $L_{FCC}$, pink indicates $L_{HCP}$. White markers indicate the Shock melt experiments, where each marker indicates a different experiment: squareBrownMcQueen, diamondYooHolmes, circleNguyen2004, downward triangleHermand, and upward triangleBalugani2024. The black dashed line indicates the $\gamma-\varepsilon$ phase transitionSwartzendruber1982. The orangeSINMYO2019, greenAnzellini2013, light greenBoehler1993 and grayAquilanti2015 circle indicates the heated DAC melt experiments. The light blue star indicates heated DAC stability measurements of $\varepsilon(HCP)$Tateno2010.
• Figure 2: The change in Gibbs free energy as a function of the interstitial concentration $\Delta G(c)$ at $200 [\text{GPa}]$ at different temperatures near melting.
• Figure 3: The Hugoniot possible trajectories near melting. The red dotted lines indicate the $\varepsilon - L_{HCP}$ and the $L_{HCP} - L_{FCC}$ phase transitions. The solid blue line represents the derived Hugoniot trajectory, while the dashed blue line indicates a metastable branch that passes through the edges of the solid stability regime. The markers represent different shock melting measurements: gray circlesBrownMcQueen, light blue diamondsYooHolmes, green trianglesNguyen2004, yellow squaresHermand, and pink starsBalugani2024.
• Figure 4: Isotherms of (a) the density $\rho$ and (b) the entropy $S$ per gram at different temperatures. The vertical lines represent phase transitions along the isotherms, where the larger jumps indicate melting.
• Figure 5: Comparison between experimental data and our EOS fits for the $\alpha$ (BCC) phase: (a) Diamond Anvil Cell (DAC) pressure–density data Dewaele2008, (b) entropy–temperature relation from heat capacity measurements NIST_JANAF, (c) principal Hugoniot pressure–density data Barker1974Brown2000, and (d) thermal expansion from 1-bar measurements Abdullaev_2020Assael2006.