Thermodynamics and Equations of State of Iron to 350 GPa and 6000 K

Abstract

The equations of state for solid (with bcc, fcc, and hcp structures) and liquid phases of Fe were defined via simultaneous optimization of the heat capacity, bulk moduli, thermal expansion, and volume at room and higher temperatures. The calculated triple points at the phase diagram have the following parameters: bcc–fcc–hcp is located at 7.3 GPa and 820 K, bcc–fcc–liquid at 5.2 GPa and 1998 K, and fcc–hcp–liquid at 106.5 GPa and 3787 K. At conditions near the fcc–hcp–liquid triple point, the Clapeyron slope of the fcc–liquid curve is dT/dP = 12.8 K/GPa while the slope of the hcp–liquid curve is higher (dT/dP = 13.7 K/GPa). Therefore, the hcp–liquid curve overlaps the metastable fcc–liquid curve at pressures of about 160 GPa. At high-pressure conditions, the metastable bcc–hcp curve is located inside the fcc-Fe or liquid stability field. The density, adiabatic bulk modulus and P-wave velocity of liquid Fe calculated up to 328.9 GPa at adiabatic temperature conditions started from 5882 K (outer/inner core boundary) were compared to the PREM seismological model. We determined the density deficit of hcp-Fe at the inner core boundary (T = 5882 K and P = 328.9 GPa) to be 4.4%.

Liquid Iron Equation of State to the Terapascal Regime From Ab Initio Simulations

“…The models that are based on an assessment of the phase diagram reproduce ϱ in the liquid and its T dependence similarly well to at least 2600 K. The model by Komabayashi () starts with slightly larger ϱ at T m and uses a constant α with a smaller slope than the fit to experimental data (Assael et al, ), moving to progressively larger ϱ values compared to the experimental trend. The model of Dorogokupets et al () follows the experimental data well with a thermal expansion coefficient that increases moderately with T up to ∼2500 K, beyond which an exponential term in their thermal contribution to Helmholtz energy leads to a rapid increase in α and an unrealistical decrease in ϱ (see also Figure S1 in supporting information for residuals between our corrected model and other EOS formulations and experimental data).…”

“…While they tend to underestimate T up to P ∼250 GPa, they approach the margins of uncertainty for higher P data. Up to ∼350 GPa, all models yield T at the lower end of or below the T uncertainty in the shock experiment which continues to hold to the highest experimental P for the DFT‐MD‐based models—that is, our model and that of Ichikawa et al ()— while the apparent T increase along the Hugoniot is larger in the models based on the thermodynamic assessment of the phase diagram (Dorogokupets et al, ; Komabayashi, ).…”

“…Here we formulate a thermodynamically self‐consistent EOS for liquid iron that is applicable to a wide range of planetary core conditions—from a few gigapascals for the Moon to 2 TPa for exosolar planets—along two directions: (i) We extend the P / T range of previous work by de Koker et al () and Wagle et al () into the TPa regime and (ii) combine it with a correction scheme by French and Mattsson () to match experimental data at ambient P . Throughout the manuscript, we compare results from our model to previously published equations of state for liquid iron by Komabayashi (), Ichikawa et al (), and Dorogokupets et al ().…”

“…The description of the thermal contribution in various EOS models for liquid iron—ours and Komabayashi (), Ichikawa et al (), and Dorogokupets et al ()—is further explored by using experimentally measured P , T , and ϱ triplets; we take the experimentally determined P and vary T in each of the EOS models until ϱ is matched (Figure ). Since our corrected EOS uses ambient P and T = 2000 K as the reference condition (i.e., near T m ) and is corrected at T m , it agrees well with experiments and is close to the thermodynamic models by Komabayashi () and Dorogokupets et al () which are based on experimental data. Our uncorrected model requires T ∼3000 K, that by Ichikawa et al () T > 4000 K. Temperature at the low P data point by Tateyama et al () is similarly well reproduced by our corrected EOS and the models of Komabayashi () and Dorogokupets et al ().…”

“…Temperatures calculated from thermodynamic models at a set of experimentally determined P ‐ ϱ ‐ T data points for liquid iron (red symbols), using ϱ and P from the experiments and computing T from the equation of state models to match ϱ for a given P . In addition to our model (uncorrected with open symbols, corrected with filled black symbols) we have evaluated the equations of state by Komabayashi () (K14, green symbols), Ichikawa et al () (I14, blue symbols), and Dorogokupets et al () (D17, purple symbols). Experimental data (red symbols) at the melting point (squares) are from Cezairliyan and McClure (), the critical point (diamonds) from Beutl et al (), at 4.3 GPa from the multianvil experiments by Tateyama et al () (T11, triangles), and along the Hugoniot by Brown and McQueen () and Brown et al () (B86/00, circles).…”