First-Principles Phase Diagrams: Iron at Earth’s Inner Core Conditions with Full Inclusion of Anharmonic and Finite-Size Effects

Phase diagrams provide an essential source of information about phase composition and stability at different thermodynamic states, which are crucial to know for many applications. However, its evaluation at extreme temperature and pressure conditions can be challenging for experimentalists. At such states, computational methods can work as a rigorous alternative. Since such assessments involve evaluation of finite temperature free energies, the computation cost can easily exceed the available resources. Therefore, advances are greatly needed to make such assessments possible with minimal computational effort.

Building the phase diagram of iron at conditions relevant to Earth’s inner core (IC) is an example of such challenging and scientifically interesting problem, and a prototype for other applications. The IC is widely believed to be in a solid state composed, mainly, of iron [1] at state of about 350 GPa and 6000 K. However, the exact crystal structure (which is important to understand the complex seismic data observed in Earth’s IC and hence understand how Earth was evolved) is not fully understood/determined to the date (both experimentally and theoretically).

Ab initiosimulation methods (e.g. DFT), provide reliable and rigorous tools that can help predicting iron crystal structure at such conditions. Unfortunately, free energy calculations based on DFT are computationally expensive — especially with the large number of Fe electrons (around 16 electrons) that contribute significantly at such high pressure. Therefore, a wide range of levels of theories have been used to approximate the free energy; e.g., static energy [2], quasiharmonic approximation [3], and self-consistent phonon calculations [4].

In this work [5], we utilize a novel machinery developed by our group (called “harmonically-mapped averaging (HMA)” method [6,7]) to tackle this long standing problem using DFT method as implemented in VASP. The speed up gained by using HMA method to measure thermodynamic properties is at least an order of magnitude faster than “direct” methods. This enabled measuring of the full (harmonic and anharmonic) Gibbs free energy of Iron at several states. In addition, the finite-size effects in free energy were estimated based on a harmonic-based technique described here [8]. Applying these techniques to the three known iron candidate structures (hcp, fcc, and bcc), we were able, to the first time, to build a completely ab initio-based phase diagram of iron at Earth’s IC conditions in the thermodynamic limit.

[1] F. Birch, J. Geophys. Res. 57, 227 (1952).
[2] P. Soderlind, J. Moriarty, and J. Wills, Phys. Rev. B 53, 14063 (1996).
[3] L. Stixrude, Phys. Rev. Lett. 108, 055505 (2012).
[4] W. Luo, B. Johansson, O. Eriksson, S. Arapan, P. Souvatzis, M. I. Katsnelson, and R. Ahuja, Proc. Natl. Acad. Sci. 107,9962 (2010).
[5] S. G. Moustafa, A. J. Schultz, E. Zurek, and D. A. Kofke, In preparation, (2016).
[6] S. G. Moustafa, A. J. Schultz, D. A. Kofke, Phys. Rev. E 2015, 92, 043303.
[7] A. J. Schultz, S. G. Moustafa, L. Weisong, S. J. Weinstein, and D. A. Kofke, J. Chem. Theory Comput., 2016, 12 (4), pp 1491-1498.
[8] T. B. Tan, A. J. Schultz, and D. A. Kofke, J. Chem. Phys. 133, 134104 (2010).