Microscopic Origins of the Anomalous Melting Behavior of Sodium under High Pressure

Abstract

Recent experiments have shown that sodium, a prototype simple metal at ambient conditions, exhibits unexpected complexity under high pressure [1][2][3][4] . One of the most puzzling phenomena in the behaviour of dense sodium is the pressureinduced drop in its melting temperature, which extends from 1000 K at ∼30 GPa to as low as room temperature at ∼120 GPa 1 . Despite significant theoretical effort to understand the anomalous melting 5-10 its origins have remained unclear. In this work, we reconstruct the sodium phase diagram using an ab-initio-quality neural-network potential 11,12 . We demonstrate that the reentrant behaviour results from the screening of interionic interactions by conduction electrons, which at high pressure induces a softening in the shortrange repulsion. It is expected that such an effect plays an important role in governing the behaviour of a wide range of metals and alloys.At ambient conditions, sodium crystallizes in the bodycentered cubic (bcc) structure, which transforms to the face-centered cubic (fcc) phase upon compression to 65 GPa 13 . At 105 GPa, the fcc phase undergoes a transformation to the cI16 phase -a distorted variant of the bcc structure 2 . Further compression results in the formation of a large variety of complex phases 3,4 , the existence of which is quite intriguing. However, the focus of this work is one of the most striking features of the sodium phase diagram -the anomalous melting curve. Diffraction measurements have revealed a maximum in melting temperature around 1000 K at ∼30 GPa followed by a pressure-induced drop, which extends to nearly room temperature at ∼120 GPa 1 and spans the regions of stability of three solid phases (Fig. 1a). This anomaly implies that in this range the liquid is denser than the solid.Although previous ab initio studies 5-10 have reproduced the reentrant melting behaviour of sodium the physical origin of this anomaly still remains unclear. The hypothesis of structural and electronic transitions in liquid sodium proposed in one of the studies 6 has not been confirmed by other authors 5 . The discrepancy between these simulation results has been attributed to a different treatment of the semicore states 8 .It is important to point out that due to the high computational cost of ab initio MD all simulations so far re- ported 5-8 have used small simulation cells and estimated melting temperatures (T m ) by heating the solid phase until it melts. In such simulations, the absence of nucleation centers and fast heating schedules delay melting relative to the thermodynamic coexistence point and, therefore, only an upper bound on T m can be obtained. A proper reconstruction of the melting curve requires comparing the free energies of the liquid and solid phases and involves long (∼10 ns) MD simulations on large systems (∼ 10 3 atoms). Performing such simulations with direct ab initio methods is computationally too expensive at present.

Solid-liquid coexistence in small systems: A statistical method to calculate melting temperatures

“…12 and Table IV very well with the result reported in Ref. 42, while the MPs are still significantly lower than results from the fast heating method, 15 as summarized in Fig. 13.…”

“…Although the MPs we compute, along with Ref. 42, are significantly lower than experiments, the good agreement with large-size coexistence calculations points to DFT errors, rather than a flaw of our method.…”

“…Our results suggest that the neutral network potential in Ref. 42 successfully mimics the interactions as comparable to DFT accuracy, while the over-heating issue, though relatively small, is still limiting the accuracy of the fast heating method employed in Ref. 15.…”

“…All 3s valence and 2p core electrons are included for electronic structure calculations, as the importance of the core 2p electrons is widely acknowledged. [42][43][44] Fermi distribution among the energy level density of states is imposed to realize the electronic temperature. The plane wave energy cutoff is set to 260 eV and it is further increased to 500 eV for pressure correction.…”

“…Raty et al 15 employed the fast-heating method and obtained a melting curve close to the experiments. Eshet et al 42 used a neural-network potential based on DFT and calculated a melting curve through the free energy method. Despite of their successful capture of the reentrant behavior, the detailed melting points in these two articles are nevertheless quite different.…”