Predicting the phase behaviors of superionic water at planetary conditions Bingqing Cheng,1, 2, ∗ Mandy Bethkenhagen,3 Chris J. Pickard,4, 5 and Sebastien Hamel6 1Department of Computer Science and Technology, J. J. Thomson Avenue, Cambridge, CB3 0FD, United Kingdom 2Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge, CB3 0HE, United Kingdom 3Ecole´ Normale Sup´erieure de Lyon, Universit´eLyon 1, Laboratoire de G´eologie de Lyon, CNRS UMR 5276, 69364 Lyon Cedex 07, France 4Department of Materials Science & Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge, CB3 0FS, United Kingdom 5Advanced Institute for Materials Research, Tohoku University, Sendai, Japan 6Lawrence Livermore National Laboratory, Livermore, California 94550, USA (Dated: March 17, 2021) Most water in the universe may be superionic, and its thermodynamic and transport properties are crucial for planetary science but difficult to probe experimentally or theoretically. We use machine learning and free energy methods to overcome the limitations of quantum mechanical simulations, and characterize hydrogen diffusion, superionic transitions, and phase behaviors of water at extreme conditions. We predict that a close-packed superionic phase with mixed stacking is stable over a wide temperature and pressure range, while a body-centered cubic phase is only thermodynamically stable in a small window but is kinetically favored. Our phase boundaries, which are consistent with the existing-albeit scarce-experimental observations, help resolve the fractions of insulating ice, different superionic phases, and liquid water inside of ice giants. Water is the dominant constituent of Uranus' and Nep- gen lattice. In the experimental studies of superionic wa- tune's mantle [1], and superionic water is believed to be ter [2,4, 11{13], sample preparation is extremely chal- stable at depths greater than about one-third of the ra- lenging, hydrogen positions cannot be determined, and dius of these ice giants [2]. Water superionicity is marked temperature measurements in dynamical compression ex- by exceptionally high hydrogen diffusivity and ionic con- periments are not straightforward [2,4]. Notably, recent ductivity, as hydrogen atoms become liquid-like while dynamical compression experiments combined with x-ray oxygen atoms remain solid-like on a lattice with a certain diffraction (XRD) found a superionic phase with fcc oxy- crystallographic order. Although superionic water was gen lattice, ice XVIII, above 2000 K [2]. Static compres- postulated over three decades ago [3], its optical proper- sion experiments combined with synchrotron XRD sug- ties (it is partially opaque) and oxygen lattices were only gest a triple point between liquid, ice VII0, and ice VII00 accurately measured recently [2,4], and many properties (a bcc superionic phase) at 14.6 GPa and 850 K [13]. of this hot \black ice" are still uncharted. Amongst the many mysteries regarding superionic wa- Theoretical computation of the high-P water phase ter, the phase behaviors over a large range of pressure (P ) boundaries is also difficult. No reliable empirical force and temperature (T ) are particularly important in plan- fields are available for this system. FPMD simulations etary science: The location of various coexistence lines are computationally expensive, and are thus confined to (including the high-pressure melting line, the insulating short trajectories and small system sizes, which may in- ice to superionic transition line, and the phase boundaries troduce artifacts leading to contradictory results on the between competing superionic phases of water) are essen- locations of phase boundaries [5,7{10, 14{16] and diffu- tial for understanding the formation, evolution, interior sivity [8,9] between various studies based on the same structure and magnetic fields of planets [1,5,6] How- assumption of the underlying electronic structure. An- ever, the locations of these coexistence lines and even other shortcoming of most FPMD simulations is that the the possible types of oxygen lattices for superionic wa- nuclei are treated as classical point masses even though ter have long been debated. Initial computational stud- they are wave-like quantum mechanical particles. Yet, ies [3] proposed a face-centered-cubic (fcc) oxygen lattice for light elements such as hydrogen, nuclear quantum ef- while early first-principles electronic structure molecular fects (NQEs) are particularly acute. Molecular dynamics dynamics (FPMD) considered superionic water with a (MD) combined with the Feynman path integral (PIMD) body-centered cubic (bcc) oxygen lattice as a high tem- can be used to treat the NQEs, but PIMD multiplies the perature analog of the ice X [7]. Later FPMD studies re- computational cost by about another 20-fold [17]. How- arXiv:2103.09035v1 [physics.comp-ph] 16 Mar 2021 proposed a fcc [8,9], suggested a close-packed (cp) [10], ever, machine-learning potentials (MLPs) can help over- and, at pressures higher than 1 TPa, a P21/c [10] oxy- come these limits by first learning a data-driven model of atomic interactions from first-principles calculations [18], and then driving large-scale simulations at an affordable computational cost. Thanks to their low cost and high ∗ [email protected] accuracy, MLPs have helped reproduce the low-pressure 2 a 100 GPa 3250 K b 300 GPa 4000 K c 600 GPa 4000 K 2 .4 4 0 0 0 4 0 0 0 4 0 0 0 3 5 0 0 b cc 3 5 0 0 fcc 3 5 0 0 h cp ) 1 3 .2 3 0 0 0 3 0 0 0 3 0 0 0 s 2 2 5 0 0 2 5 0 0 2 5 0 0 / cm T [K] T [K] T [K] H 4 .0 600 GPa 1000 K 2 0 0 0 100 GPa 1000 K 2 0 0 0 300 GPa 1000 K 2 0 0 0 (D 10 1 5 0 0 1 5 0 0 1 5 0 0 log 4 .8 1 0 0 0 1 0 0 0 1 0 0 0 5 0 0 5 0 0 5 0 0 2 0 0 4 0 0 6 0 0 2 0 0 4 0 0 6 0 0 2 0 0 4 0 0 6 0 0 P [GPa ] P [GPa ] P [GPa ] FIG. 1. The diffusion of hydrogen in water with bcc, fcc, and hcp lattice of oxygen. On the left of each panel, the diffusion coefficients of hydrogen DH are shown logarithmically as a function of pressure (P ) and temperature (T ). The dashed lines illustrate the ice to superionic transition temperatures Ts as defined in the main text. On the right, the distributions of hydrogen atoms are indicated using gray contours, and the instantaneous positions of oxygen atoms are indicated using red spheres. For bcc, fcc, and hcp lattices, both a low-temperature ice structure (lower) and a high-temperature superionic structure (upper) are shown. phase diagram of water [19, 20], as well as elucidate We analyze the hydrogen diffusion in the fcc lattice, the nucleation behaviour of gallium [21], the liquid-liquid as the results for bcc and hcp are similar (Supplemen- transition of high-pressure hydrogen [22], and the struc- tary Materials). At pressure and temperature condi- tural transition mechanisms in disordered silicon [23]. tions where the lattice remains stable, the diffusivity of In this study, we construct a MLP for high-pressure hydrogen changes rapidly but smoothly across the ice- water using an artificial neural network architecture [24] superionic transition region, as evident in Fig.2a that based on Perdew-Burke-Ernzerhof (PBE) [25] density shows DH (on the log scale in the inset) with respect to functional theory (DFT). Combining the MLP with ad- T . Above Ts, when the system is fully superionic, DH vanced free energy methods, we predict the properties increases gradually as a function of temperature, and is of superionic and liquid water at the PBE level of the- only weakly dependent on pressure (see the Supplemen- ory, using large system sizes, long time scales and con- tary Materials). sidering NQEs. We elucidate the mechanisms for ice- The detailed and microscopic picture of the hydro- superionic transitions and hydrogen transport, map the gen motion revealed in Fig.1 complements experiments, high-pressure water phase diagram, and probe the kinet- which cannot determine the positions of hydrogen at the ics of phase transition. high P; T conditions. The H diffusion coefficients can be used to derive the ionic conductivity of superionic water using a generalized Einstein formula [26], which is a key Hydrogen diffusivity in superionic phases quantity in magnetic dynamo simulations for reproduc- ing the complex magnetic field geometry of the Uranus To probe hydrogen diffusion, the aforementioned bcc and Neptune [6]. and fcc lattices of oxygen, and a hexagonal close-packed (hcp) lattice that has a low-T ice analog with the Pbcm space group (structures are in the Supplementary Mate- A physical model of ice-superionic transitions rials) are considered. The insulating structures become superionic when the temperature rises, as marked by the To rationalize the ice-superionic transition and the hy- fast diffusion of hydrogen atoms inside the oxygen lat- drogen diffusion, we start from an interstitial formation tices. The distributions of position of hydrogen are shown model [27]: Taking x to be the fraction of the conduct- in the contour plots of Fig.1: at low T hydrogen atoms ing hydrogen atoms, the free energy of the system can be are confined to their equilibrium sites, while at high T expressed as only a fraction occupy these sites at any given time. The diffusivity of hydrogen DH is shown for a wide range of thermodynamic conditions in Fig.1. At about λ 2 f(x; T ) = (0 − T s0)x − x 2000 K, there is a rapid increase of DH, which marks the 2 insulating ice-superionic transition.