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Mbar. 0.5 21, ,H O, oi tt 4 ,6 ,8 ] experimen- 9], 8, 7, 6, 5, [4, state ionic y A970 USA 94720, CA ey, Pcca ri 45,USA 94550, ornia 2 ib reeeg acltosshow calculations energy free Gibbs O 2 n H and , 1,3 s noH into ose inocr nsproi water. superionic in occurs tion C 2 2 O / ff 9 m ewudhv ihionic high a have would te O rn yrgnoye com- hydrogen-oxygen erent and , 2 4 ff n H and , tutrswl loassume also will structures .So fe,Pickard after, Soon ]. rn ueincstructure superionic erent 2 Pa O 2 ymtyta are that symmetry 3 ¯ n nhydrogen- an and ± . br This Mbar. 0.5 9 u grew hus O cwater ic form. c ensive 4 et We previously predicted a transition from a body-centered cubic (bcc) to face-centered cubic (fcc) oxygen sub- lattice in superionic water at approximately 1 Mbar [8]. In this paper, we show with ab initio simulations that other close packed structures have Gibbs free energies that are very similar to that of fcc and we can therefore no longer predict with certainty which close packed structure superionic water will assume up to 22.8 0.5 Mbar. Consistent ∼ ± with the findings in Ref. [24], we predict superionic water assume a novel structure with P21/c symmetry that is not close packed but more dense than fcc at the same P-T conditions [25]. This transformation is accommodated by changes in the oxygen sub-lattice.
DENSITY FUNCTIONAL MOLECULAR DYNAMICS SIMULATIONS AND GIBBS FREE ENERGY CALCULATIONS
All ab initio simulations were based on density functional molecular dynamics (DFT-MD) and performed with the VASP code [26]. We used pseudopotentials of the projector-augmented wave type [27] with core radii of 1.1 and 0.8 Bohr for the O and H atoms respectively. For most calculations, we used the exchange-correlation functional of Perdew, Burke and Ernzerhof [28] but we employed the local density approximation (LDA) for comparison. A cutoff energy of 900 eV for the plane wave expansion of the wavefunctions was used throughout. The Brillioun zone was sampled with 2 2 2 Monkhorst-Pack k-point grids [29]. 4 4 4 grids had been tested [8] and yielded consistent results. The occupation× × of electronic states are taken to be a× Fermi-Dirac× distribution set at the temperature of the ions [30]. The simulation time ranged between 1.0 and 5.0 ps. A time step of 0.2 fs was used. An initial configuration for a particular structure is obtained by starting from the ground state and gradually increasing the temperature during the DFT-MD simulation until the hydrogen atoms are mobile and equilibration is reached. Positions and velocities of equilibrated superionic structures were recycled to initialize simulations at other densities and temperatures after adjusting the cell parameters or velocities, respectively. We found that all ice structures transform into a superionic state at megabar pressures. Depending on the structure, the solid-to-superionic transition may be accompanied by re-arrangements of the oxygen sub-lattice that needs to be analyzed for every structure and density. All simulations were performed in supercells with between 48 and 108 molecules. We used our algorithm [31] to construct compact supercells of cubic or nearly cubic shape by starting from the primitive cell vectors ~ap, ~bp, and ~cp:
~a = ia~ap + ja~bp + ka~cp , ~b = ib~ap + jb~bp + kb~cp , ~c = ic~ap + jc~bp + kc~cp . (1)
The integer values, (ia ja ka, ib jb kb, ic jc kc), will be specified in each case. While some structures like fcc have no adjustable cell parameters, other less symmetric crystal structures may have up to five free parameters, the b/a and c/a ratios as well as the three angles, for a given volume and temperature. Most simply one can derive these parameters with ground-state structural relaxation. Since all superionic properties are neglected in such optimizations, the results need to be confirmed with finite temperature DFT-MD simulations. If there is just one free parameter, such as the c/a ratio in the hcp structure, one can perform a series of constant- volume simulations for different c/a ratios and determine the optimal value by fitting a linear stress-strain relationship, σzz (σxx + σyy + σzz)/3 = A(c/a) + B, where σi j are the time-averaged components of the stress tensor. The optimal c/a ratio− is given by B/A when the system is under hydrostatic conditions. For systems with− more free cell parameters, it may be more efficient to perform constant-pressure, flexible cell simulations [32]. All free parameters will then fluctuate around a mean value that can be estimated by taking a simple average. Since the fluctations may be quite large, one can introduce a bias into the average. By monitoring the stress value during the MD simulations, one may only consider cell parameter value for the average if the instantaneous stress coincides with the target stress. Alternatively, one may use the the stress-strain pairs from the MD trajectory to target fit a linear relationship σi j σ = Aǫi j + B. The optimal strain value is again given by B/A. We have tested all − i j − these three methods and found them to have comparable accuracy, which is primarily controlled by the length of the MD trajectory. We derive the ab initio Gibbs free energies with a thermodynamic integration (TDI) technique [33, 34, 35, 36, 37, 38, 39, 40, 41, 42] that we adapted to superionic systems in Ref. [8]. In this scheme, the difference in Helmholtz free energy between a DFT system and a system governed by classical forces is computed from,
1 FDFT Fcl = dλ VKS Vcl λ . (2) − Z0 h − i The angle brackets represent an average over trajectories governed by forces that are derived from a hybrid potential energy function, Vλ = Vcl + λ(VKS Vcl). Vcl is the potential energy of the classical system and VKS is the Kohn-Sham energy. We typically perform five independent− simulations with λ value equally spaced between 0 and 1. In order to maximize the efficiency of evaluating the integral in Eq. (2), we construct a new set of classical forces for each density and temperature. Our classical reference system consists of a pair potentials between each pair of atomic species combined with a harmonic Einstein potential for every oxygen atom. The harmonic force constants are derived first from the mean square displacements in a constant-volume simulation. The residual forces are fitted to O-O, O-H, and H-H pair potentials [43] that we present with spline functions [44]. The free energy of the classical reference system, Fcl, is obtained with classical Monte Carlo simulations where we again take advantage of the TDI technique to gradually turn off all pair forces. The free energy of an Einstein crystal of oxygen atoms and that of a gas of noninteracting hydrogen atoms is known analytically.
COMPARISON OF DIFFERENT H2O STRUCTURES
In Ref. [8], we predicted superionic water to assume a fcc structure at megabar pressures while earlier ab initio simulations had assumed a bcc oxygen sub-lattice. We begin the discussion in this section by comparing the Gibbs free energy of fcc with that of other close packed structures. In particular we will consider a hexagonal close packed (hcp) structure with an ABAB layering and double hexagonal close packed (dhcp) structure with an ABAC layering [45].
0.30 0 fcc 48 mol. Density ρ ≈ 4.0 g cm 3 O)
2 0.25
/H 0 0.20 0