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The Journal of Physical Chemistry C ized gradient approximation (GGA)34 and (ii) van der Waals- augmented density functional theory (vdW-DF).35 Including van der Waals interactions is essential for describing the behavior of sulfur, as the crystal structure of both α and β allotropes consists of discrete, covalently bonded S8 molecular units (“cycloocta”) that interact through dispersion forces. Five different vdW-DF methods were tested: these include the so- called vdW-DF1 methods having exchange functionals based on revPBE,35 optPBE,36 optB88,37 optB86b,37 and also the vdW- DF238 method. Equilibrium cell volumes were determined by fitting energy−volume data to the Murnaghan equation of state.39 Atom positions were relaxed to a force tolerance of 0.01 eV/Å. Article As previously mentioned, the existence of a persulfide phase, Li2S2, during discharge of a Li−S battery remains an open question. Such a phase does not appear in the equilibrium Li−S phase diagram, and its crystal structure is unknown. To examine the stability of a hypothetical Li2S2 phase we generated several candidate Li2S2 crystal structures using various A2B2 phases as structural templates. Here A represents an alkali metal (Li, Na, K, Rb), and B is a chalcogenide (O, S, Se). Seven structures were investigated by replacing A sites with Li and B sites with S. The template compounds included: Li O , Na S , γ= 1(Gslab−Nμ) S2AS SS 2222 K2S2, Na2O2, K2O2, Na2Se2, and Rb2S2. In all cases the unit cells (volume, shape, and atom positions) were relaxed using vdW- ⎢LiS Li ⎜LiS⎟⎥ Li2S2A⎣2 2Li2S⎝2 ⎠S⎦(7) aware DFT. Thermodynamic properties were evaluated within the harmonic approximation.27 Vibrational frequencies ωi were obtained using the direct method.40 Vibrational contributions to the enthalpy (Hvib) and entropy (Svib) are given by27 As the chemical potential of sulfur is not precisely known, surface energies were evaluated for a range of μS given by E(Li2S) − 2 μLi(BCC Li) ≤ μS ≤ μS(α-S). Here, E(Li2S) is the total energy of a Li2S formula unit; μLi(BCC Li) is the energy per atom of BCC Li; and μS(α-S) refers to the same for an atom of α-S. Cycling of a Li−S battery should result in the repeated nucleation and growth of solid-phase sulfur (charging) and Li− S particles (discharging). As these processes occur in the presence of a liquid electrolyte, the relevant “surface” energies are not solid/vacuum surface energiesas is typically assumed in atomistic studiesbut rather solid electrode/liquid electro- lyte interface energies. To explore the impact of solvation on surface energies, comparisons were made with and without a continuum solvation field (VASPsol).51,52 In these calculations the dielectric constant was set to that of dimethoxyethane (DME),53 7.55, as common electrolytes in Li−S batteries employ solvents based on DME11 or mixtures of DME and dioxolane.10,19 (The dielectric constant of dioxolane is similar to that of DME, 7.13.) ■ RESULTS Structure Analysis. Low-energy structures of each redox end member were evaluated in a comparative fashion using the GGA and five different vdW-DF methods. Turning first to the high-temperature β-sulfur phase, Figure S1 (Supporting Information) plots the total energy of the β-S unit cell as a function of cell volume. In contrast to the other functionals, which show a clear minimum in the energy vs volume data, the curves calculated with the GGA and the revPBE-based vdW- DF1 functional monotonically decrease as volume increases. Such behavior might be expected from the GGA, where the neglect of vdW interactions between cycloocto rings is a known omission. However, in the case of vdW-DF1, vdW contribu- tions are explicitly accounted for; hence, the poor representa- tion of energy−volume behavior is surprising. For this reason, the GGA and revPBE-based vdW-DF1 functionals were not used in subsequent structure calculations on S-based systems. DOI: 10.1021/jp513023v J. Phys. Chem. C 2015, 119, 4675−4683 3N−31 ⎡ ⎛ħω⎞ ⎤−1 H (T)= ∑ ħω+ħω⎢exp⎜ i⎟−1⎥ vib ii⎣B⎦ 2 ⎢⎝kT⎠⎥ i (1) 3N−3 ħω/kT ⎡ ⎛−ħω⎞⎤ S (T)= ∑ i B −ln⎢1−exp⎜ i⎟⎥ i vib exp(ħω/k T) ⎢ ⎝ k T ⎠⎥ iB⎣B⎦ surfaces, comparisons were made using the GGA and vdW-DF (optB88) functionals. In addition, seven sulfur surfaces were considered, with indices of (100), (010), (001), (011), (110), and (111). These surfaces were comprised of at least three layers of cycloocta rings and were constructed such that no S8 rings were broken when the surface was cleaved. For sulfur surfaces only the vdW-DF method was used. The Monkhorst− Pack scheme was used for both Li2S and sulfur surfaces with 4 × 4 × 1 and 1 × 1 × 1 k-point meshes, respectively. The energies of Li2S and S surfaces are given by 1 γ = (Gslab−Nμ −Nμ) Li2S 2A Li2S Li Li S S (5) (6) where Gslab is the energy of the surface slab; N is the number of i atoms of type i in the slab; and μi is the corresponding chemical potential. In the case of the Li2S surface, the surface energy can be written in terms of the energy per formula unit of bulk Li2S, gLi2S, and the chemical potential of sulfur 1⎡1⎛1⎞⎤ γ= Gslab−Ngbulk+N−Nμ where ħ is Planck’s constant divided by 2π; kB is the Boltzmann factor; and N refers to the number of atoms in the supercell. The enthalpy and Gibbs free energy are expressed as H(T)=E+H (T) vib (3) G(T) = H(T) − Svib(T)T (4) where E is the static (0 K) energy of a compound in its ground state. To estimate electronic properties, the Heyd−Scuseri− Ernzerhof (HSE06) hybrid functional41 and the non-self- consistent quasi particle G0W0 method42 were used. In the case of G0W0 calculations, vdW-DF wave functions from an earlier self-consistent calculation were used as input. A γ-point- centeredsamplingschemewith1×1×1(α,β-S),4×4×4 (Li2S), and 7 × 7 × 7 (Li2S2) k-point meshes was used. The Gaussian smearing method was applied to obtain the density of states (DOS); the band gap was estimated using energy differences between the lowest occupied and highest unoccupied eigenvalues. The surface energies of 38 distinct surfaces were evaluated. These included 31 Li2S surfaces of varying stoichiometry43 with Miller indices (100), (110), and (111). Each surface slab consisted of at least 9 Li/S planes; approximately 20 Å of vacuum was included in each surface supercell. For Li2S 4677 (2)PDF Image | First-Principles Study of Redox End Members in Lithium Sulfur
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