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The Journal of Physical Chemistry C Article Table 1. Calculated Bulk Modulus, Lattice Parameters, and Unit Cell Volume of Li and Li−S Redox End Members Using the optB88-vdW Functional Li optB88-vdW experiment Li2S optB88-vdW experiment α-S optB88-vdW experiment β-S optB88-vdW experiment 13.8 3.45 12.944 3.4845 42.6 5.70 45.746 5.6947 11.3 10.33 12.76 24.45 14.548 10.4649 12.87 24.49 10.8 10.66 10.72 10.84 - 10.6950 10.72 10.81 90 40.9 90 43.2 90 185.1 90 184.2 90 3221 90 3296 95.44 1232 95.75 1233 lattice parameters functional bulk modulus (GPa) a (Å) b (Å) c (Å) angle (deg) volume (Å3) Figure 2. (a) Calculated convex hull for the Li−S system. The inset magnifies the energetic ordering of the different Li2S2 candidate phases above the hull. (b) Gibbs free energy of Li2S + S vs Li2S2 as a function of temperature. Relaxed structures and vibrational contributions were calculated using the optB88-vdW functional, while total energies were evaluated using the vdW-DF2 functional. Table 1 compares the calculated bulk modulus, lattice parameters, and cell volume of Li−S redox end members with experimental data. The calculated values are from the optB88- vdW functional, which comparison calculations (Table S1, Supporting Information) revealed performed the best for structural properties among three other functionals: optPBE- vdW, optB86b-vdW, and vdW-DF2. Typical deviations between theory and experiment were less than 1% for the optB88-vdW. The optB86b- and optPBE-based functionals have slightly larger deviations (1−2%), whereas vdW-DF2 exhibits a range of 1−3%. Thermodynamic Properties. Figure 2a shows the convex hull for the Li−S system as a function of atomic percent lithium at 0 K. Among the various structure candidates considered for Li2S2, the one yielding the lowest energy was based on the hexagonal Li2O2 prototype (P63/MMC space group). This structure lies above the convex hull by approximately 67 meV/ atom and should therefore be metastable with respect to a two- phase mixture of S and Li2S. Of course, other hypothetical structures for Li2S2 are possible; however, identifying them will require a more extensive search (using genetic algorithms, etc.), which is beyond the scope of the present study. While such an effort might yield a structure whose energy is lower than the Li2O2 prototype identified here, the energy above the convex hull calculated for this structure, 67 meV, is sizable and in our judgement would be difficult to overcome with an alternative structure. Regarding the accuracy of this prediction, ref 54 reported that typical errors associated with DFT phase stability 4678 calculations involving oxides exhibited a standard deviation of 24 meV/atom. If the conclusions from oxides can be transferred to sulfides (to our knowledge a similar study on sulfides does not exist), then the energy above the hull for Li2S2 reported here safely exceeds the error threshold. The Li2O2 prototype for Li2S2 was used for calculations involving Li2S2 henceforth. Figure 2b compares the Gibbs free energies of Li2S2 to the two-phase mixture, Li2S + α-S, as a function of temperature. Free energies were evaluated by combining the static electronic energy from the vdW-DF2 functional with vibrational contributions obtained from the optB88-vdW. For the entire temperature range considered (0−400 K), we find that the two- phase mixture of Li2S and S has lower free energy than Li2S2. As previously mentioned, Li2S2 has not been successfully synthesized and does not appear in experimental phase diagrams.18 The calculated thermodynamic data in Figure 2 are consistent with these observations. To aid in the identification of Li2S2 during discharge of Li−S batteries, Figure S2 (Supporting Information) plots the calculated XRD pattern for Li2S2 and compares to that of Li2S. Li2S has major peaks around 27°, 31°, 45°, and 53°, which correspond well to experimental data.19 On the other hand, the peaks of Li2S2 do not match data from recent experiments.19,55 Table 2 summarizes the calculated redox potentials, formation enthalpies (ΔH), and formation free energies (ΔG) for Li2S and Li2S2 at room temperature (298 K). Comparisons are made between different functionals by DOI: 10.1021/jp513023v J. Phys. Chem. C 2015, 119, 4675−4683PDF Image | First-Principles Study of Redox End Members in Lithium Sulfur
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