Thermodynamic and Mechanical Stability of Crystalline Phases of Li2S2

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C: Energy Conversion and Storage; Energy and Charge Transport 2

2

Thermodynamic and Mechanical Stability of Crystalline Phases of LiS Qing Guo, Kah Chun Lau, and Ravindra Pandey

J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b11290 • Publication Date (Web): 10 Feb 2019 Downloaded from http://pubs.acs.org on February 11, 2019

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The Journal of Physical Chemistry

Thermodynamic and Mechanical Stability of Crystalline Phases of Li2S2

Qing Guo1, Kah Chun Lau2* and Ravindra Pandey1*

Department of Physics, Michigan Technological University, Houghton, MI 49931 Department of Physics, California State University, Northridge, CA 91330

(February 06, 2019)

Email: [email protected], [email protected]

1

ACS Paragon Plus Environment

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Abstract

In lithium-sulfur battery, Li2S2 is one of the key intermediate products which may exist as insoluble solid in a battery system. In this article, the low-energy crystalline phases of Li2S2 are investigated using density functional theory coupled with the Van der Waals correction terms. The calculated results find that the tetragonal, triclinic and hexagonal phases of Li2S2 phases are thermodynamically stable with the preference for the hexagonal phase to be the ground state. The low-energy Li2S2 phases also exhibit highly anisotropic elastic properties which can be attributed to the unique S-S bond orientations in their lattices. We believe that the elastic modulus in Li-S solids can be properly tuned if the S-S bonds orientation and distribution can be controlled during the synthesis process, which may be helpful for the development of functionalized cathode in the Li-S battery.

2


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1. Introduction Lithium-sulfur (Li-S) battery has been considered as a promising energy storage system due to its high theoretical energy density and relatively low cost in terms of main reactants (e.g. sulfur)

1-7.

During the discharge process,

various lithium polysulfides (PS) Li2Sx (x = 2~8) species are formed, and eventually converted into insoluble lithium sulfide (Li2S) as the end discharge product

5, 7.

For lithium and alpha-S8 that converted completely into Li2S, a

theoretical specific capacity of about 2500 Wh/kg can be achieved, and it is several times higher than that of today’s Li-ion batteries (~387 Wh/kg )

3-4, 6.

Although the overall redox reaction in the Li-S battery system has been studied over the last several years, the detailed reaction pathways and formation of Li2S and Li2Sx (x = 2-8) in the Li-S battery remain to be fully understood. For example, in contrast to the soluble lithium polysulfides species, it is believed that lithium disulfide (Li2S2) species is the only insoluble intermediate during the discharge process in Li-S battery

7-9.

However, the

presence and material characterization of solid Li2S2 in the discharge process remains a matter of debate in the scientific literature 8-15. One of the examples include in-situ atomic force microscopy and ex-situ spectroscopic methods which have been used to distinguish the morphology and growth processes of insoluble products, i.e. Li2S2 and Li2S 14. In an in-situ Li-S cell using the highly oriented pyrolytic graphite (HOPG) electrode set up, formation of the larger lamellae Li2S structure were observed at high 3



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discharge rate, whereas smaller nanoparticles of Li2S2 were seen at lower discharge rate14. This is in contrast to the results based on X-ray absorption near edge spectroscopy (XANES)

12

and Raman spectroscopy

8

which failed

to detect the presence of a solid Li2S2 during the charge-discharge cycles in the Li-S cells. In recent in-situ XRD measurements

13,

formation of the

crystalline Li2S2 is confirmed under certain experimental conditions suggesting the structure to be tetragonal Li2S2. On the other hand, a recent theoretical study based on classical molecular dynamics method

15

has predicted that Li2S2 and Li2Sx (x = 3-8) species are

present in various Li-S nanoparticles at room temperature, in addition to the commonly known Li2S. First principles calculations coupled with the evolutionary algorithm reported stability of a triclinic Li2S2 with the space group of P1 10. Later, a tetragonal Li2S2 with the space group of P42/mnm was found to be about 17 meV/atom lower in energy

11.

Calculations including van

der Waals (vdW) terms found high stability of Li2O2-liked hexagonal Li2S2 with the space group P63/mmc 9. It was suggested that the Li2S2 can transform to Li2S and alpha-S8 (i.e. Li2S2  Li2S + S) via a non-electrochemical pathway. This is due to the relatively low formation enthalpy (~ 0.1 eV) of Li2S compare to Li2S2 at room temperature 9. To address ambiguity in identifying the low-energy crystalline phases of Li2S2, we perform state-of-the-art density functional theory calculations coupled with the vdW corrected terms. Note that some of the previous 4


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calculations

10-11

have not included contributions from the phonons and vdW

terms which may be important in considering the energy difference between the low-energy crystalline phases to be very small within the order of room temperature energy fluctuation. Our results suggest that the hexagonal Li2S2 is the energetically-preferred crystalline phase for temperatures up to 400K. Moreover, our prediction of thermodynamic and mechanical stability of these low-energy Li2S2 crystalline phases would be expected to work as a baseline study for future developments of a functionalized cathode in the Li-S battery. 2. Computational method Electronic structure calculations were performed in the framework of density functional theory. The generalized gradient approximation (GGA) proposed by Perdew-Burk-Ernzerhof

(PBE)

16

was

employed

to

describe

exchange-correlation effects. The Grimme’s D2 correction term

17

the was

included as it was found to be important for Li-S systems 9. The plane wave energy cut-off was set to 500 eV, and the energy convergence criterial was set to 10-7 eV/atom. The force convergence limit on each atom was set to 0.01 eV/Å for the structural optimization calculations. The Brillouin zone was sampled by a (15 × 15 × 15) Monkhorst-Pack grid

18.

The program package

VASP was used for electronic structure calculations performed in this work 19-20.

To test the accuracy and reliability of the modeling elements, calculations were performed on the well-characterized bulk Li2S. The calculated structural 5



Page 6 of 31

properties are in excellent agreement with the measured values (Table S1, Supporting Information), e.g., the calculated lattice constant is 5.712 Å as compared to the experimental value of 5.708 Å

21.

The simulated X-ray

diffraction (XRD) pattern of Li2S (Fig. S1 and Table S2, Supporting Information) is also an excellent agreement with the standard Li2S powder diffraction file

22.

Similarly, the calculated elastic constants and bulk modulus

are in excellent agreement with the corresponding experimental values reported for bulk Li2S. 3. Results and discussions 3.1 Structural properties Fig. 1 displays the crystalline phases of Li2S2, namely (P42/mnm)

11,

triclinic (P1)

10,

and hexagonal (P63/mmc)

the tetragonal 9

which were

considered in this work. Table 1 lists the structural parameters calculated at PBE+D2 level of theory for Li2S2.

It appears that S atoms prefer to form a

S-S pair with RS-S of 2.11-2.16 Å in the lattice. Note that RS-S of the neutral S2 molecule is 1.89 Å

23-24.

The tetragonal Li2S2 has nearly the same density

(1.65 g/cm3) as calculated for the cubic Li2S (1.64 g/m3). On the other hand, the calculated density values of triclinic and hexagonal phases are 1.80 g/cm3 and 2.01 g/cm3, respectively. For the -S8 crystalline solid, the calculated density is 2.03 g/cm3 2, 25. Thus, we expect that a substantial difference in the density among these Li2S2 phases will lead to different responses in mechanical vulnerability of the cathode due to substantial volume change of 6


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sulfur or Li-S nanoparticles caused by lithiation and delithiation during the discharging and charging process of the Li-S cell.

Figure 1. A ball and stick model of (a) tetragonal, (b) triclinic and (c) hexagonal phases of Li2S2. Atomic color code: Li (green), S (yellow).

The calculated results find the hexagonal Li2S2 to be stable by 0.17 eV per formula unit relative to the triclinic Li2S2 at the PBE+D2 level of theory (Table 2). Interestingly, the total energy/formula unit of tetragonal and triclinic Li2S2 is nearly degenerate with the triclinic is being lower in energy by 0.01 eV. However, this is not the case at the PBE level of theory where the tetragonal Li2S2 is predicted to be relatively more stable (Table S3, Supporting Information). Thus, our results agree with the previously reported results 26

9-11,

about the importance of including the dispersion-related term for an

accurate description of the Li-based sulfides (Tables S1 and S4, Supporting Information). Note that the calculated Bader charge

27

on each S atom is

about -0.8e (or ~ -1.6 e/S2 unit) in Li2S2 phases relative to that of -1.7 e in 7



Page 8 of 31

Li2S. The predicted electronic band gaps of Li2S2 phases (i.e. ~ 1 - 2 eV) are in general smaller than that of Li2S (i.e. ~ 3.4 eV) (Tables S1 and S4).

Table 1. Structural properties of Li2S2 phases calculated using the PBE+D2 level of theory. Tetragonal (P42/mnm)

Hexagonal Triclinic (P1)

(P63/mmc)

Lattice constants a (Å)

5.07

4.40

3.97

b (Å)

5.07

4.64

3.97

c (Å)

6.11

7.48

9.47

α (º)

90

90

90

β (º)

90

90

90

γ (º)

90

109.7

120

Bond distance 8


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R (Li-S), Å

2.47

2.43-2.50

2.53, 2.63

R (S-S), Å

2.11

2.12

2.16

Volume (Å3)

157.3

143.9

129.1

Density (g/cm3)

1.65

1.80

2.01

Q (e), Li

+0.85

+0.85

+0.86

Q (e), S

-0.80~-0.89

-0.85

-0.80, -0.92

Bader’s charge,

Table 2. Total energy/ formula unit (EDFT) and Gibbs free energy/formula unit (G) for Li2S2 calculated at PBE-D2 level of theory. Zero is taken to be the hexagonal Li2S2.

Tetragonal (P42/mnm)

EDFT@0K (eV)

G@300K (eV)

0.18

0.15

9



Triclinic (P1)

0.17

0.14

Hexagonal (P63/mmc)

0

0

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Figure 2 The simulated XRD pattern of (a) tetragonal, (b) triclinic and (c) hexagonal Li2S2 in range of 22~38º.

In situ X-ray diffraction (XRD) pattern taken in the electrochemical cell with cathode consisted of S and C deposited on Al mesh displayed a broad peak at 27º associated with the (1 1 1) crystallographic orientation of cubic Li2S

13.

Besides Li2S XRD signature, additional XRD peaks at 23.8, 24.7, 29.9 31.9, and 36º were also observed. These additional XRD peaks

13

are likely to be 10


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likely associate with the triclinic Li2S2 as suggested by the simulated XRD spectra (Fig. 2). Our calculated results predict the peaks to be at 23.6, 23.8, 24.1, 24.6 and 36.4º for the triclinic Li2S2 (Fig. 2) with the most dominant peak attributed to (1 -1 0), (1 0 1) and (0 1 2). We note that a small mismatch in the DFT and XRD simulated peaks may be due to a slight difference in the lattice constants used by the XRD simulation relative to those predicted by the DFT calculations. On the other hand, calculations predict the diffraction peaks to be at 22.8, 24.8, 28.8, 29.2, and 35.4º for the tetragonal Li2S2, and 25.9, 27.6, 32.2, and 38.7 º for the hexagonal Li2S2. Due to proximity of some of these XRD peaks in these position (e.g. ~ 23-24o and 36o), the presence of the low-energy Li2S2 phases could not be ruled out in the XRD spectra. 3.2 Phonon Spectra To support the future experimental investigation associated with lattice vibrations, Fig. 3 displays the phonon dispersion curves calculated using the phononpy program package

28.

The high symmetry points for Li2S2 phases in

the first Brillouin Zone were taken from the Reference

29.

As shown, no

imaginary phonon modes were predicted suggesting dynamic stability of the low-energy Li2S2 phases throughout the Brillouin zone. The highest vibrational frequencies (max) at Γ are in the range of 449-493 cm-1 and are associated with the stretching mode of the stiffest S-S bonds as indicated by a strong peak in the projected density of states (PDOS) of sulfur atoms. max is 493 cm-1 for tetragonal, 478 cm-1 for triclinic, and 449 cm-1 for 11



Page 12 of 31

hexagonal Li2S2. The trend in max follows the trend of the calculated S-S bond lengths of tetragonal, triclinic, and hexagonal Li2S2 (Table 1). Next, we obtained the Raman spectra (see Supporting Information) of these Li2S2 phases using Quantum ESSPRESO

30-31

The signature Raman mode

associated with the S-S stretching mode are predicted to be at ~463 (tetragonal), 435 and 424 (triclinic) and 429 cm-1 (hexagonal) in vacuum (Table S4 Supporting Information). Similar results have also been reported for S22- stretch mode by previous calculations, 440 cm-1 for PBE-DFT level of theory

32

and 420 cm-1 for B3LYP-DFT level of theory33. Experimentally, the

S-S stretch mode for S22-shows lower Raman peak in vacuum (392 cm-1) than that in electrolytes (428 cm-1

in tetrahydrofurane (THF)) 8. The S-S stretch

mode at 473 cm-1 for BaS2 and 451 cm-1 for β -Na2S2 have also been reported34. In Li-S battery, Raman peaks at 174 and 514 cm-1 attributed to S-S bending and stretching vibrations were observed at the sulfide cathodes in electrolyte after cycling35. Note that our calculated value of Raman active mode, i.e. 355 cm-1 for cubic Li2S (Fig. S2 Supporting Information) is in good agreement with the reported experiment value of 372 cm-136.

12


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Figure 3. Calculated phonon dispersion curves, total phonon density of states (grey) and projected phonon density of states (green for lithium, yellow for sulfur) for (a) tetragonal Li2S2, (b) triclinic Li2S2, and (c) hexagonal Li2S2.

13



Page 14 of 31

3.3 Thermodynamic Stability: Gibbs formation energy To address the thermodynamic stability of these Li2S2 phases, we employ the harmonic approximation

37

in which the vibrational contributions to the

enthalpy (Hvib) and entropy (Svib) are determined by: 3𝑁 ― 31 2ℏω𝑖

𝐻𝑣𝑖𝑏(𝑇) = ∑𝑖

3𝑁 ― 3 ℏω𝑖/𝑘𝐵𝑇 exp (ℏω𝑖/𝑘𝐵𝑇)

𝑆𝑣𝑖𝑏(𝑇) = ∑𝑖

―1

[ ( ) ― 1] ― ln [1 ― exp( )]

+ℏω𝑖 exp

ℏω𝑖

𝑘𝐵𝑇

―ℏω𝑖 𝑘𝐵𝑇

(1) (2)

where, ωi is the vibrational frequency, ℏ is the reduced Planck constant, kB is the Boltzmann factor and N is the number of atoms in the supercell. The enthalpy and Gibbs free energy can then be expressed as: 𝐻(𝑇) = 𝐸𝐷𝐹𝑇 + 𝐻𝑣𝑖𝑏(𝑇)

(3)

𝐺(𝑇) = 𝐻(𝑇) ―T × 𝑆𝑣𝑖𝑏(𝑇)

(4)

where, EDFT is the DFT static energy at 0 K.

Assuming the correction of free energy attributed to the volume variation at the low temperature and low-pressure regime is small, the Gibbs free energy of formation can be written as: 𝛥𝐺(𝑇) = 𝐺(𝑇) ― 𝑁𝐿𝑖𝜇𝐿𝑖(𝑇) ― 𝑁𝑆𝜇𝑆(𝑇)

(5)

where μLi and μS are the chemical potentials of Li (bcc) and S (alpha-S) bulk respectively, as a function of temperature. Table 2 and Fig S3 (Supporting Information) list the calculated G (eV/Li2S2) and G (eV/Li) values of these Li2S2 phases and suggesting that the hexagonal phase is energetically preferred at 300 K relative to triclinic and 14


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tetragonal Li2S2 phases. Note that the relatively small energy difference cannot exclude the possibility of co-existence of the low-energy Li2S2 phases under the non-equilibrium operating conditions of a Li-S battery system. At equilibrium volume, variation of heat capacity, Cv, with temperature for Li2S2 together with the cubic Li2S are displayed in Fig. S4, Supporting Information. At 500 K, the Cv is 187, 188, 188 J mol-1 K-1 for tetrahedral, triclinic and hexagonal Li2S2 respectively reaching the Dulong-petit limit of 3nNAkB (~ 199 J mol-1 K-1). To relate to the Li-S discharge process, we calculate the equilibrium potentials of Li2S2 formation considering the following reaction to occur in the Li-S batteries. 1

(6)

2𝐿𝑖 + 4𝑆8↔𝐿𝑖2𝑆2 According to the Nernst equation, the equilibrium cell potential is 𝐸0 = ―(∆𝐺/𝑛𝑒),

(7)

where n is the number of electrons (e) transferred in the process and ΔG is the difference of free energy (G). The calculated results are listed in Table 3. Experimentally, the open circuit voltage of Li-S cell is found to be around 2.15 V

1, 38,

and the typical

discharge voltage that yields Li2S/Li2S2 is found to be ~ 2.05V

39.

Our

calculation results suggest that formation of Li2S2 solids can be found within the equilibrium voltage (E0) range of ~ 1.98-2.05 V, compared to Li2S at ~ 2.10 V at ambient temperature. With a controlled discharge process within a 15



Page 16 of 31

narrow voltage range of ~ 0.1 V relative to Li2S, formation of Li2S2 solids at the cathode could therefore be possible.

Table 3 Calculated cell potential E0 (V) at 300K calculated at PBE-D2 level of theory. E0@300K (V) Cubic Li2S (Fm3m)

2.10

Tetragonal Li2S2 (P42/mnm)

1.98

Triclinic Li2S2 (P1)

1.99

Hexagonal Li2S2 (P63/mmc)

2.05

3.4 Mechanical Stability: Equation of State and Elastic properties Next, we focus on determining the mechanical stability of Li2S2 phases for which the Born stability criteria was used

40-41.

Essentially, the mechanical

stability is a function of hydrostatic pressure and is derived from elastic moduli calculated/measured for a given solid. Note that softening of the shear moduli may lead to a mechanical instability of a given solid. Also, fundamental knowledge of the elastic properties, including elastic limits of Li-S solid, is critical in improving the mechanical stability and flexibility of Li-S solid in composite Li-S cathode. A detailed atomistic informed mechanical model (e.g.

16


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stress-strain relationship) of Li-S solids will help us to implement the state-of-the-art of multiscale simulation framework to include multi-phase and porous electrode models in Li-S battery

42

to improve the design of cathode

materials in Li-S battery. Fig. 4 (a) displays energy vs volume for Li2S2 which is fitted to the fourth order Birch-Murnaghan (BM4) equation of state (EOS)

43-44.

The normalized

volume (V/V0) vs Pressure (P) diagram describing the compressibility of Li2S2 phases is given in Fig. 4 (b), where V0 is the zero-pressure volume. From Fig. 4 (b), tetragonal and triclinic Li2S2 exhibit similar compressibility, while the hexagonal phase shows lower compressibility and is expected to have a higher bulk modulus. Within this pressure regime (i.e. up to 20 GPa), no high-pressure phase transition is expected from the hexagonal Li2S2 to either triclinic or tetragonal Li2S2.

Figure 4 (a) Energy vs Volume curve calculated PBE-D2 level of theory, and (b) Normalized volume-pressure diagram at 0K for Li2S2. V0 is the zero-pressure volume. 17



Page 18 of 31

To obtain a comprehensive understanding of mechanical properties of these Li2S2 solids, the complete necessary and sufficient conditions for the elastic stability in the tetragonal, triclinic, and hexagonal phases can be defined as the following by adopting Born stability criteria40-41. To obey Born stability criteria, conditions for the tetragonal and hexagonal phases are: 𝐶11 > |𝐶12|, 𝐶33 × (𝐶11 + 𝐶12) > 2 × 𝐶213 𝐶44 > 0, 𝐶66 > 0

(8)

Whereas for triclinic phase, conditions are: 𝐶11 > 0, 𝐶22 > 0, 𝐶33 > 0, 𝐶44 > 0, 𝐶55 > 0, 𝐶66 > 0, [𝐶11 + 𝐶22 + 𝐶33 +2 × (𝐶12 + 𝐶13 + 𝐶23)] > 0, (𝐶33 × 𝐶55 ― 𝐶235) > 0, (𝐶44 × 𝐶66 ― 𝐶246) > 0, (𝐶22 + 𝐶33 ―2 × 𝐶23) > 0.

(9)

Table 4 lists the elastic constants Cij, bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio ν and the universal anisotropy AU calculated for these Li2S2 phases. Applications of Eq. 8 and 9 show that the stability criteria are generally fulfilled for these Li2S2 phases. Relative to the cubic Li2S, Li2S2 is expected to be much softer and more flexible due to the presence of the S-S bonds and anisotropic Li-S interactions in the lattice. The calculated bulk modulus (B) of these Li2S2 solids is in the range of 31.2-34.7 GPa and the shear modulus (G) is in the range of 14.1-19.3 GPa. These values are significantly smaller relative to those 18


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calculated for the cubic Li2S with B and G being 41.1 and 33.3 GPa, respectively (Table S1, Supporting Information). Note that the reported bulk modulus of -S8 solid is found to be 7.7 GPa 46.

Table 4. Elastic constants Cij, bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio ν and Universal anisotropy AU for Li2S2. The expressions for B, G, E, ν and AU are taken from the reference

45.

The unit for

Cij, B, G, and E is in GPa, whereas ν and AU are unitless. Tetragonal

C11

Triclinic

Hexagonal

(P42/mnm)

(P1)

(P63/mmc)

51.6

65.9

71.1

C22

41.2

C33

45.0

38.2

95.0

C44

18.8

26.6

10.7

C55

21.7

19



Page 20 of 31

C66

15.4

14.3

22.1

C12

25.7

22.4

26.9

C13

20.7

28.6

5.4

C15

-7.2

C23

26.8

C25

4.1

C35

8.0

C46

4.9

B

31.2

32.3

34.7

G

15.8

14.1

19.3

E

40.5

37.0

48.9

ν

0.28

0.31

0.27

AU

0.14

2.65

1.60

20


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Fig. 5 displays the spatial dependence of elastic properties calculated for Li2S and Li2S2. Unlike Li2S, Li2S2 phases show anisotropic characteristics in both Young’s modulus and shear modulus. These anisotropy elastic properties are largely attributed to the S-S bonds in the crystalline lattice. The maxima of Young’s modulus can be found along [1 1 0], close to [1 1 1] and [0 0 1] directions which are the same orientations as those of S-S bonds in these crystalline Li2S2 phases. Similarly, the same trend can also be found for the shear modulus of these Li2S2 solids. Thus, the highly anisotropic Young’s modulus is dictated by the orientation and distribution of S-S bonds in the crystal lattices. For the hexagonal Li2S2 phase, the highest degree of anisotropy found along [0 0 1] in Young’s modulus can be attributed to the uniform arrangement of S-S bond direction along [0 0 1] direction in the crystal lattice. Thus based on these results, we suggest that it is possible to tune the elastic properties in functionalized cathode in Li-S battery by controlling the orientation and distribution of S-S bonds during the synthesis process.

21



Page 22 of 31

Figure 5. The spatial dependence of mechanical properties associated with cubic Li2S and Li2S2 phases. The grey regime represents the maxima of the computed shear modulus.

4. Summary First-principles (DFT) calculations coupled with the van der Waals correction terms suggest that low-energy phases of Li2S2 (i.e. triclinic, hexagonal and 22


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tetragonal) are thermodynamically stable with the hexagonal phase being the ground state at ambient conditions. The calculated equilibrium cell potential of Li2S2 is found to be around 2 eV at 300 K and it is within a narrow voltage range ~ 0.1 V relative to the Li2S. In contrast to Li2S, highly anisotropic elastic properties are predicted for these Li2S2 crystalline phases which can be attributed to the unique alignments of S-S bonds in their crystalline lattices. The calculated results are expected to provide basis to further improve the mechanical stability and flexibility of Li-S solids in composite Li-S cathodes in Li-S battery.

Supporting Information DFT parameters benchmark using crystalline Li2S. Calculated Raman and IR data, calculated thermal dynamic properties of different Li2S2 crystal structures. Acknowledgement K.C.L acknowledge the support from California State University Northridge faculty start-up fund. RAMA and Superior, high-performance computing clusters at Michigan Technological University were used in obtaining results presented in this paper. Support from Dr. S Gowtham is gratefully acknowledged.

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Thermodynamic and Mechanical Stability of Crystalline Phases of Li2S2

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