Toward a Molecular Understanding of Energetics in Li–S Batteries

Article pubs.acs.org/JPCC

Toward a Molecular Understanding of Energetics in Li−S Batteries Using Nonaqueous Electrolytes: A High-Level Quantum Chemical Study Rajeev S. Assary,*,†,‡ Larry A. Curtiss,*,†,‡ and Jeffrey S. Moore‡,§ †

Materials Science Division and ‡Joint Center for Energy Storage Research, Argonne National Laboratories, Argonne, Illinois 60439, United States § Department of Chemistry, University of Illinois at Urbana−Champaign, 505 South Mathews Avenue, Urbana, Illinois 61801, United States S Supporting Information *

ABSTRACT: The Li−S battery (secondary cell or redox flow) technology is a promising future alternative to the present lithium intercalation-based energy storage, and, therefore, a molecular level understanding of the chemical processes and properties such as stability of intermediates, reactivity of polysulfides, and reactivity toward the nonaqueous electrolytes in the Li−S batteries is of great interest. In this paper, quantum chemical methods (G4MP2, MP2, and B3LYP) were utilized to compute reduction potentials of lithium polysulfides and polysulfide molecular clusters, energetics of disproportionation and association reactions of likely intermediates, and their reactions with ether-based electrolytes. Based on the computed reaction energetics in solution, a probable mechanism during the discharge process for polysulfide anions and lithium polysulfides in solution is proposed and likely intermediates such as S42−, S32−, S22−, and S31− radical were identified. Additionally, the stability and reactivity of propylene carbonate and tetraglyme solvent molecules were assessed against the above-mentioned intermediates and other reactive species by computing the reaction energetics required to initiate the solvent decomposition reactions in solution. Calculations suggest that the propylene carbonate molecule is unstable against the polysulfide anions such as S22−, S32−, and S42− (ΔH† < 0.8 eV) and highly reactive toward Li2S2 and Li2S3. Even though the tetraglyme solvent molecule exhibits increased stability toward polysulfide anions compared to propylene carbonate, this molecule too is vulnerable to nucleophilic attack from Li2S2 and Li2S3 species in solutions. Hence, long-term stability of the ether molecules is unlikely if a high concentration of these reactive intermediates is present in the Li−S energy storage systems.

1. INTRODUCTION Successful developments of rechargeable batteries with high energy density are essential for energy storage, especially for stationary and transportation needs.1−9 Alternatives for going beyond the concept of lithium ion batteries include metal (Li/ Na/Mg)−O2 and metal (Li/Na/Mg)−S batteries for transportation requirements and the redox flow batteries (RFB) for stationary applications.10−14 All these game changing ideas require fundamental understanding of the chemical processes, cell design, and technoeconomic analysis prior to implementation for societal needs.15,16 At present, the fundamental redox chemical processes concerning all three aforementioned areas are under intense experimental and theoretical study. After three decades of research, Li−S still offers significant challenges such as cycling efficiency, formation of insulating Li−S layers on the anode, anode reactivity with electrolyte,17,18 poor understanding of solid−electrolyte interface, stability of nonaqueous electrolyte, solubility of polysulfide and conductivity, and safety issues.19−24 In spite of these challenges, this is a very © 2014 American Chemical Society

appealing battery reaction system due to the high theoretical capacity (10 times that of the present Li ion) and the fact that sulfur is relatively abundant and less toxic compared to the cathodes in the conventional lithium batteries. There are some recent developments such as semisolid lithium−sulfur flow batteries and smaller pore nanoparticle assisted sulfur reduction that are providing increased interest in the lithium−sulfur fundamental chemistry.11,12,21,25−32 There are many lithium−sulfur electrochemical studies in the literature18,33 (and references therein), however very few have probed the reaction intermediates and products under the battery operating conditions.34−36 Recently Abruna et al. suggested that reduction of sulfur in the nonaqueous solvent involves persistent anion intermediates during charge and discharge processes.35 Barchasz et al.36 has confirmed this with Received: February 12, 2014 Revised: May 7, 2014 Published: May 9, 2014 11545

dx.doi.org/10.1021/jp5015466 | J. Phys. Chem. C 2014, 118, 11545−11558

The Journal of Physical Chemistry C

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details of disproportionation/association reactions in a Li−S cell using conventional electrochemical methods. Considering the various likely combinations of intermediates between S8 (cathode) and S2− (at the end of discharge, Li2S), in the presence of Li ions, the reaction mechanisms are clearly complex and require further fundamental investigations using both experiment and theory. To our knowledge, there have been few theoretical studies of the chemical and electrochemical reactions of lithium sulfides or sulfides in the context of Li−S battery. Recently, Chen et al.37 reported DFT studies of Li2Sx clusters, x = 1 to 8, that provided insight into their geometries, stability, and their involvement in the discharge mechanism. However, there is still a lack of knowledge of thermochemistry of chemical and electrochemical reactions in solution and reaction mechanisms of likely intermediates from these reactions with the nonaqueous electrolytes that are required to promote improvements in Li−S batteries. In this paper, we present a comprehensive set of predictions based on accurate thermochemical data for possible reactions in the Li−S system using high-level quantum chemical methods. We consider an overall reaction based on the reversible formation of Li2S from lithium and sulfur (S8) cathode in a nonaqueous solvent (for example dimethoxyethane (DME),38,39 1,3-dioxolane17 (DOL), tetraethylene glycol dimethyl ether (TEGDME36,40−42)). The likely reaction sequence in the discharge reaction is shown in eq 1 and also shown in Scheme 1.

Four critical types of reactions (I to IV) shown in Scheme 1 are considered in this study to gain molecular level understanding of various chemical and electrochemical processes. They are fragmentation reactions (I), electrochemical reduction reactions of polysulfur oligomers and lithium polysulfides (II), association or dissociation reactions of polysulfides and lithium poly sulfides36 (III), and chemical reactions of polysulfides and lithium polysulfides with nonaqueous electrolytes (IV). The intermediates chosen for the chemical reactions with electrolytes (IV) are based on likely intermediates from the reactions I−III. The results of our quantum chemical calculations are discussed in terms of reaction mechanisms in Li−S battery systems. Details regarding computations are presented in section 2. In section 3, detailed results and discussion are presented.

2. COMPUTATIONAL DETAILS All calculations, except for the stability of ether (TEGDME) and PC in the presence of polysulfides and lithium sulfides, were performed using the highly accurate G4MP243,44 level of theory, which is based on coupled cluster theory. We note that G4MP2 predicts very accurate electron affinities for sulfur atom and sulfur dimer compared to density functional methods (see Table S1 of the Supporting Information). Single point solvation energy calculations were performed using the SMD45 solvation model on gas phase optimized geometries employing a dielectric medium of acetone (ε = 20.5) similar to our previous studies.46,47 We note that changing the solvent (water, DMSO, acetonitrile) dielectric to acetone does not have a significant effect on the computed reduction potentials of S2 species (Table S1A of the Supporting Information). We also note that optimizing in gas phase or in solvent dielectric medium does not affect the accuracy of computed reduction potential (Table S1B of the Supporting Information). The B3LYP/6-31+G(d)// B3LYP/6-31+G(d) level of theory was used to evaluate the solvation free energy for the reduction potential calculations. Thus, for redox potential evaluations and thermochemistry of polysulfides (sections 3.1 to 3.3), the free energy in solution is the sum of the gas phase free energy at the G4MP2 level of theory and the solvation energy at the density functional theory. All calculations presented in this paper were performed using the Gaussian 09 Software.48 The reduction potentials (Ered with respect to Li/Li+) are computed from the computation of Gibbs free energy change (ΔGred, eV) at 298 K in the solution (dielectric) for addition of an electron to the species of interest, using the equation Ered = −ΔGred/nF − 1.24 V, where F is the Faraday constant (in eV) and n is the number of electrons involved in the reduction process. The addition of the constant “−1.24 V” is required to convert the free energy changes to reduction potential (Li/Li+ reference electrode), a commonly used convention to compute the reduction potentials in solution.49,50 The change in energy of electrons when going from vacuum to nonaqueous solution is treated as zero, similar to what has been used by others.51 Details regarding the computation of redox potential can be found elsewhere.51−56 Solvation energy contributions of a species are added to the gas phase enthalpies to approximate solution enthalpies (Hsoln = Hgasphase + ΔGsolv). We make this approximation because the main free energy contributions (GCDS: cavitation, dispersion, and solvent structure terms) cancel out (or are negligible) when computing the apparent barriers (H(TS) − H(reactants)). Therefore, the dominant solvation contributions are

S8 → Li 2S8 → Li 2S6 → Li 2S4 → Li 2S3 → Li 2S2 → Li 2S (1)

Scheme 1. Schematic of the Critical Reactions in a Li−S Cella

a

(I) Fragmentation reaction of S8, (II) reduction potentials of polysulfides and lithium polysulfides, (III) association or dissociation reactions of polysulfides, and (IV) chemical reactions between polysulfides and lithium polysulfides with TEGDME solvent molecule. 11546



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Figure 1. (left) Calculated potential energies (au) and S−S bond lengths (Å) from the ADMP trajectories (B3LYP/6-31+G(d)). (right) Lowest energy optimized structure taken as the starting structure (1 fs) and selected snapshots from 100, 150, and 200 fs from the ADMP simulation.

using experiment and theory.57−63 The structures of Sn clusters (n = 2−8) were also calculated in a recent computational study by Chen37 et al. Gas phase cluster calculations by Wong et al.,59 indicate that the most stable conformer of S8 (at 0 K) is the D4d cyclic form with a S−S bond length of 2.08 Å. Wong et al.59 have also shown that there are at least five more S8 cyclic conformational isomers within a range of 0.5 eV, with the next stable isomer being an exo−endo S8 and a S8 cluster with C2 symmetry, 0.29 eV relative to the lowest energy D4d structure. To provide more detailed information about the S−S bond of S8 and its conformational flexibility we have performed ADMP64−67 simulations with a time step = 0.1 fs (fs) for 230 fs at the B3LYP/6-31+G(d) level of theory using the lowest energy structure as the starting structure. The result of the simulation is shown in Figure 1. The potential energies (au), the S1−S2 bond length (Å) during the simulation, the lowest energy D4d structure, and snapshots at 100, 150, and 200 fs are shown. As shown in Figure 1, analysis of structure and potential energy within 100−150 fs suggests that an energy change δE = 0.54 eV leads to stretching of the S−S bond by a significant 0.42 Å. Therefore, the ADMP simulations suggest that various isomers with short (1.95 Å) to long sulfur−sulfur (2.37 Å) bonds are likely within a small potential energy span of about 0.5 eV, which is in good agreement with the static cluster calculations by Wong et al.59 In terms of gas phase cluster calculations, a triplet configuration of S8 is found to be the highest energy neutral structure, which is 1.36 eV higher in energy than the lower energy D4d sulfur allotrope. In addition, studies have also shown that S6 has a cyclic structure, while S7, S5, S4, and S3 prefer acyclic (linear) geometries.37,59 We have used these results for the lowest energy structures of the Sn clusters in G4MP2 calculations to obtain accurate predictions of fragmentation energies of S8. The B3LYP/6-31G(2df,p) optimized geometries used for the G4MP2 energies are given in the Supporting Information (Figure S1).

electronic, nuclear, and polarization terms (GENP) which are included in the computation of solution enthalpies. The C−O reaction barriers presented in section 3.4 are apparent enthalpy barriers, computed as the difference between the enthalpy of the transition state structure (H†) and the sum of enthalpies of reactants in solution at 298 K. The B3LYP/6-31+G(d) level of theory is used to compute the transition state structures in the gas phase, and subsequently, a single point solvation energy calculation is performed using the SMD solvation model (with an acetone dielectric medium) at the same level of theory. In addition to the B3LYP/6-31+G(d) level of theory, a gas phase single point MP2/6-311+G(2df,p) level of theory is used to obtain more accurate energies for the barriers. In general, both levels of theories agree very well, giving confidence in the accuracy of our computations. Thus, the enthalpy barriers presented in section 3.4 are the sum of the gas phase enthalpy barrier (either from B3LYP or MP2 calculations) and the solvation energy (ESMD). TS TS reactants TS reactants ΔHsoln = (Hgas − Hgas ) + (ESMD − ESMD )

In section 3.4, free energies of proton/hydrogen abstraction reactions were evaluated at the B3LYP/6-311+G(2df,p)// B3LYP/6-31G(2df,p) level of theory, with solvation energies computed using the B3LYP/6-31G(2df,p) level of theory with the SMD solvation model (acetone dielectric)

3. RESULTS AND DISCUSSION 3.1. Sn Clusters. The energetics for the fragmentation of Sn clusters and their reduction potentials are critical in understanding the reaction mechanisms upon discharge and the dissolution of sulfur cluster species in the Li−S energy storage devices. The fragmentation of Sn clusters (reaction I in Scheme 1) can occur during the discharge process prior to reduction and affect the initial reduction step. In this section we present the G4MP2 energies for these reactions. Fragmentation Energies. Various sulfur clusters (Sn) and their isomers have been investigated by Steudel and Wong 11547



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The G4MP2 fragmentation energies (ΔEo, gas phase) of the S8 cluster into various smaller clusters are given in Table 1. Also

Table 2. G4MP2 Electron Affinities (EA, eV), Gas Phase Gibbs Free Energies for Reduction (ΔGred, eV), and Reduction Potentials (Ered in V with Respect to Li/Li+ Reference Electrode) of Sn and Li2Sn (n = 2 to 8) Species

Table 1. G4MP2 Fragmentation Energies (in eV) of S8a Sn species

ΔE(0)

ΔH(298 K)

S8 (cyclic) S6 (cyclic) + S2b S5 + S3 2S4 2S3 + S2 S4 + 2S2 S7 + Sb 4S2

0 1.39 1.78 2.05 3.33 3.28 3.01 4.52

0 1.40 1.76 2.05 3.34 3.31 3.03 4.57

a b

ΔG(298 K)

ΔG(298 K) + ΔGsolv

speciesa

EA1b

ΔGred1c

ΔGred2c

Ered1a

Ered2a

S8cyc

0.92 1.27 1.45 2.29 2.29 2.63 3.13

0 0.87 1.16 1.32 2.10 2.14 2.56 2.96

S7 S6 S5 S4 S3 S2d S8d S6d Li2S8 Li2S7 Li2S6 Li2S5 Li2S4 Li2S3 Li2S2 Li2S LiS8e LiS6e LiS3e LiSe

1.12 1.77 1.35 1.80 2.29 2.36 1.70 2.72 2.82 2.91 2.87 2.10 1.91 1.53 0.41 0.49 0.33 3.52 3.51 1.95 1.41

−1.30 −1.85 −1.49 −1.92 −2.34 −2.39 −1.69 −2.70 −2.90 −3.00 −2.96 −2.18 −2.01 −1.68 −0.42 −0.51 −0.57 −3.52 −3.50 −1.94 −1.38

−1.41 −0.90 −0.69 −0.75 −0.27 0.64f 2.02f

1.56 2.30 1.94 2.41 3.00 3.15 2.71 2.92 3.31 2.94 2.99 2.23 2.23 1.92 0.52 0.06 0.13 3.69 3.87 2.91 2.60

2.35 2.27 2.33 2.54 2.54 2.38 2.07

Structures are given in the Supporting Information (Figure S1). Triplet electronic state has the lowest energy for S2 and S.

given in the table are corresponding G4MP2 enthalpies and free energies, the latter of which include solvation effects at 298 K. The fragmentation reactions of S8, all of which involve S−S bond cleavage, are computed to be thermodynamically uphill and indicate that the cyclic S8 cluster is quite stable. The enthalpies and free energies in Table 1 including solvation effects also indicate that fragmentation of cyclic S8 is unfavorable. From Table 1, the fragmentation free energies (in solution) of cyclic S8 range from 0.87 to 2.56 eV. The most favorable fragmentation of S8 is to S6 and S2 (+0.87 eV) and S5 and S3 (+1.16 eV) and the least favorable is to S7 and S (2.56 eV). The fragmentation of S8 to two S4 species is endergonic by 1.32 eV. Therefore, based on the Gibbs free energies, the cyclic S8 structure is energetically quite stable. Based on the fragmentation energies of S8 to two fragments, the relative stability of fragments is in the following order: S8 > S6 and S2 > S5 and S3 > 2S4. Overall, the relative stability of the S8 structure indicates that the molecule is unlikely to exhibit high chemical or electrochemical reactivity compared to S6, S5, S2, S5, S3, and S4 species. Based on the formation energies, the S4 species is likely to be more reactive than other allotropes. Therefore, it is considered as an ideal starting material as a cathode provided there are effective synthetic routes.25 Reduction Potentials. The optimized gas phase geometries of the singly and doubly negatively charged Sn clusters (n = 2− 8) were computed at the B3LYP/6-31G(2df,p) level of theory for the calculation of G4MP2 energies. The optimized geometries are given in the Supporting Information (Figure S3). The computed first (Ered1) and second (Ered2) electron reduction potentials of Sn (n = 2 to 8) species at the G4MP2 level of theory are given in Table 2 along with the electron affinities and Gibbs free energies of reduction from which reduction potentials are calculated. The binding of the electron to the Sn (n = 2 to 8) in solution is favorable in all cases with all electron affinities being positive. Binding of the second electron to the monoanion in the gas phase is thermodynamically uphill (negative electron affinity), while inclusion of solvation contributions favors the binding of the second electron. The negative electron affinities result in less accurate reduction potential, but, in cases where experimental values are available, agreement is reasonable. It has been found that finite basis sets can give reasonable results in comparison to gas phase experimental results for gas phase temporary anions with negative electron affinities due to a cancelation of errors.68 The good agreement between computed second reduction

2.89 2.64

a

Structures are given in the Supporting Information (Figures S1 and S2). bGas phase first electron affinity (EA) at the G4MP2 level. cGas phase free energy change for the reduction process. dTriplet electronic state for neutral species. eDoublet electronic state for neutral species. f Second electron addition is not favorable in the gas phase, while this process is favorable in solution.

potentials of Sn species (S2, S4, S3) and experimental observations34,36 suggests that the finite basis set and level of theory used here is giving reasonable results for these −2 anions. The first reduction of cyclic S8 occurs at 1.56 V and results in the formation of an acyclic S8 anion radical, and the subsequent reduction to linear S82− occurs at 2.35 V. The reduction potentials of sulfur species (S 4 , S 3 , S 2 ) from cyclic voltammetry10,34,36,69 (2−2.5 V) are in good agreement with the computed values in Table 2. This potential range is significantly higher than the calculated first reduction potential of cyclic S8. The cyclic S8 structure may undergo ring perturbation (cyclic S8 → acyclic S8) prior to reduction at the battery operating condition. Reduction potentials of acyclic S8 are likely higher than that of cyclic structure to the higher energy of the isomers.10,34,36 For instance, the computed reduction potential of S8 (D2d boat conformer) is 2.08 V, while the computed reduction potential of S8 triplet is 2.92 V. Similar to the cyclic S8 species, the computed first reduction potential of the cyclic S6 species is also quite low, 1.94 V. The first reduction potentials of the S7 and S5 clusters occur in the region of 2.3 to 2.4 V, while the initial reductions of S4, S3, and S2 occur above 2.7 V. The computed second reduction of S2 (S21− → S22−) occurs at 2.07 V, which is consistent with a recent experimental study, where the second reduction plateau is observed around at 1.97 V with respect to Li/Li+.25 In general, potentials required for the initial reduction of sulfur allotropes are in the following descending order: S8cyc > S6cyc > S7 > S5 > S2 > S4 ∼ S3. Our calculations suggest that acyclic 11548



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Table 3. G4MP2 Reaction Energies for Dissociation/Association of Li2S8 and Possible Polysulfide Speciesa (eV) reaction no.

reactiona

ΔE(0)

ΔH(298 K)

ΔG(298 K)

ΔG(298 K) +ΔGsolvb

1 2 3 4 5 6 7 8 9 10 11 12 13d 14d 15d 16d 17d 18d 19d 20d 21d 22d 23d 24d 25d 26d

S8 + 2Li → Li2S8 Li2S8 → Li2S6 + (1/4)S8 Li2S8 → Li2S5 + S3c Li2S8 → Li2S4 + (1/2) S8 Li2S6 → Li2S4 + (1/4)S8 Li2S4 → Li2S2 + (1/4)S8 2Li2S4 → 2Li2S3 + (1/4)S8 2Li2S3 → 2Li2S2 + (1/4)S8 2Li2S2 → 2Li2S + (1/4)S8 Li2S8 → LiS6c+ LiS2c Li2S6 → 2LiS3c Li2S6 → LiS4c + LiS2c Li2S82− + 2Li+ → 2Li2S4 Li2S82− + 2Li+ → Li2S3 + Li2S5 Li2S82− + 2Li+ → Li2S2 + Li2S6 Li2S82− + 2Li+ → 2Li2S + Li2S7 Li2S72− + 2Li+ → Li2S3 + Li2S4 Li2S72− + 2Li+ → Li2S2 + Li2S5 Li2S72− + 2Li+ → Li2S + Li2S6 Li2S62− + 2Li+ → 2Li2S3 Li2S62− + 2Li+ → Li2S2 + Li2S4 Li2S62− + 2Li+ → Li2S + Li2S5 Li2S52− + 2Li+ → Li2S3 + Li2S2 Li2S52− + 2Li+ → Li2S + Li2S4 Li2S42− + 2Li+ → 2Li2S2 Li2S42− + 2Li+ → Li2S + Li2S3

−6.02 −0.11 1.11 0.18 0.29 1.14 0.96 1.32 2.10 2.36 1.77 4.10

−6.05 −0.13 1.09 0.15 0.28 1.15 0.98 1.32 2.13 2.36 1.78 4.14

−5.42 −0.21 0.59 −0.05 0.16 1.00 0.79 1.21 2.35 1.76 1.16 3.43

−5.72 −0.24 0.46 −0.17 0.07 0.62 0.54 0.69 1.54 1.47 1.37 3.17 −1.27 −1.06 −0.73 0.09 −1.29 −1.00 −0.25 −1.33 −1.26 −0.76 −1.26 −0.76 −1.51 −1.09

Structures of lithium polysulfides are given in the Supporting Information (Figure S2). bΔGsolv computed using the SMD solvation model. cRadical electronic state. dReactants are charged, therefore the gas phase energetics (∼−12 to −14 eV) are not given.

a

sulfur allotropes (S7 to S3) can be reduced at ∼2.3 V compared to cyclic allotropes (S8 and S6) that require a potential of around ∼1.9 V for the initial reduction. 3.2. Lithium Polysulfides, LimSn. If we assume the first step in the discharge mechanism involves double reduction of S8 and reaction with two Li cations to form Li2S8, there are numerous possible subsequent electrochemical and chemical reactions (dissociation and association) involving Li2S8 and its fragments that can lead to the final product Li2S (reaction type II in Scheme 1). In this section we present and discuss the computed reduction potentials of various lithium polysulfides (reaction type I, Scheme 1) and the energetics of some of the dissociation reactions of lithium polysulfides at the G4MP2 level of theory. Reduction Potentials of Lithium Polysulfides. The G4MP2 reduction potentials (1 electron) of the lithium polysulfides Li2S8, Li2S6, Li2S4, Li2S3, and Li2S2 are given in Table 2. The B3LYP/6-31G(2df,p) optimized geometries of the anions used for these computations are given in the Supporting Information (see Figure S2). The computed reduction potentials of Li2Sn, n = 4−8, are above 2.2 V and, therefore, are very likely to occur at the discharging conditions of the Li−S battery, while Li2S3 and Li2S2 have very low computed reduction potentials (below 0.5 V) and are unlikely, unless there is very deep discharge. For instance, the computed reduction potentials (first and second) of Li2S8 and Li2S6 species are around 2.9 and 2.2 V, respectively. From Table 2, radical species such as LiS3, LiS4, LiS6, and LiS8 have higher reduction potentials (>2.9 V). Therefore, these radical species will undergo reduction and the resulting anions would most likely bind with available lithium cations. Based on the calculations, lithium polysulfides except Li2S3, Li2S2, and

Li2S will undergo reduction at the battery operating potentials. The computed reduction potentials of various sulfur (Sn) clusters and lithium polysulfides together with dissociation energetics (described later) are useful to understand likely reactions in the Li−S cell. Dissociation Reactions. The optimized geometries of Li2S8 and its fragments used for the G4MP2 energy calculations were determined at the B3LYP/6-31G(2df,p) level of theory (geometries given in the Supporting Information, Figure S2). The overall reaction during the discharge of a Li−S battery is written as S8 + 16Li → 8Li2S. The computed Gibbs free energy change using G4MP2 theory for this reaction in solution is −36.05 eV (i.e., −2.25 eV per Li atom). This computed value in the solution medium is consistent with the experimental value of 2.20 V.6 We note that the computed free energy change in the gas phase is −1.64 eV per Li atom, indicating that the solvation contribution is essential to compute accurate energetics. The energetics of almost all lithium polysulfide dissociation/association reactions are significantly affected by the presence of solvent medium compared to gas phase calculations (see Table S3 and S4 of the Supporting Information). During the discharge of the Li−S cell, the elemental sulfur (S8) undergoes double reduction and the resulting dianion binds to lithium ions to form Li2S8. Subsequently, this intermediate undergoes further reduction and disproportionation according the reaction sequence Li2S8 → Li2S in eq 1 and shown in Scheme 1. The Gibbs free energies of individual steps of the discharge mechanism were computed at the G4MP2 level of theory and are listed in Table 3. The G4MP2 energies suggest that formation of Li2S8 from Li2 and cyclic S8 in 11549



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solution is exergonic by 5.72 eV (reaction 1 in Table 3). We note that the energetics could be slightly different if a different reference value for Li is used. For example, when the energy of Li8 is used as one of the reactant (S8 + (1/4)Li8 → Li2S8), the reaction is exergonic by 4.16 eV in solution. Formation of Li2S6 and (1/4)S8 from Li2S8 is also exergonic by 0.24 eV (reaction 2), while the formation of Li2S5 and S3 from Li2S8 (reaction 3) is endergonic by 0.46 eV. The formation of Li2S4 from Li2S8 (reaction 4) is computed to be exothermic in solution (0.17 eV). Formation of Li2S4 and (1/4)S8 from Li2S6 (reaction 5) is nearly thermoneutral (ΔGrxn = 0.07 eV). Formation of Li2S2 (reactions 5, 7) from Li2S4 and Li2S3 is computed to be endergonic by ∼0.5 eV. Similarly, formation of Li2S3 from Li2S4 (reaction 7) is also computed to be thermodynamically uphill by ∼0.5 eV. Disproportionation of Li2S2 to Li2S (reaction 9) is thermodynamically uphill by 1.5 eV, while fragmentation reactions (reactions 10−12) that result in the formation of radical species such as LiS3 or LiS2 from Li2S8 or Li2S6 are largely unfavorable. Thus, based on these calculations, the chemical dissociation of Li2S8 to Li2S3, Li2S2, and Li2S is highly unfavorable. In terms of reductive dissociation, the Li2S8 species can undergo double reduction and subsequent reaction with two lithium ions from the solution to form two Li2Sn fragments (n = 1 to 7). Computed free energy changes of these reactions (reactions 13−16) are also given in Table 3. Based on the energetics of computed reactions, formation of Li2S4 (reaction 13) is most likely (ΔG = −1.27 eV) and formation of Li2S (reaction 16) is least likely (ΔG = 0.09 eV). Computed free energies of formation of Li2S3 (reaction 14) and Li2S2 (reaction 15) are also exergonic by 1.27 and 0.73 eV, respectively. 3.3. Anionic Species, Sn2− and Sn1− Species. The chemical complexity of Li−S batteries arises from the formation of various polysulfide (Sn; n = 1−8) anions. The polysulfide anions may undergo disproportionation or association reactions (reaction III, Scheme 1). In this section we present the G4MP2 energies for disproportionation reactions of anions and dianions of the Sn species and association reactions of some of these species. The B3LYP/6-31G(2df,p) optimized geometries of these species are given in Figure S3 of the Supporting Information. Computed Gibbs free energies (298 K) of 48 reactions (Rn) including various disproportionation and association reactions at the G4MP2 level of theory (in solution) are presented in Table 4. Based on the free energy change we have assigned either “likely,” “no”, or “yes” to describe the thermodynamic feasibility of the reactions in the solution. Based on the computed free energies of reaction, the most favorable polysulfide anions resulting from the disproportionation of S82− are in the order S31− radical (R9) > S42− (R12) > S62− (R3). It is likely that the S31− radical undergoes reduction to form S32− under battery discharge conditions. Among the disproportionation reactions of S62−, the most likely reaction is the formation of S31− radical, R25. Disproportionation of S81− to S21− (via R20 and R21) is also thermodynamically feasible. Thus, the S8 anion radical can undergo either reduction (2.35 V) or dissociation. Note that the S21− species is likely to be reduced at 2.07 V (Table 2) to form S22−. The disproportination of S61− to S31−, S21−, and S41−, via R33, R34, and R35, respectively, is also thermodynamically likely, and these resulting anion radicals could be reduced to corresponding dianions during the Li−S discharge process. Disproportionation reactions of S52−, S42−, S41−, S31−, and S22− are thermodynamically not feasible.

Table 4. Computed Gibbs Free Energies (eV) of Disproportionation and Association Reactions (Rn) of Polysulfide Ions in Solution (298 K) at the G4MP2 Level of Theorya label

category

reactions

ΔG298K (soln)

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 R19 R20 R21 R22 R23 R24 R25 R26 R27 R28 R29 R30 R31 R32 R33 R34 R35 R36 R37 R38 R39 R40 R41 R42 R43 R44 R45 R46 R47 R48 R49 R50

S82−

S82− → S42− + (1/2)S8 S82− → S22− + S2 + S4 S82− → S62− + (1/4)S8 S82− → S62− + S2b S82− → S22− + S6 S82− → S61− + S21− S82− → 2S2 + 2S2− S82− → 2S31− + S2 S82− → 2S31− + (1/4)S8 S82− → 3S2 + S22− S82− → S5 + S32− S82− → 2S41− S82− → S22− + 2S3 S82− → S51− + S31− S82− → S52− + S3 S82− → S52− + (3/8) S3 S82− + S22− → 2S5 2− S82− + S42− → 2S62− S82− + S62− → 2S72− S81− → S6 + S21− S81− → (3/4)S8 + S21− S81− → 2S3 + S21− S81− → S61− + S2 S81− → S61− + (1/4)S8 S62− → 2S31− S62− → S32− + S3 S62− → 2S2 + S22− S62− → S42− + S2 S62− → S4 + S22− S62− → S41− + S21− 2S62− → S52− + S72− S61− → S31− + S3 S61− → S31− + (3/8)S8 S61− → (1/2)S8 + S21− S61− → (1/4)S8 + S41− S52− → S31− + S21− S52− → S3 + S22− S52− → (1/4)S8 + S32− 2S52− + 2S32− → S42− S42− → 2S21− S42− → S22− + (1/4)S8 S42− + S22− → 2S32− 2S42− → S52− + S32− S41− → S31− + Sb S41− → S21− + S2 2S41− → S82− 2S32− + 2S31− → 3S42− S31− → S21− + Sb S31− + S42− → S32− + S41− S22− → 2S1−

0.28 2.68 0.17 0.91 1.41 0.91 2.24 0.49 −0.25 3.51 1.09 0.00 3.12 0.29 1.20 0.10 −1.08 0.06 0.47 −0.28 −0.96 +0.96 +0.49 −0.25 −0.42 1.35 2.60 0.85 1.77 0.21 0.09 0.06 −0.66 −0.15 −0.53 0.15 1.18 0.50 −0.15 0.48 1.01 −0.35 0.15 2.27 1.12 0.00 −0.12 2.90 0.46 1.52

S81−

S62−

S61−

S52−

S42−

S41−

S32− S31− S22−

likely no likely no no no no no yes no no yes no likely no likely yes likely no yes yes no no yes yes no no no no likely likely likely yes yes yes likely no no yes no no yes likely no no yes yes no no no

a

The optimized structures of all species are shown in Figure S3 of the Supporting Information. bRadical state.

Based on the computed reduction potentials (Table 2), free energies of various dissociation/association reactions of lithium polysulfides (Table 3), and sulfide anions (Table 4), a detailed mechanism is proposed to explain the likely reaction network of 11550



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Figure 2. Schematic representation of reduction pathways for polysulfide anions (a) and lithium polysulfides (b) in Li−S systems, proposed based on the computed free energies of various reactions and reduction potentials (Tables 1 to 4). (a) Each row represents the mechanism of formation of small polysulfides (Sn, n ≤ 3) from longer chain sulfides (Sn, 4 ≥ n ≤ 8). In each row, adjacent yellow cells represent electrochemical processes; cells with Rn (green colored cells) represent chemical transformations (shown in Table 4), and cells with blue color (with 2Li+) represent addition of two lithium ions. (b) Each row represents mechanism of formation of small chain lithium sulfides (Li2Sn, n ≤ 3) (last column) from lithium polysulfides (Li2Sn, n ≤ 8). In each row, adjacent yellow cells represent electrochemical (two electron reduction) reactions. In each row, the green cell represents reaction with 2Li+ ions and subsequent fragmentation to give the species shown in the red cell (second last column) .The species in the red cell undergoes further reaction sequences including reduction, addition of lithium ion, and subsequent fragmentation to form the species in the final column, which are shown in the subsequent rows.

polysulfides (S82−) as well as the dissolution of lithium polysulfides (Li2S8) in solution, which is schematically shown in Figure 2. To provide a complete picture of possible chemical transformations in solution, the reaction mechanism is categorized into two groups: (1) reaction networks of polysulfide anions and (2) lithium polysulfides. In Figure 2, each row represents a sequential transformation, regardless of the categories. In each row, adjacent yellow cells represent electrochemical transformation, while the presence of green and blue cells represents chemical transformations. The green cells labeled with Rn represent the reactions described in Table 4. The blue cells labeled with 2Li+ represent addition of two lithium ions to the dianion. Note that the addition of two lithium ions is the termination step for the polysulfide anions to form lithium polysulfides ((Li2Sn, n ≤ 3), column 6 of the polysulfide reaction network). In the reaction network associated with lithium polysulfide (right side of Figure 2), the green cell labeled with 2Li+ represents addition of two lithium ions and subsequent fragmentation to lithium polysulfides ((Li2Sn, n > 3), in red cells, second last column) and shorter chain lithium polysulfides (last column). For example, in the first row, Li2S82− binds to 2Li+ and then fragments to Li2S7 and Li2S. The lithium polysulfides (Li2Sn, n > 3) undergo further reduction, lithium ion addition, and fragmentation, represented in the subsequent cells. In Figure 2, the first category is polysulfide anions, where the reaction network of S81− and S82− to S22−, S32−, and S42− is

shown suggesting that the formation of these anions is thermodynamically feasible. The formation of S22− is possible from the decomposition of S81− and subsequent electrochemical reduction. According to the proposed mechanism, the formation of S42− species is also likely from the S61− via fragmentation (R35) and subsequent reduction process. Similarly, the formation of the S31− radical is most likely via fragmentation (R25) of S62− in the solution. In general, the proposed mechanism is consistent with the experimental studies,29,35,36 and can explain the presence of intermediates such as S22−, S32−, and S31− in a Li−S cell during the discharge process. The reaction network of polysulfide anions in solution indicates that the most abundant intermediate upon complete utilization of S82− is S32−. In Figure 2, the second category is lithium polysulfide, where Li2S8 formed from S8 undergoes further transformations to shorter chain polysulfides (Li2Sn, n ≤ 3), shown in the final column of Figure 2. Based on the composition of the final column of the reaction network, it is clear that Li2S and Li2S2 are the most abundant lithium polysulfides. In the presence of lithium, Li2S2 will convert to Li2S, since the process is thermodynamically downhill (Table S3 in the Supporting Information). We note that both disproportionation and association reactions may also be controlled by the kinetics in addition to the thermodynamics. The computation of reaction barriers for various disproportionation/association reactions such as 11551



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Scheme 2. Schematic Representation of Possible Sites of TEGDME from Where the Proton/Hydrogen Abstractions Are Possible by Reactive Intermediates Such As S22−, Li2S2, S32−, LiS3, S32−, Li2S3 2−, S42−, and Li2S4 in the Solutiona

a

Example hydrogen/proton abstraction reactions are also shown.

Table 5. Computed Enthalpy of Activation (ΔH†) for the C−O Bond Cleavage by Nucleophilic Reaction of Various Species (Sulfides and Lithium Polysulfides) at the MP2/6-311+G(2df,p)//B3LYP/6-31+G(d) Level of Theorya TEGDME

PC

†

ΔH (eV) species S31−d S32− S42− S21−d S1−d S22− Li2S Li2S2 Li2S3 Li2S4

solution 2.27 1.50 1.61 1.82 1.78 1.17 1.44 0.71 1.00 1.12

(2.36) (1.82) (1.93) (2.09) (2.03) (1.66)e (1.64) (0.95) (1.17) (1.35)

gas phase 1.71 (1.80) 0.0 (0.09) 0.15 (0.47) 1.06 (1.33) 0.87 (1.12) 0 1.11 (1.31) 0.49 (0.72) 0.91 (1.07) 1.07 (1.31)

†

ΔH (eV)

solution rate constant,b k (s−1) 2.6 2.7 3.8 1.1 5.0 1.0 2.8 6.2 7.7 7.2

× × × × × × ×

10−26 10−23 10−15 10−18 10−18 10−7 10−12

× 10−5 × 10−7

half-life,c T1/2 (h) 7.3 7.1 5.1 1.8 3.8 1.9 6.8 3.1 2.5 2.7

× × × × × × × ×

1021 108 1010 1014 1013 103 107 10−7

× 102

solution 1.59 0.93 1.07 1.21 1.23 0.61 1.34 0f 0g 0h

(1.54) (0.94) (1.32) (1.19) (1.20) (0.68) (1.32)

solution

gas phase

rate constant, k (s−1)

0.75 (0.71) 0 0 0.18 (0.17) 0 0 1.23 (1.21) 0 0 0

1.59 0.93 1.07 1.21 1.23 0.61 1.34

half-life,b T1/2 (h) 8.2 × 5.8 × 5.1 × 2.2 × 1.0 × 300 1.4

10−15 10−2 10−6 10−8 10−8

a

Values in parentheses are at the B3LYP/6-31+G(d) level of theory. The solvation energy contributions were computed using the SMD solvation model at the B3LYP/6-31+G(d) level of theory using the dielectric medium of acetone. The transition state structures for the C−O bond breaking †

of PC and TEGDME are shown in Figures 3 and 4, respectively. bRate constant is computed using the Eyring equation, (T) = (kBT/hc0) exp−ΔG /RT. Here, instead of free energy of activation (ΔG†), the enthalpy of activation (ΔH†, this includes solvation energy) is used due to the uncertainties in computing entropic contributions in solution from gas phase approximations. Note that rate constants using computed activation free energies in solution are given in Table S2 of the Supporting Information cHalf-life of the reaction (hours) is based on the first order reaction kinetics. dRadical species. eThe activation barrier is 0.81 eV for 1,3-dioxolane for this reaction at the MP2 level of theory in solution (see Figure S5 in the Supporting Information). fThe reaction of PC with Li2S2 to form a PC−Li2S2 complex (Figure 3) is exothermic by 1.8 eV in solution. gThe reaction of PC with Li2S3 to form a PC−Li2S3 complex (Figure 3) is exothermic by 1.2 eV in solution. hThe reaction of PC with Li2S4 to form a PC−Li2S4 complex (Figure 3) is exothermic by 0.8 eV in solution.

described in Tables 1, 3, and 4 and a detailed micro kinetic modeling of these reaction networks is essential to understand the exact nature and concentration of polysulfides in the Li−S cell. Additionally detailed understanding of phase changes during the chemical or electrochemical transformation of sulfides and lithium polysulfides is also essential. This requires another comprehensive study, which is beyond the scope of this investigation. 3.4. Reactivity of Polysufides and Lithium Sulfides toward Ether and Carbonate Solvents. The stability of solvent molecules against various polysulfides and lithium sulfides is crucial to the cyclability of lithium−sulfur batteries. The results of the calculations in the previous sections are used

as a basis to derive likely intermediates that may react with solvents in the computations in this section. The most common electrolytes used are linear ethers such as glymes70,71 (monoglyme, diglyme, triglyme, tetraglyme), cyclic ethers (1,2-dioxolane,17 THF72), and ionic liquids.73 The stability of ethers (cyclic or acyclic) depends on the stability of C−O and C−H bonds toward likely reactive components such as anions, radicals, and the lithium sulfides. Here, we have assessed the stability of tetraglyme (TEGDME), which can be used as a model system for any glymes, by computing both the C−O bond breaking barriers by various likely and reactive sulfur intermediates (S1− radical, S21− radical, S22−, S31− radical, S32−, S42−, Li2S, Li2S2, Li2S3, and Li2S4) in lithium−sulfur battery. 11552



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Figure 3. Computed transition state structures for the C−O bond breaking of propylene carbonate (PC) by various sulfides and lithium sulfides at the B3LYP/6-31+G(f) level of theory in solution. Computed activation barriers are shown in Table 5. Reactions of PC withLi2S2, Li2S3, and Li2S4 result in the formation of PC−LinSm complexes. The complex formation is exothermic (barrierless, the enthalpies of complex formation are given in Table 5).

estimation of entropy) in gas phase calculations;74−78 therefore instead of activation free energies, the enthalpies of activation are computed and discussed throughout this section. We note that solution phase enthalpies (these include solvation free energies) are used in the Eyring equation79 to approximate the rate constants of selected decomposition reactions. (Enthalpies, free energies, and rate constants are also shown in Table S2 of the Supporting Information.) Preliminary calculations using S22− suggest that the enthalpy of activation in solution required to break any C−O bond (out of four possible C−O single bonds) is almost identical. Therefore, for subsequent studies, only one possibility is explored, which is explained in detail. Based on the computations (Table 5), the gas phase barriers for the C−O bond breaking process are quantitatively less than that in the solution due to the high energy of the sulfide or lithium sulfide anions in the gas phase compared to that in the solution. In solution, the anions will be stabilized by the solvation energy. Computed reaction barriers (enthalpy of activation) in solution are discussed throughout. Among the polysulfide anions, the computed reaction barriers by both the S22− and the S32− in solution are 0.61 and 0.93 eV, respectively, to break a C−O bond of the PC molecule. The optimized transition state structures are shown in Figure 3. Under battery operating conditions both reactions are kinetically possible due to the relatively small activation barriers (less than 1 eV). Nucleophilic attack of S22− and S32− with the PC results in the formation of linear carbonates with C−S bonds, and these processes are computed to be exothermic (∼1 eV) in solution

These results are compared with the similar reaction profiles for the propylene carbonate (PC), given that this is not an ideal solvent to explore Li−S chemistry. In addition, thermochemistry of the C−H bond breaking reactions from various TEGDME sites (Scheme 2) was computed to assess the stability of these bonds. The C−O Bond Cleavage. Recent reports suggest that carbonate-based electrolytes are not suitable for Li−S battery applications due to the decomposition of electrolyte.15,35 Ether solvents such as dimethoxyethane (DME), tetraethylene glycol dimethyl ether (TEGDME), 1,3-dioxolane, etc. are recommended due to their relative solvation ability of polysulfides.42 In order to investigate the relative reactivity of polysulfide anions and lithium sulfides toward the carbonate-based and ether-based solvents, we have computed the C−O bond breaking reaction barriers (ΔH†) in solution. The propylene carbonate (PC) and TEGDME solvents were investigated, and the activation enthalpy for the C−O bond breaking of the solvent molecule by S anion radical, S2 anion radical, S22−, S31− radical, S32−, S42−, Li2S, Li2S2, Li2S3, and Li2S4 was computed at the MP2/6-311+G(2df,p)//B3LYP/6-31+G(d) level of theory. The computed enthalpy barriers (ΔH†) for the C−O bond breaking of PC and TEGDME in solution and gas phase are tabulated in Table 5.We note that the computed free energy barriers are about ∼0.5 eV larger than the enthalpy barriers, and they are tabulated in Table S2 of the Supporting Information. Accurate description of free energies of activation for the reactions with significant entropy change (e.g., association and dissociation reactions) in solution is difficult based (over11553



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Figure 4. Computed transition state structures for the C−O bond breaking in tetraglyme at the B3LYP/6-31+G(d) level of theory. Only a portion of the TEGDME is shown for clarity. Computed reaction barriers are given in Table 5.

(See Figure S4 of the Supporting Information). The S21− radical, S31− radical, and S42− have relatively high activation barriers (>1 eV) for the C−O bond cleavage. The computed rate constants and half-life based on a first order kinetic approximations for all reactions are shown in Table S5 of the Supporting Information. The reaction barrier for the C−O bond breaking by the Li2S molecule is 1.34 eV in solution. The lithium polysulfides such as Li2S2, Li2S3, and Li2S4 make stable complexes with PC in the ring-opened form, as shown in Figure 3. In the presence of lithium polysulfides the C−O bond cleavage by the nucleophilic reaction does not require reaction barriers due to the exothermic reaction (>1 eV) and the weakening of the C−O bond due to binding of Li+ ions to the oxygen atoms of the PC molecule. The resulting complexes are stable, and the removal of Li2CO3/CH2S/CH3−CHS moieties from these complexes are computed to be thermodynamically uphill, indicating that the ring-opening results in the formation of alkyl carbonates with C−S bonds, as shown in Figure 3. Based on the exothermicity of the reaction of PC with the lithium sulfides, the reactivity of lithium sulfides is in the following order: Li2S2 > Li2S3 > Li2S4. The computed reaction barriers for the C−O bond breaking in TEGDME by various polysulfides and lithium sulfides are also presented in Table 5. The optimized structures of transition state geometries are shown in Figure 4 (only part of TEGDME is shown for clarity). Based on the computed reaction barriers (>1 eV), the C−O bond of TEGDME is likely to be stable against polysulfide anions or radicals compared to the carbonate solvents. This conclusion is likely to be true for all linear glymes, such as monoglyme, diglyme, and triglyme. However, this is not true for cyclic ethers such as 1,3-dioxolane,

another commonly used solvent molecule for Li−S cells. For the 1,3-dioxolane, the computed enthalpy barrier for the C−O bond cleavage by S22− is 0.81 eV (Figure S5 of the Supporting Information). Therefore, the cyclic ether solvents are less stable compared to the linear ethers but relatively more stable compared to the carbonate-based solvents. This is consistent with the experimental study by Barchaz et al.42 Additionally, the rate constant evaluated using the activation enthalpy in solution suggests that the 1,3-dioxalone (DOL) may undergo reaction (ΔH† = 0.81 eV) in the presence of S22− ions, while the free energy barrier (ΔG† = 1.23 eV) suggests that this reaction is very slow (half-life ∼1 month at 298.15 K). A recent experimental study42 suggests that the decomposition of DOL is likely during Li−S battery operating conditions consistent with the approach we used for calculating the rate constants. From the computed reaction barriers (Table 5) for the C−O bond cleavage in TEGDME by lithium sulfides, the Li2S2 and the Li2S3 species have the lowest activation barriers, 0.71 and 1.00 eV, respectively. A relatively low barrier of 0.71 eV suggests that this reaction can occur under the battery operating conditions and Li2S2 is the most reactive among lithium polysulfides toward breaking the C−O bond of the ether molecule (note that Li2S2 does not require a reaction barrier to break the C−O bond of PC). Second, the C−O bond breaking catalyzed by Li2S3 species in solution requires an activation barrier of 1.00 eV. This barrier corresponds to a halflife of 2.5 h at room temperature, based on a first-order reaction kinetics approximation (see Table S2 of the Supporting Information). Therefore, this reaction is also likely under the battery operating conditions. Third, the reaction barrier required for the C−O bond breaking by Li2S4 is 1.12 eV and 11554



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Figure 5. Computed enthalpy profile (solution) for the reaction between TEGDME and Li2S2. The relative enthalpies (MP2) of transition state and the products are also given. The product has characteristic features including a CH3−O−Li group and C−S bonds, which are highlighted.

Table 6. Computed Energetics for Hydrogen/Proton Abstraction Reactions from Various Sites (see Scheme 2) of TEGDME by S22−, Li2S2, S32−, LiS3, S32−, Li2S3 2−, S42−, Li2S4, Li2S6, and Li2S8 at the B3LYP/6-311+G(2df,p)//B3LYP/6-31G(2df,p) Level of Theory in Solutiona proton abstraction (eV) site

S22−

A B C D E

1.93 0.55 1.85 2.13 1.89

Li2S2

S3

2−

3.13 1.74 3.05 3.32 3.09

2.28 0.89 2.20 2.47 2.24

hydrogen abstraction (eV)

Li2S3

S4

2−

Li2S4

LiS3

Li2S2

Li2S3

Li2S4

Li2S6

Li2S8

3.48 2.10 3.40 3.68 3.44

2.47 1.08 2.39 2.66 2.43

3.71 2.32 3.63 3.90 3.66

1.65 1.51 1.48 1.66 1.51

1.83 1.69 1.66 1.84 1.69

1.45 1.31 1.28 1.46 1.31

1.36 1.22 1.19 1.38 1.23

1.97 1.83 1.80 1.99 1.84

1.61 1.47 1.43 1.62 1.47

a

The solvation contributions were computed using the SMD solvation model at the B3LYP/6-31G(2df,p) level of theory using a dielectric medium of acetone at room temperature.

a barrier of this magnitude is equivalent to a reaction with a half-life of approximately 270 h at room temperature, indicating that this reaction is unlikely. Additionally, the computed reaction barrier for the C−O bond breaking by Li2S is also significantly high (1.44 eV), and, therefore, this reaction is unlikely to occur. Thus, the computed reaction barriers for the C−O bond cleavage of TEGDME by Li2S2 and Li2S3 indicate that these reaction intermediates likely cause solvent decomposition. These bond breaking reactions result in the production of CH3OLi and alkyl−sulfur compounds as shown in Figure 5. This could ultimately result in the failure of the cell. Therefore, further developments are required for improving the electrochemical stability of ether-based electrolytes and controlling the solubility of polysulfide species in these solvents.18 Alternatives to ether based electrolytes are blended electrolytes (using additives), ionic liquids, and solid state electrolytes.18 The C−H Bond Cleavage. Similar to Li−O2 battery chemistry, solvent decomposition initiated by proton/hydrogen abstraction reaction is probable due to the presence of anions and radicals.80 To understand the energetics of the proton (H+) and hydrogen (Hradical) abstraction reactions from TEGDME, we have computed the Gibbs free energies for proton/ hydrogen abstraction reactions from various positions of TEGDME. There are five potential sites on TEGDME from which potential hydrogen/proton abstraction reactions are

possible, which are denoted as A, B, C, D, and E, as shown in Scheme 2. Some possible hydrogen and proton abstraction reactions are also shown in Scheme 2. We have selected LiS3, Li2S2, Li2S3, and Li2S4 species for hydrogen (radical) abstraction and S22−, Li2S2, S32−, Li2S3, S42−, and Li2S4 species for proton abstraction from the five positions of TEGDME as shown Scheme 2. The computed free energies for the hydrogen/proton abstraction reactions from various sites of TEGDME in solution are tabulated in Table 6. Based on the computations, the proton abstraction from any sites of TEGDME by reactive intermediates such as S22−, Li2S2, S32−, Li2S3, S42−, LiS3, Li2S4, Li2S6, and Li2S8 are computed to be endergonic by more than 0.5 eV. Proton abstraction from TEGDME by S22− is thermodynamically uphill by ∼1.9 eV, except from site B (see Scheme 2). Proton abstraction from site B leads to the formation of CH3− OCHCH2 and O−R (R = CH3−OCH2CH2OCH2CH2OCH2CH2−O) anion, where the anion makes a coordinate complex with the carbon−carbon double bond (alkoxy−ene complex, see Figure S6 of the Supporting Information). In terms of kinetics, it can be assumed that a minimum of 1.9 eV is required to initiate the C−H bond cleavage by S22−, therefore this reaction is unlikely to occur in the bulk solution. Similarly, proton abstraction by Li2S3, S32−, Li2S3, S42−, and Li2S4 are endergonic by more than 2 eV; therefore, these reactions are also unlikely to occur. 11555



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Calculations presented in Table 6 also indicate that the hydrogen abstractions from TEGDME by LiS3 (radical), Li2S2, Li2S3, Li2S4, Li2S6, and Li2S8 are thermodynamically uphill in solution. The computations indicate that the most likely hydrogen abstraction is from site C of the TEGDME by Li2S4 species, which is endergonic by 1.19 eV. This suggests the existence of reasonably high reaction barriers (>1.19 eV) for all hydrogen abstraction reactions, and it is assumed that the reactions of this category are less likely under the battery operating conditions.

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term stability of ether molecules is unlikely in the presence of reactive Li2S2 and Li2S3 species in the Li−S energy storage systems.

ASSOCIATED CONTENT

* Supporting Information S

Assessment of methods and models (Tables S1A and S1B). Free energies, enthalpies, rate constants, and half-life of all C− O bond breaking reactions (Table S2). Optimized structures of sulfur allotropes (Figure S1), lithium polysulfides (Figure S2), and polysulfide anions (Figure S3a−c). Computed energetics of selected reactions of lithium polysulfides (Tables S3 and S4). Energy profiles for ring opening of propylene carbonate by S22− and S32− (Figure S4) and ring opening of 1,3-dioxolane by S22− (Figure S5). The optimized structure of tetraglyme upon on the deprotonation of site B (Figure S6). This material is available free of charge via the Internet at http://pubs.acs.org.

4. CONCLUSIONS A molecular level investigation of the chemical reactions in the Li−S batteries (secondary or redox flow) is presented using high-level electronic structure methods. Quantum chemical methods (G4MP2, MP2, and B3LYP) were utilized to compute reduction potentials of lithium polysulfides and polysulfide molecular clusters, energetics of disproportionation and association reactions of likely intermediates, and their reactions with commonly used nonaqueous electrolyte, tetraglyme. The following conclusions are drawn from this study. I. Detailed fragmentation energies of S8 and Li2S8 to small sulfur clusters (Sn, n = 1 to 7) and lithium sulfides are computed in solution. Based on the fragmentation energies of S8 to two fragments, the relative stability of fragments is in the following order: S8 > S6 and S2 > S5 and S3 > 2S4. For the fragmentation reaction of Li2S8, the formation of Li2S6 and Li2S4 is computed to be exergonic, while further fragmentation reactions of Li2S6 and Li2S4 are endergonic in solution. II. The reduction potential computations indicate that the second reduction of S2 occurs at 2 V with respect to Li/ Li+, while other polysulfides and lithium sulfides except Li2S3 and Li2S2 occur under the lithium−sulfur battery discharge conditions of 2.5−2.0 V. The computed reduction potentials of Li2S3 and Li2S2 of 0.5 and 0.06 V, respectively, indicate that deep discharge is required to reduce these molecules. III. Based on the computed Gibbs free energies of the dissociation/association of polysulfide anions and lithium polysulfides in the solution, we have explored a detailed reaction network that explains the likely reaction mechanism during the discharge process. The mechanism explains the energetics of reactions associated with the discharge process and the existence of the major intermediates (S22−, S32−, S42−, S31−) in the Li−S battery systems. The reaction network of polysulfide anions in solution indicates that the most abundant intermediate upon complete utilization of S82− is S32−. Also, the reaction network of lithium polysulfides suggests transformation of Li2S8 to Li2S and Li2S2 upon the discharge. IV. Calculations of the stability and reactivity of propylene carbonate against the various polysulfides such as S1− (radical), S21−, S22−, S31−, S32−, S42−, Li2S, Li2S2, Li2S3, and Li2S4 suggest that the propylene carbonate molecule is unstable against the polysulfide anions such as S22−, S32−, and S42− (ΔH† < 0.8 eV) and highly reactive toward Li2S2 and Li2S3. In contrast, the tetraglyme solvent molecule exhibits stability toward polysulfide anions. However, tetraglyme is also vulnerable to nucleophilic attack from Li2S2 and Li2S3 species in solution, with Li2S2 being more reactive in solution. Therefore, the longer

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Tel: 630-252-3536. Fax: 630-2529555. *E-mail: [email protected]. Tel: 630-252-7380. Fax: 630-2529555. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported as part of the Joint Center for Energy Storage Research, an Energy Innovation Hub funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences. We gratefully acknowledge the computing resources provided on “Fusion”, a 320-node computing cluster operated by the Laboratory Computing Resource Center at Argonne National Laboratory. Use of the Center for Nanoscale Materials was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.

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