Introduction

Lithium-sulfur batteries (Li–S) fulfill the urgent demands on the storage and transport of renewable energy, owing to their overwhelming theoretical energy density and relatively low cost.1,2,3,4,5,6,7 However, one of the fatal issues that restricts the commercialization of Li–S batteries is the notorious “shuttle effect”, which stems from the dissolution and the endless transport/shuttle of lithium polysulfides (LiPSs) between cathode and anode during the redox process. These phenomena greatly reduce the utilization of sulfur and Coulomb efficiency, causing the self-discharge behavior and the rapid decrease of battery capacity.8 For the practical applications of Li–S batteries, the shuttle effect has to be greatly suppressed in liquid electrolyte.9

An effective strategy for overcoming the shuttle effect is to introduce anchoring materials in cathode to trap LiPSs. Conductive mesoporous carbon framework was first used to encapsulate the sulfur cathode.10 Subsequently, functionalized carbon materials were developed to enhance the polarity of carbon materials.11 Recently, two-dimensional (2D) materials like polar metallic oxides,12,13 transition-metal sulfides,14 and metal nitrides,15 were utilized to alleviate the shuttle effect, but their applications are limited by their bad conductivity. Encouragingly, 2D transition-metal carbides, carbonitrides, and nitrides (known as MXenes)16 have been demonstrated to greatly promote the electrochemical performance for energy storage devices.17,18,19,20 Because of their prominent advantages, such as metallic conductivity, a plastic layer structure, and the hydrophilic nature of its functionalized surface,21 MXenes have been intensively investigated as anchoring materials in Li–S batteris.22,23,24

The accurate prediction of binding energies between anchoring materials, electrolytes, and LiPSs is a prerequisite for uncovering the anchoring mechanism and design cathode in Li–S batteries.25,26,27 In this contribution, we apply the many-body dispersion (MBD) method to investigate two prototypal anchoring materials and find that the many-body effect significantly affects the binding energies of LiPSs on Ti2CF2. In this system the larger many-body effect originates from the large polarization and the domination of van der Waals (vdW) forces. In contrast, for N-doped graphene (NG) systems, the contribution of the many-body effect for binding strength is only 2%, since the chemical bonding predominates the whole process of lithiation. Our work paves the way for more reliably screening anchoring materials to control the shuttle effect and to design high-performance cathode materials for Li–S batteries.

Results

Performance of FeS and SnS2 for trapping Li2S6

Zhou et al.28 have experimentally investigated the anchoring behavior of metal sulfides for trapping Li2S6 by comparative tests (different metal sulfides with equivalent total surface area were added into originally yellow-colored 1, 2-dimethoxyethane/1, 3-dioxolane (DME/DOL) solution with Li2S6). According to the discoloration of solution, they found that the metal sulfides have different adsorption capability for Li2S6 and several of them completely suppressed the shuttle effect. The competition between anchoring materials and electrolytes for trapping soluble LiPSs has been concerned to evaluate anchoring materials.29,30 However, in their density functional theory (DFT) calculation the anchoring factor ΔE \((\Delta E = E_{{\mathrm{AM}}}^{\mathrm{b}} - E_{{\mathrm{so}}}^{\mathrm{b}})\) was not considered. Here \(E_{{\mathrm{AM}}}^{\mathrm{b}}\) is the binding energies of LiPSs absorbed on anchoring materials, and \(E_{{\mathrm{so}}}^{\mathrm{b}}\) denotes the binding energies of LiPSs with electrolyte molecules. The positive ΔE indicates that the interaction between polysulfides and anchoring materials is stronger than polysulfides with electrolyte molecules; the negative ΔE means that polysulfides tend to dissolve into electrolyte rather than absorb on anchoring materials.

To compare theoretical prediction from different vdW methods, we apply a criterion (ΔE) to evaluate the performance of FeS and SnS2 for trapping Li2S6 and compare with experimental observation.28 The optimized configurations of Li2S6 absorbed on FeS and SnS2, and its binding with DME/DOL molecule are shown in Supplementary Fig. 1. Recently, Brandenburg et al.31 reported that the MBD method performs better than most other approaches, such as DMC and CCSD(T), for the fundamental system. Meanwhile, it has been demonstrated that it is essential to include the many-body effect in adsorption systems for the evaluation of binding energies.32,33,34 In this context, here we apply the MBD method to predict the interaction between Li2S6 and FeS/SnS2. From the results of PBE + MBD method in Table 1, ΔE (FeS-DME) and ΔE (SnS2-DME) are negative values (−0.10 eV and −0.07 eV), ΔE (FeS-DOL) and ΔE (SnS2-DOL) are positive values (0.02 eV and 0.05 eV). We predict that a part of Li2S6 species will be anchored on the surface of FeS and SnS2 and the rest of Li2S6 still binds with DME molecules. In other words, the yellow DME/DOL solution with Li2S6 just has an incompletely discoloration when FeS and SnS2 are added as anchoring materials. This conclusion is fully compatible with the experimental observation.

Table 1 The binding energies (eV) between Li2S6 and anchoring materials (FeS and SnS2) as well as electrolyte molecules (DME and DOL) from several DFT methods, the anchoring factor ΔE are also listed.
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However, from the PBE + D2 method, the ΔE (FeS-DOL) is 0.04 eV and other ΔE values are negative, which suggests that SnS2 cannot trap Li2S6 species efficiently. Therefore, the yellow DME/DOL solution with Li2S6 should keep the color invariable when SnS2 is used as anchoring materials. From the PBE + vdWsurf, PBE + D3, vdW-DF2 and optB86b-vdW methods, all ΔE are positive values, implying that Li2S6 is absorbed on FeS and SnS2 firmly and scarcely binds with DME/DOL molecules. Thus, the yellow DME/DOL solution with Li2S6 should fade completely. Nevertheless, the experiment clearly showed that when FeS and SnS2 are added as anchoring materials, the originally yellow-colored polysulfide solution becomes lighter in color rather than become colorless in experimental observation. Therefore, the conclusion from those computational methods is inconsistent with experimental observation.

Performance of Ti2CF2 and NG for trapping LiPSs

According to the above analysis, the MBD method is more accurate than other methods to describe the interaction between LiPSs and metal sulfides. However, many studies also got reasonable conclusions of the anchoring materials for suppression of the shuttle effect by pairwise-based methods.26,30,35 Therefore, to attain an in-depth understanding of the many-body effect on binding strength of LiPSs with anchoring materials, we selected Ti2CF2 and NG as platforms and investigated their intrinsic different anchoring mechanisms. Ti2C-based materials are the lightest Mxenes and atom-multilayered 2D materials, NG is a typical heteroatom-doped anchoring material with one-atomic-layer structure. As for Ti2CF2, we note that the F-termination has different adsorption sites on Ti2C surface36,37 and the HCP site is chosen in our work because of the stronger binding strength of LiPSs (Supplementary Fig. 2 and Supplementary Table 1). Firstly, the configurations of soluble LiPSs with electrolyte molecules are investigated. Notably, the real electrolyte system is very complicated, which usually consists of solvents, lithium salts, and additives.38 Nevertheless, the major component causing the shuttle effect is solvent molecules in electrolyte, which competes with anchoring materials and introduces solution of high-order LiPSs. As such, in the present work, and the same in previous work,29,30 we used DME and DOL molecules to explain the interaction between LiPSs and electrolyte. We considered that one LiPSs cluster interacts with one or two DME (DOL) molecules (Supplementary Fig. 3). From Supplementary Table 2 we find that the \(E_{{\mathrm{so}}}^{\mathrm{b}}\) are very similar whatever soluble LiPSs binding with one or two DME (DOL) molecules from the vdWsurf and MBD methods. For systematically examining the \(E_{{\mathrm{AM}}}^{\mathrm{b}}\) between LiPSs and Ti2CF2/NG, various initial adsorption configurations of Li2Sn (n = 1, 2, 4, 6, and 8) clusters are considered to find the optimized structures. The most stable structures with the lowest energy are shown in Fig. 1a. Comparing with Li2S6 and Li2S8, Li2S, Li2S2, and Li2S4 prefer to adsorb vertically on Ti2CF2, with two Li atoms binding with Ti2CF2. We also search the most stable configurations in NG systems (Supplementary Fig. 4). The calculated \(E_{{\mathrm{AM}}}^{\mathrm{b}}\) of most stable configurations from PBE + vdWsurf, PBE + MBD, and other methods at different lithiation stages are shown in Table 2, Fig. 2a, b, and Supplementary Table 3.

Fig. 1: The adsorption configurations and charge transfers of LiPSs adsorbed on Ti2CF2.
figure 1

a Side views of the most stable configurations of Li2Sn (n = 1, 2, 4, 6, 8) anchoring on Ti2CF2. The shortest length of Li–S bond is also labeled. b Charge transfers between LiPSs and Ti2CF2. Here, red (green) is the spatial regions gain (loss) in charge. In all configurations LiPSs lose charge and Ti2CF2 obtain charge. The value of charge transfer is labeled. The purple, gray, sky blue, yellow, and green spheres represent the Ti, C, F, S, and Li atoms, respectively.

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Table 2 Computed binding energies (eV) between LiPSs and Ti2CF2/NG from the PBE + vdWsurf and PBE + MBD methods.
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Fig. 2: The binding energies and ratio for many-body effect and vdW interaction of Ti2CF2 and NG systems.
figure 2

Computed binding energies for a LiPSs on Ti2CF2 and b LiPSs on NG with the PBE + vdWsurf and PBE + MBD methods. The ratio for many-body effect (RMBD) and vdW interaction (RvdW) of binding energies in (c) Ti2CF2 systems and d NG systems.

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Figure 2a shows that the PBE + MBD method reduces the \(E_{{\mathrm{AM}}}^{\mathrm{b}}\) over 20% compared to the PBE + vdWsurf method at high-order lithiation stages for Ti2CF2 systems. In contrast, the binding energies for the NG systems from the two methods are almost identical (Fig. 2b). To assess the contribution of MBD to the \(E_{{\mathrm{AM}}}^{\mathrm{b}}\), we determine the ratio of MBD contribution (RMBD) as follows,

$$R_{{\mathrm{MBD}}} = (E_{\mathrm{vdW}}^{{\mathrm{b}}\quad {\mathrm{}}} - E_{\mathrm{MBD}}^{\mathrm{b}})/E_{\mathrm{MBD}}^{\mathrm{b}}$$
(1)

where \(E_{\mathrm{vdW}}^{{\mathrm{b}}\quad {\mathrm{}}}\) and \(E_ {\mathrm{MBD}}^{\mathrm{b}}\) denote the binding energies of Ti2CF2 (NG) from the PBE + vdWsurf and PBD + MBD methods, respectively. The magnitudes of RMBD for LiPSs absorbed on Ti2CF2 and NG are shown in Fig. 2c, d. From Table 2, when Ti2CF2 are selected as anchoring materials, we find that RMBD varies from 16.8% to 33.3% with the delithiation of LiPSs. The many-body effect affects Li2S6-Ti2CF2 system largely, which has weakened the binding strength by 33.3% and may lead to the dissolution of Li2S6 due to the small ΔE (0.02 eV). While for the NG system, we find that the RMBD (range from 0 to 1.4%) is slight and the many-body effect could be ignored.

To attain an in-depth understanding why the MBD method reduces the binding strength of LiPSs with anchoring materials, we divide the EbAM into two aspects: the vdW part corresponds to the physical interaction, and the PBE part (termed as \(E_{{\mathrm{PBE}}}^{\mathrm{b}}\)) corresponds to the chemical interaction using pairwise-based method vdWsurf, because the MBD effect mainly contributes to the part of physical interaction. To clarify how these two aspects affect the anchoring behavior, we define the ratio of the vdW interaction (RvdW) as follows,

$$R_{{\mathrm{vdW}}} = (E_{\mathrm{vdW}}^{{\mathrm{b}}\quad {\mathrm{}}} - E_{{\mathrm{PBE}}}^{\mathrm{b}})/E_ {\mathrm{vdW}}^{{\mathrm{b}}\quad {\mathrm{}}}$$
(2)

In Eq. (2), \(E_{\mathrm{vdW}}^{{\mathrm{b}}\quad {\mathrm{}}}\) denotes \(E_{{\mathrm{AM}}}^{\mathrm{b}}\) from the pairwise PBE + vdWsurf method. The ratio of vdW contribution to \(E_{{\mathrm{AM}}}^{\mathrm{b}}\) for Ti2CF2 systems is shown in Fig. 2c, the vdW interaction shows different weights towards different lithiation stages. During the delithiation from Li2S to Li2S8, the Li-F bond length increases from 1.86 to 2.01 Å, suggesting that the chemical interaction is gradually weakened. The bond length is 1.86 and 1.88 Å for Li2S and Li2S2 absorbed on Ti2CF2 (within the distances of covalent bonds), and the RvdW is 53% and 59%, respectively. This suggests that both chemical interaction and vdW interactions play important roles in these two adsorption systems. However, the physical interaction dominates the \(E_{{\mathrm{AM}}}^{\mathrm{b}}\) of high-order Li2Sn species (n = 4, 6, 8) absorbed on Ti2CF2 (the RvdW nearly 100%). In contrast, although the RvdW for the NG systems increases with the increasing size of LiPSs, the RvdW of NG systems (ranges from 18% to 45%) is smaller than those of Ti2CF2 systems especially for high-order LiPSs, so the chemical interaction (the bond length of Li-N ranges from 1.86 Å to 1.90 Å) dominates the whole process of lithiation (Fig. 2d).

Static polarizability of adsorption systems and charge transfer

Comparing with pairwise vdW-correction methods, the MBD method computes the correlation energy of long-range electrostatic screening through the coupled harmonic oscillator model Hamiltonian, and considers dipolar vdW interactions to all orders in perturbation theory.39 The reduction of \(E_{{\mathrm{AM}}}^{\mathrm{b}}\) from the PBE + MBD method must come from the collective inclusion of the long-range electrostatic screening. Polarizability is central to the description of the long-range electron correlation, which indicates the willingness of a material to respond under the influence of a perturbation of the electric field. It is known that five atomic layers of Ti2CF2 are stacked along the c-axis direction, middle Ti atoms experience the dynamic electric field created by the upper and lower atoms, which gives rise to polarization effects.40 To provide a quantitative description of the polarizability, we define the changes in static polarizability of the adsorption systems Δα by

$$\Delta \alpha = \Delta \alpha _{{\mathrm{total}}} - \Delta \alpha _{{\mathrm{AM}}} - \Delta \alpha _{{\mathrm{LiPSs}}}$$
(3)

where Δαtotal, ΔαAM, and ΔαLiPSs represent the screened static polarizabilities of the adsorption systems, the anchoring materials and the free-standing LiPSs, respectively. Considering the similar polarizabilities along x and y coordinates, we present the Δα along x and z directions, termed as Δα‖ and Δα⊥, which are parallel and perpendicular to the surface of 2D anchoring materials, respectively. The calculated Δα (Δα|| and Δα⊥) are listed in Table 3. We find that for Ti2CF2 systems, contrary to the magnitude of Δα‖ that varies slightly, the magnitudes of Δα⊥ are very different from the delithiation of LiPSs. We find that Δα⊥ are positive for Ti2CF2 systems, which means that the adsorption of LiPSs increases the Δα⊥ of the systems. However, the values of Δα⊥ in N-doped graphene adsorption systems are negative, indicating that the Δα⊥ are decreased after LiPSs adsorbed on N-doped graphene.

Table 3 The total static polarizability (Δα), and the Δα along x (Δα‖) and z (Δα⊥) directions of Ti2CF2 adsorption configurations.
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We notice that the magnitudes of Δα⊥ of Li2S6 and Li2S8 absorbed on Ti2CF2 decrease to 10 bohr3. The decreasing values of Δα⊥ suggest the weakening of the polarization along perpendicular direction. The characteristics of anisotropic polarizability can be simply estimated by atomic polarizability matrix, which exhibits both spatial distributions and orientations of the polarizability in LiPSs cluster. For Li2S4 adsorbed on Ti2CF2 in Supplementary Fig. 5, the polarizability matrixes show different anisotropic characteristics and the xy-orientational polarizability is much larger than that of Li2S6 and Li2S8, consistent with the optimized adsorption structures. Figure 2a shows that the Li2S, Li2S2, and Li2S4 species “stand” on the surface with two Li atoms parallel to the surface of Ti2CF2, along with a charge transfer of 0.73–1.08 e. For Li2S6 and Li2S8, however, the configurations transform to “lying-in-plane” configuration, with only one Li atom binding with Ti2CF2, and a decreased charge transfer (0.42 e for Li2S6 and 0.58 e for Li2S8).

In light of the charge distribution analysis41 shown in Fig. 3, we find that all F atoms in the F-terminated surface have a negative charge of 0.75 e. As for NG, N atoms obtain a large charge of 1.2 e, but other C atoms are nearly electroneutral. Since the vdW forces arise from the electrostatic interactions between the constantly fluctuating electron clouds, the spontaneous polarization in Ti2CF2 make the polarizability sensitive to electron exchange. The above analysis shows that reduction of electron transfer is probably the major reason that causes the sharply reduction of the Δα⊥. Meanwhile, from the density of states (DOS) of NG and Ti2CF2 (Supplementary Fig. 6) we find that they maintained its metallic properties after adsorption of Li2S6, implying that they can supply electrons directly to the redox reaction of the anchored LiPSs. To further understand the many-body effect in MXene systems, we calculated the anchoring behavior of Ti3C2F2 (Supplementary Table 4), which was also applied as an anchoring material in Li–S batteries.42 Compared with five atomic-layer Ti2CF2, we find that the polarizability perpendicular to plane (Δα⊥) and the ratio of MBD contribution (RMBD) are both enhanced with the increasing thickness of anchoring materials.

Fig. 3: Charge distribution analysis of LiPSs adsorbed on Ti2CF2 and NG.
figure 3

a–c The charge transfers (e) of S and Li atoms in Li2S2, Li2S4 and Li2S8 and adjacent F atoms in Ti2CF2 adsorption configurations. d–f The charge transfers between S/Li atoms and adjacent N atoms in NG adsorption configurations.

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From the above analysis, we find that Ti2CF2 systems has large polarization (22.2–41.3 bohr3) and adsorption of LiPSs is dominated by the vdW interactions, which generates stronger many-body effect (large RMBD). We also investigate other 2D materials such as TiS2, Ti2CO2,43,44 and Ti3C2F220,28 and the binding energies from two methods (PBE + vdWsurf and PBE + MBD) are listed in Supplementary Table 5. We find that the many-body effect also exists in these systems. Therefore, for this type of anchoring materials with large RMBD, pairwise-vdW methods tend to overestimate the anchoring performance (ΔE) even get an inverse conclusion of anchoring materials. Conversely, chemical bonding plays an important role during the whole process of lithiation in NG systems, and NG has a small polarization (4.3–12 bohr3). As such, RMBD is very small and pairwise-vdW and MBD method have the same prediction about ΔE.

In Li–S batteries, the adsorption of LiPSs on loading materials or solvent molecules is a process of competition, thus positive ΔE is the prerequisite for the anchoring materials.4 It is clearly seen that NG has a large ΔE, which agrees with the experimental results that NG can effectively anchor lithium polysulfides. However, the limited number of adsorption sites discourages high energy density of sulfur cathodes. In addition, the ΔE of Li2S6 is as small as 0.02 eV, which demonstrates that Ti2CF2 would not effectively trap Li2S6 species, and the desorption and dissolution of Li2S6 species has a high probability to take place during lithiation.

Discussion

We find the many-body dispersion interaction has a substantial impact on anchoring mechanism through binding energies of adsorption systems, especially for Ti2CF2 with strong polarization. According to the anchoring factor ΔE from the PBE + MBD method, we theoretically predict that Li2S6 partly dissolves into electrolytes solution and the DME/DOL solution with Li2S6 discolors incompletely when FeS and SnS2 as anchoring materials, which agree well with the experiment. We demonstrated that the MBD approach can accurate predict the anisotropic polarizability and the anchoring behavior in the Li–S batteries. By systematically investigating the many-body effect on Ti2CF2 and NG systems, we found that the PBE + MBD method reduces the \(E_{{\mathrm{AM}}}^{\mathrm{b}}\) over 20%, as compared to the PBE + vdWsurf method at high-order lithiation stages for Ti2CF2 systems. While for NG systems, the energy difference from two methods is slight. Our work deepens the understanding of the anchoring mechanism, and provides a more accurate criterion for screening anchoring materials to control shuttle effect and to design more reliable cathode in Li–S batteries.

Methods

Computational details

Our density functional theory (DFT) calculations were performed using the Fritz-Haber-Institute ab initio molecular simulations (FHI-aims) package45 and the Vienna Ab initio Simulation Package (VASP) code.46,47 The geometric optimization and the calculations of the charge transfer were performed within the generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof (PBE) as implemented in VASP. For geometric optimizations, all atomic positions were relaxed until the maximal forces per atom were less than 0.03 eV/Å and energy difference was smaller than 10−5 eV. The plane-wave cutoff energy was set to 450 eV. A 3 × 3 supercell of monolayer FeS (SnS2), a 5 × 5 supercell of Ti2CF2 with 5 × 5 × 1 Γ-centered k-mesh,48 and a 7 × 7 supercell of NG with 3 × 3 × 1 Γ-centered k-mesh were employed to model the adsorption systems. The distance of vacuum region was set to 20 Å along the z-direction (vertical to the interface) to avoid the interaction between adjacent layers.49

The FHI-aims code was applied to calculate density of states and energies of all configurations. We used the “tight” settings for numerical atom-centered orbital basis sets. The “tier2” standard basis was utilized for F, O, H, N, Li and C, and “tier1” for S, Fe, Ti, and Sn. The convergence criteria of 10−5 electrons per unit volume for the charge density and 10−6 eV for the total energy of the systems were used.

Van der Waals inclusive methods

For the description of the van der Waals (vdW) interactions we applied two methods in DFT calculations. The PBE + vdWsurf method50 were employed to account for the pairwise van der Waals (vdW) interactions. The PBE + vdWsurf total energy includes the dispersion energy by considering a sum of \(- C_6^{ab}R_{ab}^{ - 6}\) terms, where Rab is the distance between atoms a and b, the C6 coefficients and vdW radii (vdW parameters) are determined by the collective screening of the substrate electrons, which are captured by mean of the vdWsurf method. We employed the PBE + MBD method to describe the many-body dispersion (MBD) effect. The MBD method computes the correlation energy of long-range electrostatic screening through the coupled harmonic oscillator model Hamiltonian, and considers dipolar vdW interactions to all orders (many-body effect) in perturbation theory beyond the vdWsurf method.