Lithiation mechanism of sulfur surfaces during discharge of Li–S batteries from quantum chemical calculations

Abstract

We present a computational study based on quantum-chemical calculations to investigate the initial lithiation reactions on the (001) surface of α-sulfur. The study aims to explore the possible emerging structures during consecutive lithiation steps and to analyze their reaction enthalpies. Our results show that during the first lithiation reactions, S₈ rings in the lower layers of the (001) surface are preferentially lithiated. In subsequent lithiation steps, we find that S₈ rings on the upper layers, adjacent to previously lithiated molecules, may also undergo lithiation. Once Li₂S₈ dimers are formed, further reactions on the surface can proceed, leading to the formation of Li₂S₈ trimers in a lower/upper/lower layer arrangement or lower-order Li-polysulfides, such as Li₂S₆/Li₂S₂ and Li₂S₅/Li₂S₃. Notably, in contrast to sulfur reduction reactions in the electrolyte, the formation of Li₂S₄/Li₂S₄ does not occur on the (001) surface, likely due to the surface morphology, which prevents complete exposure of S₈ rings to lithium ions. This suggests that surface lithiation predominantly leads to the formation of high-order polysulfides in the early stages of discharge, while the dissolution of these higher-order polysulfides into the electrolyte may facilitate their reduction to Li₂S₄, a process observed experimentally. Our study provides an atomistic mechanism for the discharge process of Li–S batteries with a crystalline α-sulfur cathode, contributing to a deeper understanding of both solid- and liquid-phase reactions during the early discharge stages.

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1. Introduction

The ever-increasing global energy demand calls for more efficient energy-storage devices. Currently, lithium-ion batteries are the most commonly used in electronic devices such as mobile phones, tablets, and electric cars etc.¹ However, the energy density of these batteries has only incrementally increased in the past decade approaching a saturated energy density of 250–400 W h kg⁻¹.^2,3 This has been made possible only by using rare metals, which increases the net weight of Li-ion batteries as well as their cost. As an alternative, lithium–sulfur (Li–S) batteries have the potential to replace Li-ion batteries due to their very high energy density of around 2600 W h kg⁻¹.^4,5 Moreover, sulfur is an abundant material, and therefore considerably cheaper⁶ compared to the transition metals required for Li-ion batteries. Nevertheless, Li–S batteries still face technical issues that prevent their everyday use. These issues are partially related to the formation of soluble lithium polysulfides which result in shuttle effect during the discharge process, which is believed to be the main reason for the poor cycle life of Li–S batteries.^1,7–12

Among the proposed cathode materials to mitigate the shuttle effect, sulfur/carbon copolymers^13–23 and covalent organic frameworks^24–30 have recently attracted much attention. Despite the promising performance of these materials to immobilize the lithium polysulfides (or even to facilitate sulfur reduction reactions), the overall electrochemical performance of the whole cathode, is determined to a great extent by that of the crystalline sulfur to which the copolymers or covalent organic frameworks are added. Therefore, to prevent the shuttle effect, it is important to gather an atomistic insight into the lithiation mechanism of crystalline sulfur. Although the main reactions during discharge are believed to occur in the solvent,^{10–12,31,32} the initial lithiation reactions occur primarily on the cathode surface, as octa-sulfur is nonpolar. Additionally, previous quantum-chemical studies have addressed lithiation reaction at sulfur/solvent interface including explicit solvent molecules using ab initio molecular dynamics (AIMD) which suggest the occurrence of consecutive lithiation reactions at the sulfur surfaces forming lower-order Li-polysulfides, before their dissolution into the electrolyte.³³ Therefore, it is important to understand the morphological changes of the sulfur surfaces during early stages of discharge.

The purpose of this study is to investigate the lithiation mechanism of the (001) surface of the crystalline α-sulfur, determine the most stable products formed during this process, and study the structural changes of the surface during lithiation using quantum chemical calculations. Despite several studies to elucidate the lithiation mechanism of solvated S₈ and on sulfur surfaces,^33–35 an atomistic understanding of the underlying processes and structural changes remains incomplete due to the complex nature of the reactions.

2. Computational details

To represent the sulfur surface, certain steps were taken to create a suitable system for geometry optimizations using density functional theory (DFT) calculations.³⁶ The calculations were performed on the orthorhombic α-sulfur, considering its (001) surface which, along with the (100) surface, has the lowest energy compared to other α-sulfur surfaces.³³ The original unit cell structure was extracted from X-ray diffraction measurements.³⁷ We employed periodic boundary conditions (PBC) along the x and y axes to represent crystallinity. In addition, a vacuum buffer along the z-axis was implemented to expose the (001) surface for lithiation reactions. To prevent the formation of radical electron structures, a modification was made to the top layer in the z-direction to ensure the absence of broken S₈ rings. Detailed information can be found in Fig. 1 and Table 1. To study lithiation reactions, we used CP2K/QUICKSTEP package,³⁸ incorporating dual plane-wave and Gaussian basis functions. A double-zeta plus polarization basis set optimized for molecular systems (DZVP-MOLOPT-SR-GTH)³⁹ was used for the valence electrons together with Goedecket–Teter–Hutter-type pseudopotentials,^40,41 and the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional.⁴² To account for dispersion interactions, the DFT-D3⁴³ method was incorporated. Electronic structure calculations employed an SCF convergence criterion set at 10⁻⁶ and the atomic coordinate optimizations were conducted using the BFGS optimizer.^44–47 Our calculations including an implicit solvent model, did not show significant structural differences in the product structures of lithiation reactions in electrolyte (see ESI†). Besides, the main concern here is to understand the deformation of the sulfur surface during lithiation reactions rather than the solubility or dissociation of the lithiated products. Therefore, here we did not include an implicit solvent model for the electrolyte.

Fig. 1 Unit cell with S₈ slab. Perpendicular to the z axis are (001) surfaces, while PBC as used along x and y axes.

Table 1 Details on the structure of the model (001) α-sulfur surface in this study

Number of atoms	578
Unit cell size [Å]	(a, b, c) = (31.39, 25.73, 60.00)
Angles [°]	(α, β, γ) = (90, 90, 90)

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The sulfur surface structure was prepared for the lithiation reaction by initially relaxing the sulfur slab in a vacuum. This involved relaxing the upper three layers while keeping the lower three layers fixed, as shown in Fig. 2. The relaxation of the upper layers represents the surface flexibility, which is important for the lithiation process. Meanwhile, the constraint of the lower layers mimics the crystallinity of the sulfur system. Fig. 2(a) shows the z-displacement of each layer during the relaxation process while (b) highlights the corresponding layers, encircled using the same color coding. The obtained relaxed structure is the starting point for the lithiation process. To study the product structures in each lithiation reaction, distinct positions for two lithium atoms were chosen on the sulfur surface. Always two lithium atoms with a distance of around 3 Å were added to different positions on the surface at each lithiation step to ensure that no radical systems are present in the process of the lithiation. For each position two lithium atoms were arranged once parallel (p) and once orthogonal (o) to the (001) surface. The product structures were then obtained by optimizing the entire system, including the two lithium atoms, while constraining the lower three ring layers. Consequently, for each subsequent lithiation step, the lowest-energy product from the previous step was used as the starting point to define new distinct positions for additional lithium atoms.

Fig. 2 (a) z-Displacement of the upper layers, (b) the final structure and the marked upper layers.

The reaction energy for each lithiation step was calculated via

where ΔE_R is the reaction energy per lithium atom. Here, neglecting entropic effects, we approximate the free energies with DFT total energies. As such, E_Product, E_Educt, and E_2Li are the DFT energies of the optimized product structures, educt structures, and twice the energy of an isolated lithium atom, respectively. For example, in the first lithiation reaction,

S₈ + 2Li⁺ + 2e⁻ → Li₂S₈, (2)

the product energy, E_Product, refers to the total energy of the system with one Li₂S₈ formed on the surface, and the educt energy, E_Educt, refers to the total energy of an α-sulfur slab in Fig. 1.

3. Results and discussion

In particular, we try to provide answers to the following questions:

1. How does the surface topography affect the lithiation reactions?

2. How comparable are the product structures to the known gas/liquid-phase products at each lithiation step?

3. Is the lithiation of the surface a homogeneous process or happens through nucleation?

To provide answers to these questions, we focus on the first few lithiation reactions corresponding to the early stages of discharge.

3.1. First lithiation

The periodicity of the (001) surface along x and y axes reveals two distinct S₈ rings on the surface [Fig. 3(a)]. Notably, in Fig. 3(b), one S₈ ring resides in the upper layer, while the other is positioned in the lower layer. These layers alternate on the surface and are highlighted in Fig. 3(a) and (b) as “up” and “down”. The repeating unit of two S₈ rings [Fig. 3(a)] results in the recurrence of distinct positions for the lithiation reactions [Fig. 3(c)]. These distinct positions serve as representative sites for two Li atoms in the reactant structures and are expected to break one S–S covalent bond in the S₈ rings after the reaction. As such, S(1)/S(2) in Fig. 3(c) denote the upper S₈ ring, while S(3)/S(4) represent the distinct positions on the lower S₈ ring.

Fig. 3 First lithiation step: (a) repetitive unit of the S₈ surface from top view, (b) upper and lower layers of the S₈ surface from side view, (c) distinct positions as reaction centers.

Fig. 4(a) presents the product structures of the first lithiation reaction (only the affected S₈ ring is shown), while Fig. 4(b) shows the reaction energy distribution of all the possible products. In Fig. 4(b), for example, 4p refers to the S₈(4) structure in (a) where the two Li atoms were initially placed in a parallel configuration with respect to the (001) surface. The difference in reaction energies between the product structures is significant, namely up to 0.7 eV per lithium atom. This is due to the fact that the molecular structure of each product is sizably different. The energy distributions of the subsequent lithiation steps (i.e. second and third) were not shown as the reaction energy ranges were found to be similar to the energy distribution in Fig. 4(b). Despite their similar configuration (Table 2), the 2p product structure on the upper ring corresponds to a lithiation reaction with a reaction energy ∼0.21 eV higher than that of the 3o product structure on the lower ring. Based on these results, it can be concluded that lower S₈ rings are energetically preferred for lithiation compared to upper ones. This conclusion was additionally validated for the following lithiation steps with a broader sample size of initial geometries.

Fig. 4 First lithiation step: (a) calculated product structures after the first lithiation (only the affected S₈ ring is shown). (b) Distribution of the reaction energies. In (b), for example 4p refers to the S₈(4) structure in (a) where the two Li atoms were initially positioned in parallel with respect to the (001) surface.

Table 2 Sulfur–lithium distances [Å] of Li₂S₈ computed on the (001) α-sulfur surface and in the gas phase⁴⁸

Structure	S(1)–Li(1)	S(2)–Li(1)	S(1)–Li(2)	S(2)–Li(2)
2p	2.35	2.38	2.45	2.38
3o	2.40	2.41	2.38	2.41
Ref. 48	2.44	2.44	2.44	2.44

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Another important point to note is that the optimal structures closely resembled the Li₂S₈ structure in the gas/liquid phase, as shown in Table 2. The previous gas/liquid-phase results,⁴⁸ were obtained on a ring-shaped structure where a mesomeric rhombus forms via two similar Li–S bonds after the lithiation. These observations also align with the 2p and 3o structures obtained here. Additionally, we calculated the product structures corresponding to the gas- and liquid-phase lithiation reactions and are reported in the ESI.† Our results also showed a diamond-shaped structure for Li₂S₈ with slightly different bond angles and distances compared to those reported in ref. 48. Besides, the structural differences between gas- and liquid-phase structures are marginal (see Fig. S5 and Table S2 in the ESI†). On the (001) surface, the Li₂S₈ structures of 2p and 3o also exhibit equidistant Li–S bonds and form a mesomeric rhombus structure. They also result in a ring-shaped product structure. Here, the 3o structure, corresponding to a lithiation reaction with the lowest reaction energy, was considered as the initial structure for the next lithiation step.

3.2. Second lithiation

For the second reaction, two possibilities can be considered: (i) a neighboring ring undergoes a lithiation; or (ii) the Li₂S₈ becomes further lithiated to two lower-order polysulfides. These correspond to the following two reactions,

S₈ + 2Li⁺ + 2e⁻ → Li₂S₈

Li₂S₈ + 2Li⁺ + 2e⁻ → Li₂S_8−x/Li₂S_x, (3)

where x in the equation can have any value between 1 and 7 as these are the known conformations of Li-polysulfides,⁴⁸ and “/” represents a cluster-type structures, meaning they can not be seen as two isolated structures.

For the second lithiation, two different classes of distinct positions were selected. Firstly, positions surrounding the previously lithiated Li₂S₈ ring, and secondly, positions around intact S₈ rings. The positions surrounding the Li₂S₈ molecule were selected in such a way that the new Li atoms are allowed to attack S–S bonds of the Li₂S₈ molecule which are accessible on the surface. Due to its asymmetrical geometry, the accessible bonds on both sides of every Li₂S₈ molecule were considered. For the intact S₈ molecules, positions were chosen to ensure that the rings are not adjacent to the lithiated ones. The chosen positions are identical to those used in the previous lithiation step, with the exception that an additional position was considered closest to the already lithiated center. Moreover, positions on both sides of each S–S bond were considered to account for all lithiation possibilities. Two S₈ molecules in the lower layer (L) and two S₈ rings in the upper layer (U) were chosen [Fig. 5(b)]. It is worth mentioning that, in the second lithiation, the use of ‘U’ for upper and ‘L’ for lower sulfur layers in Fig. 5 refers only to the initial structures. This implies that the position of the two Li atoms on an upper layer could result in a polysulfide product structure formed on the lower layer and vice versa. Although such an occurrence was observed to happen rarely, it was taken into account for the average energy calculation of upper- and lower-layer conformations, as well as for comparing structures.

Fig. 5 Second lithiation step: distinct positions considered for the second lithiation. (a) Distinct positions around an Li₂S₈ molecule. (b) Distinct positions around an intact S₈.

Fig. 6 shows the lowest-reaction energy structures formed on the upper and lower sulfur layers. We observe that the two Li atoms around the Li₂S₈ ring do react with the upper neighboring rings but not with the already lithiated one. While the direct subsequent lithiation of the already lithiated ring is in principle possible, it appears to be less likely. The underlying reason could be that the lithiated ring is strongly protected by the surrounding pristine S₈ rings of the surface, and the only exposed site is where the two lithium atoms from the first lithiation step are located. Hence, the reactivity of the Li₂S₈ ring in a second lithiation step is low, and the second lithiation step is more likely to occur at neighboring S₈ rings. Li₂S₈ 1p describes the product of sulfur ring in an upper layer [Fig. 6] interacting with the adjacent Li₂S₈ ring from the lower layer. Therefore, Li₂S₈ emerges as the only stable product of the second lithiation step, as the lithiation around an Li₂S₈ molecule from the first lithiation step leads to the formation of a Li₂S₈ dimer. As such, only the first reaction in (3) seems to be thermodynamically likely.

Fig. 6 Second lithiation step: comparison of the lowest-reaction energy structures formed on the lower (left) and the upper (right) sulfur layers. Li₂S₈ 1p refers to the product of the lithiation reaction on an adjacent sulfur ring to the already lithiated one. The others are isolated rings.

Table 3 represents Li–S distances in these product structures, namely L₁ 1p, L₂ 3o, and Li₂S₈ 1p, for comparison with a Li₂S₈ molecule in the gas phase. All products have comparable Li–S bond distances, with an average bond length of ∼2.40 Å. Their atomic configurations are also similar to an Li₂S₈ molecule in the gas phase. The reaction energies of all structures are also similar, with the product structures L₂ 3o and L₁ 1p being around 50 meV lower in energy than Li₂S₈ 1p obtained through lithiation around Li₂S₈. A closer look reveals that structures L₂ 3o and L₁ 1p (two Li₂S₈ rings far away from each other) are the products of lithiation of two distant S₈ rings both in lower layers. We also observe that the L₂ 5p product structure consisting of two Li₂S₈ rings both on lower layers shows about 200 meV higher reaction energy than L₂ 3o, L₁ 1p, and Li₂S₈ 1p. One possible reason for this is that in the Li₂S₈ 1p product structure, the lithium atoms can interact strongly with the lone-pair electrons on the terminal sulfur atoms from the neighboring Li₂S₈ ring [Fig. 6, top-right panel].

Table 3 Lithium–sulfur distances [Å] of Li₂S₈ formed on the (001) α-sulfur surface in the second lithiation step, and comparison with those in the gas/liquid phase

Structure	S(1)–Li(1)	S(2)–Li(1)	S(1)–Li(2)	S(2)–Li(2)
Li2S8 1p	2.43	2.42	2.37	2.35
L2 3o	2.42	2.39	2.38	2.43
L1 1p	2.39	2.42	2.38	2.38
Ref. 48	2.44	2.44	2.44	2.44

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Compared to the first lithiation, the reaction energies of L₁ 1p and L₂ 3o are similar to that of the product of the first lithiation. This is expected since they occur on isolated S₈ rings. For the next lithiation step, the product structure Li₂S₈ 1p was selected, because even though the other two structures (L₁ 1p and L₂ 3o) correspond to a slightly lower reaction energies, they would only represent a similar starting structure as for the second lithiation.

3.3. Third lithiation

Here, all possible reaction paths for a third lithiation step are considered as following

S₈ + 2Li⁺ + 2e⁻ → Li₂S₈

Li₂S₈ + 2Li⁺ + 2e⁻ → Li₂S₇/Li₂S

Li₂S₈ + 2Li⁺ + 2e⁻ → Li₂S₆/Li₂S₂

Li₂S₈ + 2Li⁺ + 2e⁻ → Li₂S₅/Li₂S₃

Li₂S₈ + 2Li⁺ + 2e⁻ → Li₂S₄/Li₂S₄ (4)

As before, several S₈ rings and prelithiated Li₂S₈ structures were considered as potential reaction sites. The positions for the two additional Li atoms around the preexisting Li₂S₈ are shown in Fig. 7(a). The structures of the two neighboring Li₂S₈ rings are sizably different from a single Li₂S₈ molecule in the gas phase. Therefore, different positions for the two Li atoms are evenly distributed around the Li₂S₈ rings. Here, the positions 1–4, 11–12 are around the upper Li₂S₈ ring and 5–9 around the lower one [Fig. 7(a)]. For the distinct positions around an intact S₈ in Fig. 7(b), lower layer rings (L₁, L₄) adjacent to the two Li₂S₈ molecules and those farther away (L₂, L₃) from them are considered. The same procedure was applied for the upper S₈ rings in Fig. 7(c). The lithiation process on isolated rings (L₂, L₃ and U₂, U₃) only result in Li₂S₈ molecules similar to the previous lithiation steps. However, a third lithiation reaction adjacent and on the two Li₂S₈ rings (L₁, L₄ and U₁, U₄, Li₂S₈) results in more complex structures, such as Li₂S₇/Li₂S, Li₂S₆/Li₂S₂ and Li₂S₅/Li₂S₃ clusters and bridged structures like Li₂S₈/Li₂S₈. Atomic coordinate optimizations of the reactant structures mostly result in Li₂S₇/Li₂S and Li₂S₆/Li₂S₂ complexes while clusters of Li₂S₄/Li₂S₄ are not observed. One explanation for the absence of Li₂S₄/Li₂S₄ clusters is that the S–S bonds are not accessible to the two Li atoms due to their location beneath the reachable surface. This is in stark contrast to the gas/liquid-phase reactions. Without exception, the structures of the isolated or non-interacting Li₂S₈ molecules with the lowest reaction energies are those that are most similar to Li₂S₈ in the gas phase [Fig. 8]. This is consistent with the first and second lithiation steps.

Fig. 7 Third lithiation step: distinct positions on the (001) α-sulfur surface considered for the third lithiation reaction, (a) around Li₂S₈ dimer obtained in the previous lithiation step, (b) on the lower layers, and (c) on the upper layers around the dimer.

Fig. 8 Third lithiation step: comparison of the structures with the lowest reaction energies formed on the lower (left) and the upper (right) layers of the (001) surface.

We observe that in cases where an adjacent S₈ ring is lithiated, the atomic configuration of the emerging Li₂S₈ cluster is very different from the monomers in the gas phase. Therefore, the comparison of the lowest-reaction energy structures to the Li₂S₈ molecules in the gas phase is only possible when the Li₂S₈ molecule is not directly interacting with other Li-polysulfides. This is illustrated in Fig. 9, where a distorted structure called Li₂S₈ 7p is shown corresponding to a lower reaction energy than a similar gas-phase structure. The formation of Li₂S₈ 7p might be an indicator that it is thermodynamically preferred that the lithiated structure forms on a lower adjacent S₈ ring where the two Li atoms are exposed to the terminal S atoms of the neighboring Li₂S₈ ring [Fig. 9].

Fig. 9 Third lithiation step: comparison of two Li₂S₈ adjacent structures formed during the third lithiation reaction.

Quite comparable in reaction energies to the product structures such as Li₂S₈ 7p and L₄ 3p [Fig. 9] are the cluster/bridged structures shown in Fig. 10.

Fig. 10 Third lithiation step: mesomeric structures.

Of all the cluster and bridged structures, only those with the lowest reaction energies are presented in Fig. 10. For example, the Li₂S₇/Li₂S family of product structures [see reactions (4)], which exhibit considerably higher reaction energies, are not shown. All of these structures display a specific mesomeric pattern, as indicated by the dashed lines in Fig. 10. The results highlight two distinct types of lithiation products: those involving two lithiated Li₂S₈ rings (L₄ 2o) and those where an already lithiated Li₂S₈ molecule undergoes further lithiation (Li₂S₈ 8p and L₄ 1p). Additionally, some cluster structures, such as Li₂S₆/Li₂S₂ (Fig. S1 in the ESI†), exhibit similar mesomerism but have higher reaction energies, likely due to strong electrostatic repulsion between closely positioned lithium atoms.

Despite these observations, the third lithiation step exhibits limited predictive insights onto the preferred conversion pathway due to comparable reaction energies across multiple product configurations. While mesomeric structures and lower adjacent Li₂S₈ products are thermodynamically favored, Li₂S₇/Li₂S formation is energetically unfavorable on the sulfur surface. Additionally, Li₂S₄ dimerization is precluded by steric hindrance, as simultaneous Li⁺ diffusion beneath Li₂S₈ structures to access buried S–S bonds is geometrically restricted. Reactivity trends further indicate that lower adjacent S₈ rings exhibit a higher lithiation propensity compared to isolated or upper-layer rings. This is likely due to reduced Li⁺ migration barriers and cooperative interactions with neighboring intermediates. While definitive reaction pathways remain unresolved, these insights help narrow plausible mechanisms by eliminating improbable reactions (e.g., Li₂S₇ formation) and identifying preferred structural motifs (e.g., adjacent Li₂S₈ interactions). Ultimately, our findings suggest that lower-lying Li₂S₈ rings interacting with already lithiated S₈ molecules, as well as Li-polysulfide cluster-type structures such as Li₂S₆/Li₂S₂ or Li₂S₅/Li₂S₃, are the most favorable products of the third lithiation step.

4. Conclusions

Our study provides a detailed atomistic insight into the initial lithiation process on the (001) surface of -sulfur, offering a fundamental understanding of the early-stage reactions in Li–S batteries. The findings suggest that lithiation preferentially begins with S₈ rings on the lower layer of the surface, leading to the formation of Li₂S₈ molecules. These surface-stabilized Li₂S₈ molecules exhibit strong similarities to their counterparts in gas/liquid-phase lithiation, indicating that early surface lithiation plays a critical role in determining subsequent reaction pathways. As lithiation progresses, further reactions around Li₂S₈ dimers lead to the formation of Li-polysulfides, including Li₂S₆, Li₂S₅, and lower-order species like Li₂S₃ and Li₂S₂. Notably, the formation of Li₂S₄ is sterically hindered on the surface, which may explain its preferential formation in the liquid phase. The dissolution of higher-order polysulfides into the electrolyte could further drive their reduction, reinforcing the experimentally observed discharge mechanism. These insights contribute to a deeper understanding of solid–liquid interactions during Li–S battery discharge and highlight the significance of surface lithiation in shaping polysulfide evolution.

Therefore, the current study presents a detailed atomistic picture of the first lithiation reactions on the (001) surface of α-sulfur and suggests a discharge mechanism of Li–S batteries with crystalline sulfur cathode which can contribute to a better understanding of such a complex process involving both solid- and liquid-phase reactions.

Author contributions

J. L. contributed to formal analysis, investigation, methodology, visualization, and writing (original draft, review & editing); D. S. contributed to supervision, validation, and writing (review & editing); P. P.-A. contributed to conceptualization, funding acquisition, methodology, supervision, validation, and writing (review & editing).

Data availability

All data and code supporting this study are publicly available at https://github.com/jolu75/Data. The data were curated and maintained by J. Lührs.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was primarily supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the RTG BEAM (project number 329482734). The authors gratefully acknowledge DFG funding via projects PA3141/3 (project number 420536636), PA3141/5 (project number 446879138). The authors also acknowledge the computing time made available on the high-performance computer at the NHR Center of TU Dresden via the project ‘p_oligothiophenes’. This center is jointly supported by the Federal Ministry of Education and Research and the state governments participating in the NHR (https://www.nhr-verein.de/unsere-partner). P. P.-A. also gratefully acknowledges Rafael Müller for fruitful discussions.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5ya00050e

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