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Review Article

# Mesoscale Physicochemical Interactions in Lithium–Sulfur Batteries: Progress and PerspectivePUBLIC ACCESS

[+] Author and Article Information
Zhixiao Liu

Energy and Transport Sciences Laboratory,
Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77840
e-mail: liuzhixiao1985@tamu.edu

Aashutosh Mistry

Mem. ASME
Energy and Transport Sciences Laboratory,
Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77840
e-mail: aashutoshmistry91@tamu.edu

Partha P. Mukherjee

Mem. ASME
Energy and Transport Laboratory,
Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77840
e-mail: pmukherjee@tamu.edu

1Corresponding author.

2Present address: School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907

Manuscript received May 15, 2017; final manuscript received August 28, 2017; published online October 4, 2017. Assoc. Editor: William Mustain.

J. Electrochem. En. Conv. Stor. 15(1), 010802 (Oct 04, 2017) (10 pages) Paper No: JEECS-17-1049; doi: 10.1115/1.4037785 History: Received May 15, 2017; Revised August 28, 2017

## Abstract

The shuttle effect and poor conductivity of the discharge products are among the primary impediments and scientific challenges for lithium–sulfur batteries. The lithium–sulfur battery is a complex energy storage system, which involves multistep electrochemical reactions, insoluble polysulfide precipitation in the cathode, soluble polysulfide transport, and self-discharge caused by chemical reactions between polysulfides and Li metal anode. These phenomena happen at different length and time-scales and are difficult to be entirely gauged by experimental techniques. In this paper, we reviewed the multiscale modeling studies on lithium–sulfur batteries: (1) the atomistic simulations were employed to seek alternative materials for mitigating the shuttle effect; (2) the growth kinetics of Li2S film and corresponding surface passivation were investigated by the interfacial model based on findings from atomistic simulations; (3) the nature of Li2S2, which is the only solid intermediate product, was revealed by the density functional theory simulation; and (4) macroscale models were developed to analyze the effect of reaction kinetics, sulfur loading, and transport properties on the cell performance. The challenge for the multiscale modeling approach is translating the microscopic information from atomistic simulations and interfacial model into the meso-/macroscale model for accurately predicting the cell performance.

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## Introduction

The consumption of fossil fuel brings the energy crisis as well as the environment pollution to our planet. One solution to the contemporary energy issues is developing sustainable and renewable energy resources to reduce the use of fossil fuel in ground transportation and achieving the vehicle electrification [17]. The application of electric vehicles demands the improved energy storage techniques, and lithium-ion batteries (LIBs) hold the leading position in this field [8,9]. The bottleneck of commercialized LIBs is the limited specific capacity and energy density due to the nature of the active materials. The maximum specific capacity of graphite anode cannot exceed 372 mAh/g, and the LiCoO2 cathode can only deliver a specific capacity as high as 272 mAh/g.

Beyond LIBs, the lithium–sulfur (Li-S) battery is considered as the promising energy storage system in the visible future [10]. The sulfur (S) element as the cathode material can reach the specific capacity of 1672 mAh/g if $α$ -S8 can be ultimately reduced to Li2S, and the corresponding gravimetric energy density is 2567 Wh/kg. The power density can exceed 11,000 W/kg according to Nagata et al. study [11]. Also, the abundant S in the Earth crust makes the Li–S system economically competitive in the future market [12]. However, the commercialization of the Li–S battery is prevented by some critical challenges. The final discharge product Li2S has the poor electrical conductivity and is insoluble to the electrolyte [13,14]. The precipitation of Li2S can passivate the active surface of the carbon cathode framework, resulting in the loss of the discharge capacity [15]. The intermediate products, long-chain polysulfides (PSs), are soluble in the organic electrolyte. PSs can migrate to the anode side due to the concentration and potential gradients, directly react with the Li metal anode, and form the Li2S film. This process is known as the “shuttle effect,” which consumes active materials and leads to the irreversible capacity loss [16].

The shuttle effect can be suppressed via tuning the electrolyte system. Fan et al. found that diglyme as solvent could be a superior choice for the electrolyte system when the active surface area of the carbon cathode is limited [17]. Chen et al. reported that lithium trifluoromethyl-4,5-dicyanoimidazole as supporting salt in the electrolyte system could decrease the solubility of long-chain PSs and result in minimizing the shuttle mechanism [18]. Camacho-Forero studied the effect of salt concentration on solid electrolyte interphase (SEI) formation on the Li metal anode surface, and found that the salt can decompose and then form LiF as the main SEI product [19]. Using X-ray photoelectron spectroscopy and ab initio molecular dynamic (AIMD) simulation, Nandasiri found that the solid SEI evolution involves three stages: (1) the formation of stable Li compounds (Li2S, LiF, Li2O, etc.); (2) the formation of a matrix-type phase due to cross interaction between reaction products and electrolyte molecules; and (3) the precipitation of a highly dynamic monoanionic PS (i.e., LiS5) [20].

Fundamentally, the shuttle effect is attributed to the high solubility of PSs and the weak PS–carbon interaction [16,21]. The surface decoration is a promising method to trap PSs. It has been found that the heteroatoms and functional groups on the carbon surface can significantly enhance the affinity to PSs [2241]. Researchers are also interested in finding new materials to replace carbon-based cathode. It is found that transition mental dichalcogenides [4244] and Mxene [45] as cathode host materials can reduce the capacity loss. Understanding the interaction mechanisms between PSs and substrates is the key challenge for designing novel materials, and the density functional theory (DFT) calculation would provide valuable suggestions for the material design [46,47].

The performance of the Li–S battery is not only affected by the PS–substrate interactions. The Li–S battery is a complex system involving multistep electrochemical reactions, insoluble PS deposition in the cathode, mass transport between two electrodes, and self-discharge with formation of Li2S film in the anode (schematically shown in Fig. 1). These phenomena happen at different temporal and spatial scales. In addition, Li2S precipitation also varies the cathode microstructure significantly and affects the cell performance. It is a critical challenge to monitor an entire working Li–S battery without missing microscopic information by experimental techniques. In this regard, the multiscale modeling approach could be a probe to gauge physico-electrochemical phenomena in the Li–S batteries. This paper will review the progress of multiscale modeling studies on Li–S batteries.

The multiscale modeling strategy is a powerful tool for exploring the Li–S electrochemical system. The first-principles simulation can elucidate PS–cathode interactions at the atomic scale, which is helpful for designing alternative cathode materials to prohibit the shuttle effect and catalyze electrochemical reactions. Using first-principles calculations, Cui and collaborators found that two-dimensional (2D) materials with the polarized surface could immobilize PSs [21]. Following this concept, Nazar's group found that Ti2C nanosheets and MnO2 nanosheets as host materials can significant improve the discharge capacity [45,48]. Based on the reaction barriers obtained from the first-principle calculation, an interfacial model can be constructed to investigate the morphology evolution of the final discharge product precipitating on different cathode material surfaces [4951]. From the view of the mass and charge transport, the precipitation can significantly change the tortuosity and porosity of the porous cathode microstructure, which can potentially affect the cell performance [52]. The reaction kinetics and diffusion kinetics are sensitive to the intrinsic properties of materials in the cell [17]. Macroscale numerical models can be used to decrypt what (reaction limitation or diffusion limitation) determines the cell performance [53,54]. The reported macroscale models are always neglecting the effect of the precipitation morphology evolution on the cell performance, and the reaction kinetic rates as input parameters are also assumed values. In the near future, the macroscale modeling approach can be refined by incorporating reaction kinetics (estimated from first-principles calculations) and the information of precipitation growth (obtained from the interface model).

## Polysulfide–Substrate Interaction Mechanism

In the year of 2013, researchers started to investigate the effects of heteroatoms on the PS retention by using atomistic simulations. Wang et al. calculated the energy barriers of Li and S atoms penetration nitrogen (N) doped carbon nanotube [55]. They found that the N dopant can significantly reduce the Li penetration barrier from 8 to 10 eV to 0.70 to 1.77 eV; on the other hand, the N dopant can block the S penetration with a barrier higher than 9 eV. Song et al. studied the effect of N dopant on functionalized carbon substrates, and found that the S atom does not directly interact with the foreign N atom. Indeed, the N dopant promoted the attraction between the S atom and the oxygen functional group on the carbon surface [35]. These early studies provided some knowledge for introducing heteroatoms to trap the S atom. However, these simulations are far away from clearly understanding the interaction mechanisms between discharge products and the cathode surfaces. It is well known that the soluble intermediate products are long-chain PSs. The key feature of the cathode material should be the strong affinity to PSs rather than the strong binding force to a single S atom. Cui and coworkers first studied the adsorption behaviors of Li2Sn molecules at different lithiation stages (n = 1–8) [21]. The most important information from their study is that the nonpolar carbon cathode surface does not possess strong affinity to polar Li2Sn molecules (the adsorption energies are less than 1 eV with considering van der Waals correction). The reason is that the chemical binding forces between Li and C atoms are very week. The PS can only interact with the carbon substrate via van der Waals forces contributed by S atoms. As shown in Fig. 2(a), the chemical binding energies between PSs and carbon substrate are less than 0.4 eV. Cui et al. also studied the interactions between PSs and polar V2O5 surface as well as polar transition metal disulfide surfaces, and found that polar surfaces provide strong chemical attractive forces to PSs. The electronic structure analysis (Figs. 2(b) and 2(c)) clearly demonstrates that chemical bonds form between Li atoms and O (or S) atoms from the substrate surface.

Following Cui et al. concept, a variety of studies are performed for searching new cathode materials. Zhao et al. investigated the adsorption behaviors of PSs on the phosphorene monolayer with considering van der Waals correction [56]. It was found that the adsorption energies were dependent on the length of the sulfur chain, and varied between 1 eV and 2.5 eV. Comparing the adsorption energies on the phosphorene monolayer and those on the carbon substrate, it can be inferred that phosphorene as the cathode material can provide stronger affinities to PSs to mitigate the shuttle effect. Zhao et al. analyzed the density of states (DOS) of Li2S4/phosphorene, and found that both Li-S bonds and P–S bonds appeared between the adsorbate and the substrate, and the S 3p-P 3p hybridization was much stronger than the Li 2s–P 3p hybridization. Experimental efforts have also been performed for seeking alternative cathode materials, which can trap PSs. Nazar's group found that the discharge capacity can exceed 1400 mAh g−1 with C/20 rate when the Ti2C sheet was used as the cathode material. The reason is that the Ti2C nanosheet has good electrical conductivity and the surface is hydrophilic to soluble PSs [45]. Her group also reported that MnO2 nanosheets can serve as the host material in Li–S battery cathode for enhancing PS retention [48].

Silicene is a new monolayer material in group IV [5759]. The silicene bandgap is much narrower than the phosphorene bandgap [60], which indicates that the silicene cathode host will possess better electrical conductivity. Mukherjee and collaborators evaluated the silicene as the host material of the Li–S battery cathode [49]. They found that the adsorption energy of Li2S4 on the pristine silicene was 1.81 eV with considering the van der Waals correction, which indicated that silicene could also be the promising host material for the Li–S battery cathode. The N dopant can further enhance the attractive interactions between PSs and the silicene substrate. An interesting finding from this work is that the silicene does not only provide the strong affinity to the soluble Li2S4 but can also facilitate the dissociation of Li2S4. The S-S bond length is around 2.10 Å in the isolated Li2S4 molecule. After adsorption on the silicene monolayer, the S4 chain dissociates to two shorter S2 chains, and the distance between SII and SIII is 3.84 Å according to theoretically calculated results [49].

The theoretical studies discussed earlier predicted several 2D materials as cathode host materials for immobilizing soluble PSs. However, the models developed in these studies are too simple to represent the real chemical environment in the cathode. In a real Li–S battery, the electrode is always saturated by the electrolyte. The organic molecules can interact with PS molecules and the solid substrate. Great efforts have been performed to understand electrolyte-polysulfide-electrode interplays in the anode side. Liu et al. found that the compact Li2S film can form on the Li (110) surface and Li (111) surface according to the conventional first-principles calculation, and they also found that the (111) surface is more active to reduce long-chain PSs according to AIMD simulation [61]. Camacho-Forero et al. studied the decomposition behavior of solvent and Li salts on the anode surface using AIMD simulation, and found that LiF is the main SEI product [19,62]. However, the electrolyte-polysulfide-substrate interplay in the cathode side has not been well understood. Recently, Balbuena and collaborators employed the AIMD simulation to investigate the PS dissociation on transition metal chalcogenides saturated by the electrolyte consisted of 1,3-dioxolane, 1,2-dimethoxyethane and bis-(trifluoromethane)sulfonimide lithium salt [63]. They found that the layered MnO2 could oxidize the long-chain PS and increase the adsorption energy. This oxidizing process could assist the breaking of long chain to short chains by creating the polythionate complex as hypothesized by Nazar and coworkers [64,65]. Using the electrolyte system with low PSs' solubility could be a promising way to suppress the shuttle effect [18]. Seeking the alternative cathode material, which can catalyze the long-chain PS dissociation, can also prohibit the shuttle effect [50].

Recent atomistic simulations focus on evaluating the attractions between PSs and cathode surfaces. However, it should be noticed that the alternative cathode materials are expected to be inert to the electrolyte in order to avoid the side reactions, which consume the electrolyte. The AIMD simulations can track reactions among the electrolyte, PSs, and the cathode surface. The results could provide new knowledge for selecting the appropriate cathode material and electrolyte components. Also, the existence of electrolyte molecules affects the PS–cathode interactions. If the electrolyte can provide much stronger binding forces to PSs than the cathode substrate, the shuttle effect may not be prohibited. In this regard, it is necessary to understand the synergistic effect of the electrolyte and the cathode substrate on trapping the PSs.

## Li2S Film Growth and Surface Passivation

The precipitation of insoluble Li2S significantly affects the cathode microstructure and corresponding cell performance. The lateral growth of Li2S film, on the other hand, can passivate the active area of the cathode surface and lead to the death of the discharge process [15]. This “sudden death” caused by surface passivation is also observed in the Li–O2 battery and Na–O2 battery [6668]. In this regard, it is necessary to understand the growth kinetics of Li2S film and find the way to control the surface passivation during the discharge process.

Mukherjee's group developed a bottom-up mesoscale modeling approach (from DFT simulation to coarse-grained kinetic Monte Carlo) for investigating the Li2S film growing on different electrode substrates [49,50,61,69]. Using first-principles electrochemical thermodynamics, Mukherjee and collaborators found that the (111) surface has the lowest surface energy around the open circuit potential (Fig. 3(a)), which indicates that the (111) surface dominates facets of crystalline Li2S. They also proposed different reaction paths for Li2S growth along the normal direction of the (111) surface at the atomistic scale, and found that direct deposition is energetically more favorable than the Li2S2 deposition-and-reduction process [69]. Their work also revealed the atomistic structure evolution of Li2S/carbon interface. It was found that the Li2S (111) plane is more stable on the carbon substrate than the (110) plane (Fig. 3(b)) [52].

Based on findings from DFT simulations, Mukherjee's group developed an interfacial model to investigate surface passivation on the carbon cathode. Three stages of the Li2S film growth were identified by the interfacial model, which were the formation of nucleation seeds, isolated nanoisland growth, and island coalescence, as shown in Fig. 3(c) [50]. This Li2S growth process was also validated by the scanning electron microscope (SEM) measurement [15]. Mukherjee and coworkers found that a mediate-to-high temperature window is beneficial for mitigating the surface passivation [50]. The morphology evolution of Li2S film growth at different temperature conditions is depicted in Fig. 4. For 10% coverage, some small nanoislands are observed on the carbon cathode surface at T = −20 °C, fewer nanoislands are observed at T = 20 °C, but only one nanoisland appears when T$≥$ 40 °C. For the same coverage, the larger number density of nanoisland and smaller island size can provide more active sites for capturing Li2S molecules from the electrolyte to facilitate the lateral growth, which results in the fast passivation. The low temperature should be avoided for delaying the surface passivation. In addition, the mediate-to-high temperature condition can enhance the thickness growth, which is helpful for tolerating more Li2S precipitation and improving the discharge capacity. Recently, Gerber et al. reported another method to control Li2S growth by using benzo[ghi]peryleneimide (BPI) as the redox mediator [70]. They found that the discharge capacity was significantly improved by using BPI to control the morphology of the Li2S precipitation. BPI can facilitate the film growing along the thickness direction. Thereby, the surface passivation could be alleviated, and a higher discharge capacity could be achieved.

The interfacial model developed by Mukherjee's group was also extended to study the surface passivation of other cathode materials. As discussed in Ref. [49], silicene as the cathode host materials can provide much stronger affinities to PSs than carbon, which indicates that silicene is a competitive candidate for PS retention. However, the strong attractive interaction between the Li2S molecule and the silicene substrate can facilitate the formation of nucleation seeds and the lateral growth of nanoislands, leading to the fast surface passivation. The growth of Li2S film on the silicene substrate undergoes a homogeneous growth mode; while for the carbon substrate, the Li2S film growth yields the heterogeneous growth mode. For carbon-based cathode, the increase of operating temperature can delay the surface passivation. However, the silicene-based cathode still suffers from fast surface passivation due to the strong attraction between the silicene and Li2S molecule. Beyond using the alternative cathode materials, the decorated carbon cathode surface with functional groups can also trap PSs very well. Mukherjee and collaborators found that the increase in the number density of functional groups also leads to the severe surface passivation [50]. Mukherjee's work points to an important open question, hitherto unanswered in Li–S batteries: What dominates, shuttle effect or surface passivation, and what are the potential trade-offs on the cell performance?

The downside of the interfacial model presented here is that it assumes that the reactants concentrations are constants. However, in a working cell, especially at the initial stage of the discharging process, Li+ and S2− concentrations vary significantly according to the previous study [71]. Hence, it is necessary to couple the interfacial model with the macroscale model. The macroscale-level simulation is expected to provide the time-dependent concentration, which can be introduced into the interfacial model to investigate how the discharge current affects the surface passivation. On the other hand, the growth of the Li2S island also reduces the active surface area for electrochemical reactions, which affects the overpotential and voltage profile.

## Nature of Solid Li2S2

Li2S2 is the only solid intermediate product in the Li–S battery. However, the nature of the crystalline Li2S2 is not clearly understood. Xiao et al. detected Li2S2 by using an in situ nuclear magnetic resonance technique [72], but Li2S2 is not a thermodynamically stable phase according to the equilibrium phase diagram [73]. Siegel and collaborators found that Li2S2 could be disproportionate into Li2S and solid sulfur to lower the bulk Gibbs free energy of the system [74]. But the reduction reaction of Li2S2 to Li2S is kinetically slow [10]. The atomistic structure of crystalline Li2S2 was not clearly understood. In Siegel's study, the Li2S2 was assumed to have the same atomistic structure as crystalline Li2O2 (space group p63/mmc) [74]. Feng et al. predicted a Li2S2 crystal structure belonging to space group p1 by combining the evolutionary algorithm and DFT simulation [75]. Yang et al. also predicted a Li2S2 crystal structure with space group p42/mnm [76]. Yang et al. reported that the energy of p42/mnm structure was lower than the p1 structure by 17 meV per atom. However, they did not consider the contribution of atom vibration to the total energy. Mukherjee's group calculated Gibbs free energy of p1 structure and p42 structure by using first-principles atomistic thermodynamics, and found that p1 structure is more stable at the room temperature [77]. The recent experimental study detected p1 Li2S2 in the Li–S battery cathode by using an operando X–ray diffraction (XRD) technique [78].

The charge transport mechanism in p1 Li2S2 was elucidated in Liu et al. study [77]. The DOS demonstrated that perfect Li2S2 is a semiconductor with the bandgap of around 2.8 eV, which indicates that Li2S2 cannot transfer free electrons. Moradabadi and Kaghazchi investigated the effect of neutral Li vacancy on DOS distribution of Li2S [79]. They found that the Li vacancy can convert the insulating Li2S to a p-type conductor. However, the neutral Li vacancy does not work for Li2S2. The Fermi level of crystalline Li2S2 is still between the valence band maximum and the conduction band minimum according to Liu et al. study [77]. But Liu et al. found that unique crystal structure of Li2S2 is helpful for forming and transferring polarons. The mechanism of polaron formation is schematically shown in Fig. 5. In Li2S2, the $σp*$ orbital of $S22−$ is empty and can localize a negative charge element by accepting an extra electron (electron polaron); on the other hand, the $πp*$ orbit is fully occupied and can localize a positive charge element by losing an electron (hole polaron). In Li2S2, the predominant defects are negatively charged Li vacancy and the positively charged hole polaron. Liu et al. estimated the conductivity corresponded to vacancy diffusion (ionic conductivity) and polaron diffusion (electronic conductivity), and found that the electronic conductivity of Li2S2 is 15 orders of magnitude larger than the ionic conductivity at room temperature [77]. The electronic conductivity of Li2S2 is around 10−12 S cm−1, which is higher than the conductivities of Li2O2, LiO2, and Na2O2 but lower than the conductivity of NaO2 [80]. Liu et al. work demonstrated that hole polarons could transfer charges from the cathode to the electrolyte through Li2S2 film to support electrochemical reactions during discharge. If Li2S2 can be stabilized as the ended discharge products, the recharging over potential and the capacity retention could be improved with sacrificing the theoretical specific capacity. The theoretically estimated energy density for Li2S2 is about 1300 Wh/kg. In addition, converting $α$ -S8 to Li2S2 is accompanied with only 5% volume expansion, which indicates that Li2S2 precipitation will generate less mechanical damage in the carbon-based cathode framework than Li2S precipitation [77,81]. The challenge is that the solid Li2S2 is not stable according to the binary Li–S phase diagram [74,82]. A previous theoretical study on sodium oxides demonstrated that the Na2O2 phase is transferred to NaO2 phase as the particle size deceases [83]. For Li–S batteries, the size effect on the stability of discharge products should be understood for realizing the Li2S2-based system. The size of Li2S2 particles is expected to be controlled by the nanoconfined structure.

As discussed earlier, Li2S2 as the final discharge product can decrease the charging overpotential and alleviate the mechanical degradation caused by the lithiation-induced volume expansion, although the specific capacity is sacrificed. Given that Li2S2 is not a stable phase according to equilibrium Li–S binary phase diagram, stabilizing the Li2S2 in the nanoconfined structure will be an interesting avenue of study, and we should not forget that nanoconfined structures may also be helpful in trapping the soluble PSs.

## Performance Prediction

The DFT simulation and AIMD simulation are powerful tools to elucidate the physiochemical interactions at the nanometer scale, and the simulation time cannot even exceed picoseconds [61,62]. The interfacial model reported in Ref. [51] can predict the morphology evolution of the Li2S precipitation in several hours. However, the length scale of this model cannot exceed 1 μm. These low spatial-temporal scale models cannot directly predict cell performance (e.g., capacity, voltage, and cycling stability).

White and collaborators developed a one-dimensional (1D) macroscale model to describe electrochemical/chemical interactions in the entire cell [71]. White's model included the mass transport in the electrolyte and could be used to predict the PS concentration evolution and distribution. However, this model only worked at the very low discharge rate (1/50 C rate). Based on White's model, Ghaznavi and Chen systematically studied the effects of discharge current, cathode conductivity, reaction kinetics, and sulfur loading on the cell performance [53,54]. They found that the cathode conductivity significantly affects the capacity loss at the high C rate, and a minimum conductivity is required to functionalize the battery. Also, their model predicted that the increase in sulfur loading could reduce the specific capacity, which agrees well with experimental results [84]. Later, Ghaznavi and Chen extended the model to simulate the charging behavior and investigated the effect of transport properties. They found that the low PS diffusion coefficients could lead to the capacity loss and the specific capacity also tended to decrease with increasing the cathode thickness. This model could reproduce the typical two-plateau charging profile with assuming a large Li2S solubility. The assumption did not make sense in a real Li–S battery. In Yoo et al. one-dimensional model, the charging process and reactions on the Li anode surface were included [85]. Yoo et al. model was successfully validated by experimental results. It was found that the high reaction rates of PS reduction in the anode led to a significant overcharge problem, and the high PS diffusivity resulted in the poor cycling performance. Zhang et al. modified White et al. model and studied the effect of slow Li+ transport on the cell performance [86]. In this study, a Li–S pouch cell (OXIS Energy Ltd., Abingdon, UK) with a rated capacity of 3.4 Ah at 0.2 °C was used to calibrate and validate the discharge curves generated by the model. It was found that improving the Li+ transport property and optimizing electrolyte-to-sulfur ratio were key points to improve the rate capability. They also found that the capacity loss caused by high discharge current could be recovered by 1 h relaxation before charging [86].

Although models discussed earlier were successfully employed to investigate the reaction limitation and transport limitation on the battery performance, they still miss some important features of the real Li–S batteries. The cathode framework is always a porous structure. Previous models only simply considered the global porosity of the cathode, but missed the microstructure properties such as particle size and pore size distributions. Franco's group developed a microstructure dissolved model, which included the information of the size of carbon particles, the size of interparticular pores, and the size of mesopores in the carbon particle [87]. For the cathode with large carbon particles, the capacity is limited by the clogging of mesopores caused by the Li2S formation on the particle surface. This model provided new insights for how to improve the cell performance by tailor the cathode microstructure. Mukherjee and collaborators stochastically reconstructed the virtual three-dimensional (3D) cathode microstructures based on SEM images (Figs. 6(a)6(d)). It was found that the increase in sulfur loading results in reducing the porosity and increasing tortuosity of the cathode microstructure (Fig. 6(e)). The transport properties from analyzing cathode microstructure could be translated to the macroscale model to predict the cell performance (Fig. 6(f)).

Another approach to improve the accuracy of the macroscale model explicitly involves the stochastic Li2S precipitation and film growth. As demonstrated in Mukherjee's interfacial model, the Li2S film growth should start from the nucleation of Li2S clusters, and the low number density of clusters (the number density is dependent on the S2-concentration and temperature) can lead to the heterogeneous growth [50]. Recently, Zhao's group developed a one-dimensional model with considering Li2S nucleation kinetics, and found that the higher discharge rate can suppress the isolated Li2S particle growth and result in the larger capacity loss [88]. This finding agrees with the result from Mukherjee's interfacial model. Both the interfacial model and Zhao's macroscale model point to the concept that the morphology evolution of Li2S precipitation should be controlled for prohibiting the capacity loss.

## Summary and Outlook

This paper comprehensively reviewed the progress of multiscale modeling studies on Li–S batteries. A hierarchical modeling philosophy (as shown in Fig. 7) can be developed based on these studies: revealing the PS/substrate interaction mechanisms at the atomistic level; using the interaction parameters to elucidate the growth kinetics of the Li2S film and predict the growth rate at the interfacial level; reconstructing the 3D cathode microstructure based on SEM micrograph and predicting the microstructure evolution according to the growth rate of the Li2S film; and finally translating the transport properties of the cathode microstructure into the macroscale model for performance prediction.

The DFT simulation revealed that one main reason of the shuttle effect is the relatively weak attractive interaction between the polar PSs and the nonpolar carbon surface. Based on this concept, great efforts were conducted to seek novel materials for trapping PSs by using the DFT simulation and AIMD simulation. It was found that two-dimensional transition metal dichalcogenides and silicene can provide strong affinities to PSs by forming chemical bonds between Li atoms and the surface. On the other hand, the interfacial model demonstrated that the strong PS–substrate attractive interactions could facilitate the nucleation and lateral growth of Li2S islands, resulting in the fast surface passivation and the sudden death of the discharge process. Hence, it is necessary to carefully tune the PS–substrate interactions to control the growth of Li2S film for achieving the improved cell performance. Several macroscale models have been developed to analyze the effect of reaction kinetics, sulfur loading, and transport properties on the cell performance. These models can somehow reproduce the two-plateau voltage profile during the discharge process, but there are still unreasonable assumptions. For example, all reaction rates in macroscale models were assumed values, and the reliability of the rates cannot be justified. In addition, the macroscale models neglected the resistance of the insulating discharge products, which makes the voltage profile less accurate. It has been demonstrated that DFT and AIMD simulations can estimate the PS dissociation behaviors and the conductivity of the product. The coupling between atomistic simulations and macroscale models is expected to provide new insights for improving the cell performance.

Currently, the irreversible capacity loss is attributed to the shuttle effect. Decorated carbon or two-dimensional nanosheets can trap PSs and mitigate the shuttle effect. However, the strong affinity to the Li2S molecule provided by these novel cathode materials can also facilitate the lateral growth of Li2S film. Given the poor conductivity of the solid Li2S, the spread of Li2S film can passivate the active surface of the substrate and lead to the “sudden death.” According to simulation results from Mukherjee's group, it can be inferred that mitigating the shuttle effect usually results in the fast surface passivation. We hypothesize that the cell performance can be further improved by balancing the PS retention and the surface passivation. The macroscale models discussed previously can be employed to quantify the shuttle effect on the cell performance. However, the current macroscale modeling approach cannot describe the stochastic Li2S precipitation on the cell performance. On the other hand, the interfacial model developed by Mukherjee's group only focuses on the chemical reactions on the cathode surface, which cannot reflect the shuttle effect. A more detailed macroscale model with considering stochastic Li2S precipitation is needed to quantitatively answer the question about the competition about the PS retention and the surface passivation.

The cathode architecture plays an important role in determining the performance of the Li–S battery [14]. The cathode microstructure should have an appropriate tortuosity for enhancing the effective Li+ ion conductivity in the electrolyte phase. On the other hand, the microstructure also should provide enough specific surface area for mitigating the surface passivation as discussed. In addition, the pore size distribution of the cathode microstructure should also be carefully designed for tolerating mechanical degradation caused by S8-to-Li2S phase transition, which is related to a huge volume expansion [81]. A wide variety of microstructures have been synthesized to develop the performance of the battery [8992]. A desirable cathode microstructure should effectively obstruct the dissolution of polysulfide, supply a large conductive area for insulating Li2S deposition, and facilitate Li+ ion transport. The microstructure evolution during the discharge process could bridge the gap between atomistic simulations and macroscale models. Elucidating the microstructure-transport-performance interplays should be an important perspective for achieving the high-performance Li–S battery.

## Acknowledgements

This work was funded in part by the Office of Energy Efficiency and Renewable Energy (EERE), U.S. Department of Energy, under Award DEEE0006832. The authors also acknowledge the American Chemical Society, the Electrochemical Society, the Royal Society of Chemistry, and John Wiley and Sons for the figures reproduced in this article from the referenced publications of their respective journals.

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## Figures

Fig. 1

(a) Schematic diagram of a Li–S battery with main challenges. (b) A typical voltage profile and corresponding species evolution during the discharge process.

Fig. 2

(a) Adsorption energies of polysulfides on the graphene substrate. The black curve indicates that the adsorption energy is calculated without considering van der Waals interaction, and the red curve indicates that the adsorption is calculated with considering van der Waals interaction. The data points are adopted with permission from Ref. [21]. (b) Difference charge density distribution between Li2S4 molecule and V2O5 surface. Snapshots demonstrate the difference charge density of (b) Li2S4 on V2O5 surface and (c) Li2S4 on TiS2 surface. The difference charge density demonstrate electrons are transferred from the Li2S4 molecule to the cathode substrate. (Snapshots are adopted with permission from Ref. [21].)

Fig. 3

(a) Li2S surface energy as the function of applied potential (adopted with permission from Ref. [69]). The lowest surface energy indicates the most stable surface. (b) Energy profile of Li2S (111) plane formation on the graphene substrate (adopted with permission from Ref. [52]). The downhill profile represents an exothermic reaction path. (c) Snapshots in the upper row depict the simulation results of Li2S growth on carbon substrate with the coverage of 30%, 50%, and 90%. SEM images in the bottom row depict the morphology evolution of precipitated Li2S on carbon fiber cathode after 2.5 h, 4 h, and 6 h with potentiostatic discharge at 2.02 V. (Snapshots are adopted with permission from Ref. [51]. SEM images are adopted with permission from Ref. [15].)

Fig. 4

Effect of temperature on the morphology evolution of Li2S precipitation. (Snapshots are adopted with permission from Ref. [51].)

Fig. 5

Schematic diagram of polaron formation in Li2S2 (adopted with permission from Ref. [77])

Fig. 6

(a) SEM micrograph of carbon microstructure without sulfur loading; (b) stochastic regeneration of carbon microstructure; (c) SEM micrograph with sulfur loading; and (d) stochastic reconstruction of microstructure with sulfur loading. (e) The increase in sulfur loading can reduce the porosity and increase the tortuosity. (f) Discharge voltage profile and the porosity evolution predicted by the macroscale model. (Figures are adopted with permission from Ref. [52].)

Fig. 7

The schematic illustration of the hierarchical modeling philosophy for understanding multiscale interplays in a Li–S battery

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