Rate Constants of Electrochemical Reactions in a Lithium-Sulfur Cell Determined by Operando X-ray Absorption Spectroscopy

The separator/electrolyte films were prepared using a block copolymer of polystyrene-b-poly(ethylene oxide) (SEO) synthesized using methods described in the work by Hadjichristidis et al.⁴⁰ and purified using methods described in the work by Teran et al.⁴¹ The molecular weights of polystyrene and poly(ethylene oxide) are 200 kg/mol and 222 kg/mol, respectively. Lithium perchlorate (LiClO₄, Sigma-Aldrich) was dried for 24 hours under vacuum at 90°C before use. The separator/electrolyte films containing SEO and LiClO₄ were prepared according to the method described in the work by Wujcik et al.⁴² The thickness of separator/electrolyte film used was 22 μm.

Cathode slurries containing S₈ (Alfa Aesar), Li₂S (Sigma-Aldrich) carbon black (Denka), LiClO₄, and SEO (identical LiClO₄/SEO composition to that of the electrolyte separator) was mixed in n-methylpyrrolidone (NMP). The slurry was composed of 89 wt% of NMP. S₈ and Li₂S were mixed in a 256:46 weight ratio to produce Li₂S_x with an average x value of 8 as the starting material. Due to the insulating properties, both ionic and electronic, of S₈, Li₂S₈ was used as the starting material. Since Li₂S₈ is soluble in the slurry, we expect a uniform distribution of the sulfur-containing species in the cathode (as opposed to insoluble S₈), and we posit that this leads to better contact between the active material, the electrolyte and carbon black in the dry cathode. The slurry was mixed overnight at 90°C and subsequently mixed using a homogenizer (Polytron) set to 15,000 RPM. Homogenization was done for five minutes and repeated three times, with two minute rests between each cycle to prevent the solution from heating up to undesirable temperatures. The resulting slurry was then casted onto an 18 μm thick aluminum foil current collector using a doctor blade. The film was dried under Argon at 60°C for 10 hours and then placed under static vacuum overnight at room temperature. The resulting cathode had an average thickness of 16 μm, with the resulting composition: 12.8 wt% Li₂S₈, 51.4 wt% SEO, 5.5 wt% LiClO₄, and 30.3 wt% carbon. Our use of a relatively thin sulfur cathode with low sulfur loading was motivated by our desire to minimize self-absorption in the XAS experiments.

A pouch cell was prepared according to the method described in the work by Wujcik et al.³⁷ The electrolyte film was placed on the cathode. The lithium metal anode was then placed over the electrolyte film. The cathode-electrolyte-anode stack was tabbed and sealed in a pouch cell was kept at rest at room temperature in an argon environment for 48 hours before taking measurements. The cell was then taken out of the argon-filled glove box and placed on a sample holder connected to a heating source. It was then held at a temperature of 90°C for 1.5 hours to ensure good electrical contact between the cathode, electrolyte, and anode layers. The cell was then charged to partially form S₈, and then discharged at 90°C at a C/20 rate using a VMP3 Potentiostat (Bio-Logic). High temperature operation is necessary due to the limited conductivity of polymer electrolytes at low temperatures.⁴³ Figure 1 shows a schematic of the assembled cell. The discharge and charge rate was calculated using the measured mass of the cathode electrode, the known weight percent of sulfur in the cathode, and assuming a theoretical capacity of 1672 mA-h/g for sulfur. The voltage window was kept between 1.5 V and 3.0 V.

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XAS measurements were performed at beamline 4–3 of the Stanford Synchrotron Radiation Lightsource. Preliminary XAS experiments were performed at beamline 5.3.1 of the Advanced Light Source. Measurements were taken in fluorescence mode using a four element Vortex detector, with 0.1 eV energy resolution around the absorption K-edge. One scan took roughly 10 minutes to collect, equivalent to roughly 13.9 mA-h/g of capacity passed per scan. The beam spot size was 2 mm² and was not moved during cycling. The cell holder was inside a helium-filled chamber during the in operando measurements. Calibration of the X-ray energy was performed using sodium thiosulfate (Sigma-Aldrich), setting the first peak maximum to 2472.02 eV.

All spectra were analyzed using the Athena X-ray absorption spectroscopy program. Raw XAS spectra were used to calculate the "total sulfur" intensity based on methods described by our previous work.⁴² For peak deconvolution and product analysis, all spectra were normalized and self-absorption corrected using the Athena XAS analysis package. The initial spectra were fitted with 4 Gaussian peaks and a step function. After 50 mAh/g the spectra were fitted with 6 Gaussians to account for the increasing skewness in the main-edge peak due to blueshift of the main-edge peak for mid-chain and short-chain polysulfides. Example of fitting an experimental spectra with 6 Gaussian peaks and a step function is shown in Figure S1.

Theoretical XAS spectra for different lithium polysulfides were presented by Pascal et al. in a previous publication,⁴⁴ and the results are summarized in Figure 2a. In the inset of Figure 2a, we show a typical molecular conformation of one of the polysulfides, Li₂S₈. Polysulfides with chain length between 3 and 8 have two charged terminal sulfurs and the remainder of the internal sulfurs are uncharged. The two kinds of sulfurs give rise to two distinctive XAS features: a pre-edge peak corresponding to the two charged end-chain sulfurs and a main-edge peak corresponding to the internal sulfurs. The area under the theoretical pre-edge peak of each polysulfide is denoted by A^Th_p. Similarly the area under the theoretical main-edge peak of each polysulfide is denoted by A^Th_m. The spectral features of the polysulfides are approximated as a sum of Gaussian peaks and the areas under selected peaks were used to compute A^Th_p and A^Th_m as outlined in Figure S2. In Figure 2b we plot the ratio, A^Th_m/A^Th_p, as a function of polysulfide chain length, x in Li₂S_x (3 ≤ x ≤ 8). The line in Figure 2b is a least squares linear fit. We use this linear fit as a "calibration" to determine the average chain length of polysulfides in our cell, x_avg, using measured values of pre-edge and main-edge areas, A_p and A_m. The straight line in Figure 2b can be represented as

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In Figure 2c we plot the sum, (A^Th_p + A^Th_m), as a function of x in Li₂S_x (4 ≤ x ≤ 8). To a good approximation, (A^Th_p + A^Th_m) is 6.61, independent of x. The theoretical spectrum of Li₂S contains a unique peak at 2476 eV that is not present in any of the polysulfides. The area under this peak, A^Th_s, was calculated by approximating the theoretical Li₂S spectrum by a sum of Gaussian peaks as shown in Figure S3. The value of A^Th_s is 3.07.

Thus,

We use this to estimate the moles of Li₂S. in our cell is determined by estimating the area under the peak at 2476 eV, A_s.

The XAS cell was made with Li₂S₈ in the cathode. Our use of Li₂S₈ facilitated dispersion of the sulfur species in the cathode. Our main objective is to determine the state of the sulfur-containing cathode as the cell is discharged. We used a relatively thin cathode and adjusted the sulfur content in the cathode to ensure that all of the sulfur-containing species in the cell could be detected by XAS. The cell was prepared 48 hours before the XAS experiment, stored at room temperature in an argon glove box, placed in the XAS sample stage, heated to 90°C for 1.5 h, charged at C/20 until the voltage reached 3.0 V, and then discharged at C/20. Figure 3a shows all of the raw XAS spectra during these experiments. The magnitude of the high energy plateau attained between 2500 and 2575 eV is indicative of the total amount of sulfur detected. We define I₀ to be the average value of the raw XAS signal between 2500 and 2575 eV obtained just prior to discharge. We define I_n as the average value of the raw XAS signal in the same energy range obtained during other scans. The time dependence of the cell potential during these experiments is shown in Figure 3b. The corresponding values of I_n/I₀ versus time shows are shown in Figure 3c.

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If our cell was perfectly designed, then I_n/I₀ would be independent of time. In our case, I_n/I₀ increased by about 12% during the heating step, and increased by about another 16% during the charging step. This is attributed to the dissolution of Li₂S₈ into the separator during the heating and charging steps. Because the anode side faces the incoming X-ray source, the incident intensity on the sulfur-containing species in the separator is higher than that on the sulfur-containing species in the cathode. Similarly, the fluorescence signal from the sulfur-containing species in the separator is more efficiently detected because the anode side also faces the detector. Thus, the diffusion of sulfur-containing species into the separator is expected to increase I_n/I₀. During the discharge step, however, I_n/I₀ remained approximately constant, varying between 1.05 and 0.95. The constancy of I_n/I₀ during discharge indicates that all (or nearly all) of the products of sulfur reduction were detected by XAS experiment. We therefore conclude that there is no further change in the concentration of polysulfides in the separator during the discharge step.

The raw spectra shown in Figure 3a were normalized and corrected for self-absorption. All of the normalized spectra exhibited a pre-edge peak around 2471 eV and a main-edge peak around 2473 eV. This enables calculation of the areas under the pre-edge, A_p, and main-edge peak, A_m. These areas can be used to determine the average polysulfide chain length in the cell, x_avg,cell (x for Li₂S_x), using Equation 8. After the heating step, x_avg,cell equals 7.0. After the charging step, x_avg,cell reached 8.1.

An ideal cell would be one wherein all of the Li₂S₈ remained in the cathode during storage prior to the XAS experiment and during the heating step. In other words, x_avg,cell would equal 8.0 in the ideal cell after the heating step. It is evident that our cell is not ideal as x_avg,cell is 7.0 at the end of the heating step. This departure from ideality is attributed to the dissolution of Li₂S₈ into the separator, subsequent reactions with the lithium metal anode, and shuttling of the resulting shorter polysulfides back into the cathode. We posit that during storage and the heating step, 0.29 moles of Li from the anode per mole of Li₂S₈ is consumed to reduce the average chain length from 8 to 7.0, as indicated in Equation 10.

The cell with x_avg,cell = 7.0 was then charged at a C/20 rate. In an ideal cell, all of the sulfur-containing species would be converted to S₈ after charging. If this were true, x_avg,cell would equal infinity after the charging step. Instead we find that the average chain length increased from 7.0 to 8.1 during the charging step. During the charging step, 1.02 moles of electrons were delivered to the anode per mole of S₈ in the cell (t = 1.27 h in Equation 7). If all of these electrons participated in the oxidation of Li₂S_7.0, then x_avg,cell at the end of charging step would have been 14.2. The observed departure from ideality during the charging step is attributed to the reduction of polysulfide species at the anode/separator interface instead of complete conversion into Li metal. We conclude that these side reactions consume 0.73 moles of electrons per mole of S₈. The remainder participated in the oxidation of Li₂S_7.0 and the concomitant reduction of Li⁺ to Li metal:

Figure 4a shows the self-absorption-corrected normalized spectra during discharge. Figure 4b shows the dependence of cell potential versus capacity, Q, during discharge. The XAS spectra in Figure 4a contain standard signatures of polysulfides: a main-edge peak with area A_m and a pre-edge peak with area A_p. Using methods described above and Equation 8 we determined x_avg,cell as a function of capacity, and the results are shown in Figure 4c. (The spectra do not contain signatures of polysulfide radicals that are sometimes observed in Li-S cells.^34,37,45) The relatively low discharge capacity, 503 mAh/g, of our cell is due to non-idealities discussed above. During discharge, x_avg,cell decreased monotonically from 8.1 to 3.0. In the early stage of discharge, Q < 100 mAh/g, x_avg,cell decreases rapidly with increasing Q. In the late stage of discharge, Q > 100 mAh/g, x_avg,cell decreases slowly with increasing Q.

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The measured XAS spectrum at the end of discharge is shown in Figure 5a. In addition to the pre-edge and main-edge peaks at 2471 eV and 2473 eV, an additional peak is observed at 2476 eV. The three dashed lines in Figure 5a correspond to the characteristic energies of these peaks. As discussed above, the theoretical spectra in Figure 2a show that the peak at 2476 eV is a unique signature of Li₂S and it arises due to the crystalline nature of this compound.⁴⁴ In addition to determining A_p and A_m, we also determined A_s for each of the spectra shown in Figure 4a. We define as the moles of Li₂S formed per mole of polysulfides. In theory, is given by

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Where the constant 0.46 is based on analysis of the theoretical spectra and Equation 7. Note that in this analysis, Li₂S is not considered as a polysulfide. In Figure 5b, we plot on the left axis and , on the right axis versus Q. The moles of Li₂S formed is low in the early stage of discharge, Q < 100 mAh/g, but increases rapidly in the late stage of discharge, Q > 100mAh/g. Whether or not Li₂S forms in the early stage of discharge remains an interesting, open question. We suspect that the values we have obtained are due to limitations of our spectral fitting procedure. In our cell, remains small reaching a maximum value of 0.24 at the end of discharge. Note that the theoretical spectrum of Li₂S contains a feature at 2474 eV. In principal, we should correct the measured values of A_m to account for the fact that some of the signal at the main-edge peak is due to Li₂S. This correction is small because remains small in our experiment.

The dependence of x_avg,cell on n_e during discharge is shown in the inset in Figure 6. We have assumed that all of the electrons delivered to the cathode are consumed by the Li₂S₈ molecules; side-reactions such as the formation of the solid electrolyte interphase (SEI) are ignored. The curve in the inset represents the theoretical predication, Equation 2. The theoretical value of n_e corresponds to a cathode that contains pure S₈ at the beginning of discharge (see Equation 1). In the experiments however, our cathode to a good approximation contains Li₂S₈ at the beginning of discharge. The data points in the inset in Figure 6 represent experimental values of x_avg,cell and n_e. x_avg,cell was obtained from measurements of A_p and A_m using Equation 8. To account for the fact that the discharge begins with Li₂S₈, we set n_e to a value close to 2 at the beginning of discharge and it is incremented based on Equation 7. The actual value used was 1.97 to obtain a perfect match between the experimental data and the theoretical prediction at the beginning of discharge. It is evident that the decrease in the average chain length of sulfur-containing species in the cell is in reasonable agreement with Equation 2.

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Our analysis above indicates that some of the Li₂S₈ molecules located in the cathode when the cell was made diffuses into the separator and reacted with Li metal. This results in an average composition of Li₂S_7.0 before charging. The polysulfides in the separator not in contact with electronically conducting materials cannot participate in charge or discharge reactions. Their presence also affects our ability to detect the nature of the sulfur-containing species inside the cathode. We posit that these effects are responsible for the deviations between theory and experiment in the inset of Figure 6. We define x_avg,cathode as the average length of sulfur-containing species in the cathode. We assume that the average length of the sulfur-containing species in the separator is fixed at 7.0 during the discharge process. Our cell thus contains two layers with different concentrations of sulfur. Given the agreement seeing in the inset of Figure 6, we conclude that most of the sulfur is in the cathode. Specifically, in our model, we assumed that 90% of the sulfur atoms are in the cathode and 10% of the sulfur atoms are in the separator. This enables calculations of the transmission coefficients of the two layers of our cell based on the known absorption coefficients of sulfur and the other elements in our cell. These calculations indicate that the transmission coefficient of the separator layer, T_sep = 0.623, while that of the cathode layer, T_cathode = 0.398. The distance between the two layers is set to 19 um based on the geometry of our cell. (We assume for simplicity that all of the sulfur-containing species are located in the middle of each layer.) The measured value of x_avg,cell reflects the length of sulfur-containing species in both the cathode and separator (x_avg,cathode, x_avg,sep) with a weighting function that depends on the sulfur content and the transmission coefficient of each layer. This is quantified by Equation 13.

where D_cathode and D_sep reflect the weighting functions as shown in Equations 14 and 15.

Since x_avg,sep = 7.0, we can calculate x_avg,cathode corresponding to each value of x_avg,cell. Figure 6 shows the dependence of x_avg,cathode versus n_e. The agreement between theory and experiment reflects the fact that the data are consistent with our assumption that 10% of the sulfur atoms are lost in the separator and hence not available for redox reductions. Our analysis indicates that x_avg,cathode at the start of discharge is 8.28 while x_avg,cathode at the end of discharge is 2.28 (see Figure 6).

The XAS peak at 2476 eV enables detection of Li₂S. It is therefore helpful to distinguish between Li₂S and other sulfur-containing species, namely polysulfides (Li₂S_x, 2 ≤ x ≤ 8). We define x_avg,PS as the average length of polysulfides. We calculate x_avg,PS using the following equation:

We arrive at this equation based on the sulfur mole balance in the cathode. For each mole of polysulfides () in the cathode we have moles of Li₂S, and together these compounds gives moles of .

The final result of our analysis of the XAS data is given in Figure 7 where x_avg,PS and are plotted as a function of n_e.

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It is not possible to identify a particular pathway that is consistent with the data in Figure 7. We used the principal of parsimony to interpret these data. In particular we used a model presented in the introduction beginning with Equation 4 and ending with Equation 6. We define C₈, C₄, C₂, C₁ to be the molar concentrations of Li₂S₈, Li₂S₄, Li₂S₂, and Li₂S, respectively, and assume that the reactions are limited by the concentrations of the sulfur-containing species. We expect this to be true at extremely low C rates. The simplest rate expressions for Reactions 4 through 6 are given below:

Since electrons are consumed in all three reactions,

The measured quantities, x_avg,cathode, x_avg,PS, and , are related to the molar concentrations of the sulfur-containing species:

Equations 17 through 24 were integrated numerically for specific values of k₁, k₂, and k₃., with initial conditions C₈ = 1, C₄ = C₂ = C₁ = 0. The solved C₈, C₄, C₂, C₁ at each t are used to predict x_avg,cathode, x_avg,PS, , and n_e at each t. The symbols in Figure 8 show the experimentally determined values of x_avg,cathode, x_avg,PS, , and n_e, respectively, as a function of time, t. The experimental values of in Figure 8c were subtracted by a constant so that at t = 0 is zero. This subtraction is necessary as the spectral signal at any given energy is not identically zero even if the species is absent due to factors such as contributions from neighboring excitations and background subtraction inaccuracies. The curves in Figure 8 show results of the numerical integration for k₁ = 0.368 h⁻¹, k₂ = 3/4 k₁, and k₃ = 1/6 k₁. It is evident that the measurements are consistent with the proposed model. Our analysis indicates that the rate of reduction of sulfur-containing species decreases with decreasing chain length. To our knowledge, these are the first estimates of reaction rate constants for discharge reactions in the cathode of a Li-S cell.

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Figure 9 plots the predicted concentrations of Li₂S₈, Li₂S₄, Li₂S₂, and Li₂S versus discharge capacity, based on our model, Equations 17–21.

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The results presented in Figures 8 and 9 represent the first step in quantifying the rates of reactions that occur in a sulfur cathode. Our simple discharge reaction models is also consistent with the reaction mechanism proposed by Hagen et al.²⁴ Most other studies suggest that the reduction of sulfur in the cathode during discharge is likely to follow more complex schemes. For example, Barchasz et al.²¹ have proposed the following reaction for the reduction of Li₂S₄:

Such reactions require concerted action on several reactant molecules. In the example above, three Li₂S₄ molecules must react with two Li⁺ and two e⁻ to yield the stated product. In contrast, the proposed reaction for Li₂S₄ (Equation 5) only involves one reactant molecule. The additional complication with Equation 25 is the fact that the reaction must involve many steps wherein the Li₂S₄ molecules are cleaved and then recombine to give four Li₂S₃ molecules. For these reasons, Equation 5 is more likely to proceed than Equation 25.

The reaction rates that we present are only applicable to the regime 0 ≤ Q ≤ 500 mAh/g (the discharge range covered by our experiments). It is likely that these rates will change as Li₂S becomes the dominant species in the cathode. The shorter-chain polysulfides such as Li₂S₂ are insoluble^46–48 and thus their concentration near reaction sites in the cathode may be significantly different from the bulk concentration. In addition to electrochemical reactions, polysulfides can interconvert through chemical reactions. Sophisticated models that include transport are needed to account for complications arising from polysulfide dissolution and concomitant shuttling effects. Further work is needed to explore the effects.

The subject of reaction mechanisms in sulfur cathode is of considerable current interest.^{21,27,51–57,28,31,32,35,38,42,49,50} Our detection of Li₂S at the very early stage of discharge (as seen in Figure 8c) is consistent with the findings of Waluś et al.,^20,58 Cuisinier et al.,³³ and Conder et al.⁵² Similarly, the formation and subsequent consumption of Li₂S₄ up to 500 mAh/g of discharge in Figure 9 is similar to the findings of Dominko et al.,³⁶ Zhang et al.⁵⁵ and Zheng et al.⁵¹ Our results in Figure 9 also indicated a significant amount of Li₂S₂ inside the cathode at a depth of discharge of 500 mAh/g, which is consistent with the results of Kawase et al.⁵⁴ Reaction mechanisms in the sulfur cathode have also been studied using computational simulations by Burgos et al.⁵⁹ They found that a variety of radical and dianion species were formed in their simulation cell. However, S₈²⁻ dianions were formed at the early stage of discharge, S₄²⁻ dianions dominated the intermediate stage of discharge, and S₄²⁻ dianions dominated the late stage of discharge at low applied current density. Our experimental findings and approach are consistent with these results.