Probing Impedance and Microstructure Evolution in LithiumâSulfur

Article pubs.acs.org/JPCC

Probing Impedance and Microstructure Evolution in Lithium−Sulfur Battery Electrodes Chien-Fan Chen,† Aashutosh Mistry,† and Partha P. Mukherjee*,# Department of Mechanical Engineering, Texas A&M University, College Station, Texas 77843, United States ABSTRACT: Lithium−sulfur chemistry, despite being a promising candidate for energy storage due to its higher theoretical capacity, is faced with several critical challenges. Practical operation of Li−S batteries demonstrates lower capacity, poor rate capability, and insufficient cycle life, which can be related to the underlying physicochemical interactions at the electrodes. Typical carbon-based porous cathodes undergo coupled electrochemical, chemical, and microstructural evolution during operation. In this work, the mesoscale interaction resulting from the underlying chemical/electrochemical complexations and microstructural evolution is studied in order to elucidate the transient impedance behavior in Li−S battery mesoporous carbon cathodes. The discharge product (e.g., Li2S) precipitation is shown to affect impedance evolution with correlational dependence on the porous cathode microstructure attributes. This mesoscale impedance analytics can be a valuable virtual probing tool for Li−S battery electrochemical performance.

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INTRODUCTION Lithium−sulfur (Li−S) chemistry based energy storage is a center of active research1−7 owing to the high theoretical specific capacity (e.g., on a gravimetric basis, theoretical capacity is 1675 mAh/g of sulfur), as compared to the stateof-the-art lithium-ion chemistries. However, the practical performance of Li−S batteries is far from the theoretical promise, due to incomplete electrochemical operation8 (theoretical capacity is not achieved even for low rate first discharges), poor rate capability,8,9 and insufficient cycle life.10 Polysulfide shuttle effect was believed to be a source of these challenges as it relates to the chemical reduction of polysulfides at the lithium metal anode. However, accurate quantification of shuttle current11−13 revealed that it cannot, on its own, account for the reduced first discharge capacity. More recent investigations emphasized the requirement for closer probing of precipitation dynamics of solid products (reaction kinetics as well as deposition morphology14−16). In addition, a detailed understanding of the electrolyte transport,17−20 reaction pathway,21−26 and side reactions27 is missing. An important characteristic of a Li−S system is that the cathode continuously evolves during electrochemical operation, given the precipitation/dissolution reactions of solid sulfur (S8) and lithium sulfide (Li2S). This leads to considerable changes, in proportion to initial sulfur loading, for the cathode microstructure. Porosity reduction leads to increased porephase transport resistance (tortuosity increase is positively related to porosity reduction) as well as resistance to reaction kinetics (electrochemically active area decreases with porosity). The chemically active solid phases (S8 and Li2S) are electronically insulating, thus, giving rise to additional interfacial resistance. Such microstructural changes have been discussed28−31 in the literature but not studied in sufficient detail. © 2017 American Chemical Society

Electrochemical impedance spectroscopy (EIS) is often employed to characterize the impedance behavior of systems.32−38 It helps identify different sources of resistance present in the cell and provides many detailed insights compared to internal resistance measurements.39 Since microstructure evolution is directly related to different sources of impedance, appropriate interpretation of EIS response can offer insights into physical changes taking place at Li−S cathode. Despite its usefulness in identification of electrochemical state, EIS has not been used to examine Li−S cells, except a few experimental studies.40−45 The present manuscript proposes a porous-electrode-theorybased impedance model that accounts for different physicochemical events taking place inside the Li−S cell such as microstructure evolution and reaction kinetics. The impedance description is associated with a cell performance model46 to elucidate the interdependency of electrochemical + physical state of the cell and corresponding impedance response. Unlike the circuit-based models usually employed to interpret the impedance measurements, the physics-based formulation, discussed henceforth, attributes each of the observed features on impedance spectra to their pore-scale events.

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IMPEDANCE CHARACTERISTICS IN LITHIUM−SULFUR BATTERIES Electrochemical operation of a Li−S cell involves multiple reactions. Figure 1a schematically outlines different electroactive species present at the cathode. A Li−S cell is prepared with impregnated solid sulfur (S8). Solid sulfur dissolves into Received: July 22, 2017 Revised: September 7, 2017 Published: September 7, 2017 21206

DOI: 10.1021/acs.jpcc.7b07245 J. Phys. Chem. C 2017, 121, 21206−21216

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The Journal of Physical Chemistry C

Figure 1. Overview of different physicochemical processes taking place in a Li−S cathode that affects impedance behavior: (a) pore-scale reactions, (b) state of precipitate film and formation of double layer, (c) imperfect contact between the precipitate film and carbon substrate, and (d) electrode microstructural length scale. (e) Interfacial impedance contributions are summarized in terms of an equivalent circuit. (f) Schematic illustration of an impedance profile, (g) along with validation against experimental data.41 charge

the electrolyte on the basis of solubility. To begin the electrochemical operation, this cell is first discharged. At the onset of discharge, this dissolved sulfur (S8(l)) undergoes electrochemical reduction at the surface of carbon and progressively (electrochemically) transforms into polysulfide anions (Sx2−, x = 8, 6, 4, 2, 1). Eventually, the lower order sulfides combine with Li+ ions coming from anode and precipitate as Li2S. Concurrently, the electrode microstructure undergoes changes in porosity and equivalently in the pore network due to quantitative changes in solid species. The corresponding electrochemical and physical state of the cathode is estimated on the basis of the mathematical description published earlier.46 Charge-Transfer Resistance. In a porous electrode, charge transport takes place as a flow of ions in electrolyte phase and in terms of electrons in the solid electrode phase. Electrochemical reactions describe charge transfer across electrode−electrolyte interface and thus conversion between electronic and ionic charges. At equilibrium, net charge transfer is zero, and the amounts of ionic and electronic charge stay individually constant. A penalty is to be paid in order to carry out electrochemical reaction in either direction, and manifests as an overpotential. This can be expressed in the form of an interfacial resistance, more commonly known as charge-transfer resistance. For example, consider the electrochemical reduction of S82− into S62−. Corresponding (balanced) reaction equation is

2S62 − HooooooooI discharge

3 2− S8 + e− 2

(1)

Thermodynamic voltage associated with this reaction (open circuit potential) and reaction kinetics (Butler−Volmer relation) are quantified as per the following expressions: U = U0 −

RT log[(CS62−)2 /(CS82−)3/2] F

(2)

⎧ ⎛ Fη ⎞ ⎛ Fη ⎞⎫ ⎟ − exp⎜ − ⎟⎬ i = k0 (CS62−)2 (CS82−)3/2 ⎨exp⎜ ⎝ 2RT ⎠⎭ ⎩ ⎝ 2RT ⎠ (3)

where η = ϕc − ϕe − U is the overpotential required to carry out finite amount of electrochemical reaction in either direction. Note that the definitions of U and i are appropriately re-expressed to obtain a concise mathematical description (derived by suitable rearrangement of terms in previously published models29,46). OCP and reaction current density, i, are to be quantified for each of the reactions at cathode. Expression (3) is linearized about the equilibrium state to obtain charge-transfer resistance. (EIS is carried out when the cell is in electrochemical equilibrium. Experimentally this happens when no current passes through the cell and cell voltage stabilizes. Just after the current is stopped, the reactants rearrange internally to bring the cell to electrochemical 21207


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The Journal of Physical Chemistry C equilibrium, and therefore, there is a finite relaxation time from current interruption until equilibrium.) [linearization] ⇒ δi = k0 (CS62−)2 (CS82−)3/2 ·

F δη RT

The porous nature of electrode and interfacial precipitation gives rise to two distinct forms of electron conduction. The first is long-range electron conduction that results from electron transport at electrode scale and describes the availability of electrons in the electrode. In other words, it relates the ease of electron conduction from current collector to electrode and vice versa. Another form of conductivity, referred to as a shortrange conductivity, quantifies the availability of electrons at the electrochemically active interface to partake in electrochemical reactions. This is correlated to coverage of the carbon− electrolyte interface with insulating solid phase. This pore network also represents a solid−fluid interface and thus gives rise to capacitive behavior due to local electric double layers47 (in nanopores). Mathematically, accompanying capacitance, cf, per unit electrode−electrolyte area is described by the relation:

(4)

Equation 4 describes the perturbation in reaction current (δi), i.e., Faradaic current, in response to perturbation in voltage−overpotential (δη). Corresponding charge-transfer resistance is obtained when this description is transformed from time domain (eq 4) to frequency domain (by either taking a Laplace transform or a Fourier transform): rct =

RT 1 · F k0 (C 2−)2 (C 2−)3/2 S6 S8

(5)

The electrolyte phase concentrations (C’s) also change locally when an EIS is being carried out, but their perturbations create secondary effects on charge-transfer resistance and are usually ignored. Double-Layer Capacitance and Contact Resistance. A solid−fluid interface gives rise to local charge accumulation depending on a joint contribution of surface phenomena, e.g., adsorption (Figure 1b schematically illustrates this charge separation). When a voltage perturbation is applied, it changes the amount of stored charge in the double layer and equivalently leads to current flow. Since this current does not describe the electrochemical reactions but rather the charging (or discharging) of double layer, it is often referred to as nonfaradic current, to distinguish it from reaction (faradic) current. A double layer’s contribution to impedance is characterized by double layer capacitance, cdl. The precipitate−carbon interface does not necessarily represent a perfect contact. Due to surface irregularities, the precipitate phase may not penetrate all the surface cavities. This would lead to entrapped third phase at the precipitate−carbon interface and would appear on the impedance spectra as an additional resistance, rc. Figure 1c schematically illustrates this imperfect contact. Impedance of Discharge Product Precipitate. As outlined in Figure 1a, deposition of solid products (S8 and Li2S), takes place at the carbon-electrolyte interface. Both these solids have very poor electronic conductivity and effectively behave as an insulator. Therefore, with the presence of precipitate phase, charge-transfer reaction can only take place at exposed (available) carbon−electrolyte interface. This would manifest as an additional resistance for the ionic transport in the electrolyte phase. As a first approximation, here it is assumed that the precipitation takes place in the form of a uniform film with nanoporosity (Figure 1b). These pores in the precipitate film are filled with electrolyte and provide pathways for ionic transport to electrochemically active surface. Let ρf be resistivity associated with the precipitate film (i.e., ion transport resistance related to this precipitate phase pore network), then a film of thickness δf introduces a resistance rf (per unit interfacial area) quantified as rf = ρf δf

c f = εf /δf

(7)

Here εf is effective permittivity of the precipitate film. In a nutshell, the impedance behavior of a precipitate film is quantified by the following expression: rf zf = 1 + jωrf c f (8) Interfacial Impedance. The electrode−electrolyte interface in Li−S cathodes is quite interesting. The aforementioned different components contributing to this interfacial impedance have to be appropriately combined to quantify the overall interfacial dynamics. Begin with interfacial current density i, which expresses the amount of current exchanged between electrode and electrolyte phases per unit area of electrode− electrolyte interface (Figure 1d). This current has both faradic (reaction current) and nonfaradic (e.g., capacitive) components. Figure 1d explains current distribution and relative configuration of different interfacial impedance components. The current distribution helps identifying appropriate seriesparallel connections. Electrode Impedance. The bulk transport of electrons and ions experience finite resistance. This transport resistance is related to volume-averaged effective properties of the porous electrode, namely, effective electronic conductivity, σeff, and effective ionic conductivity, κeff. The effective electronic conductivity (long-range conductivity with regards to the previous discussion) is related to bulk properties of the solid phases and their relative arrangement in Li−S cathode. On the other hand, the efficiency of ionic conduction correlates to electrolyte phase as well as pore network. Mathematically, κeff is expressed as multiplication of pore network effects (ε/τ) and intrinsic ionic conduction that relates to concentrations of different constituents (eq 9 and 10). ε κ eff = κ (9) τ κ=

∑ zi2F 2 i

Di Ci RT

(10)

Following the available modeling studies, a dilute-solutiontheory-based description is adopted for ionic conduction, and summation is performed over all the present charged species in the electrolyte phase. Since species concentrations evolve during the operation of a Li−S cell, the intrinsic conductivity also changes simultaneously. Moreover, the electrode microstructure evolves in time. Thus, both microstructural (cathode)

(6)

The dimensions of such a film varies from hundreds of nanometers to couple of microns. Since this resistance is correlated to ionic transport in the nanopore network, identical resistivity can be assigned to both sulfur and Li2S deposits (assuming that they form similar nanoporous phases). 21208


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Figure 2. Microstructural characterization of mesoporous carbon structure in terms of effective properties. (a−d) Microstructural growth upon precipitation (here uniform precipitation is assumed; precipitation amount is presented on a volumetric basis), (e−h) Visually depict the corresponding evolution in percolation pathways. Related quantitative microstructural evolution is expressed in terms of (i) tortuosity, (j) pore-size distribution, and (k) pore-phase percolation pathways. (i) Comparison against effective property evolution as predicted from pore-scale simulations and as suggested by assumed relations in literature. (j) Pore-size distribution gradually shift toward smaller mean pore size as precipitation takes place, while maintaining its qualitative nature.

and material (electrolyte) aspects of ionic conduction (eq 9) vary over the electrochemical operation and in turn leads to different impedance buildup. At any cross-section of porous electrode along the direction of current flow, summation of ionic and electronic current remains invariant, and the value is the same as the operating current. In other words, inside a porous electrode, the current changes its form (i.e., identity of charge carriers) going from one location to another. At the cathode−separator interface, current is completely ionic in nature, while at the cathode− current collector interface, it has transformed into electronic current. The interfacial impedance accounts for this barrier to interconversion between charge carriers. Note that the interfacial impedance described earlier is per unit interfacial area. When it is scaled up to a porous electrode length scale (Figure 1e), electrochemically active area appears alongside. Meyers et al.48 developed a model for impedance of a porous electrode. According to this description, the total electrode impedance is expressed as

z=

⎧ σ eff 2 + cosh v eff + ⎪ κ L ⎨1 + eff v cosh v +σ ⎪ ⎩

(

κ eff

⎛ κ eff + σ eff ⎞ a where v = L ⎜ eff eff ⎟ ⎝ κ σ ⎠ z interface

κ eff σ eff

) ⎫⎪⎬ ⎪ ⎭

(11)

A typical impedance response based on formula 11 is sketched on Figure 1f. It has three characteristic regions. Capacitive effects reduce as excitation frequency is increased. The lowest resistance point on the real axis (with almost zero capacitance/imaginary part) corresponds to this highest measurement frequency. The first semicircle (intermediate frequency range) refers to the precipitate phase contributions, while the second semicircle (lowest frequency range) accounts for charge-transfer kinetics. The model predictions are compared with experimental impedance spectra for sulfur cathode on Figure 1g. It shows that the impedance model described here can account for different features observed experimentally. The values of different physical parameters, e.g., electrode thickness, are kept the same as experiments (wherever reported) or assumed to obtain a match. 21209


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MICROSTRUCTURE EVOLUTION DUE TO DISCHARGE PRODUCT PRECIPITATION IN THE CATHODE The evolution of electrode structure upon precipitation is characterized in terms of effective microstructural properties. These simulations are carried out on mesoporous carbon cathodes. The fidelity of such microstructure generation in the context of Li−S cells has been justified earlier via comparison against SEM images.46 To illustrate the process, a mesoporous carbon structure with 60% pristine porosity is studied hereafter (Figure 2a). Note that pristine porosity refers to porosity of the carbon backbone while only accounting for the open pore space. A mesoporous carbon structure has two types of pores: (i) Open pores form an interconnected pore network from one end of the electrode to the other and are saturated with electrolyte. Sulfur species that are present in open pores participate in the cell reactions. (ii) Closed pores (also known as blind voids) are completely surrounded by solid carbon backbone and electrolyte cannot access them, thus rendering them inactive during electrochemical, chemical and microstructural transitions discussed here. At the onset of discharge, the cathode has a specific amount of solid sulfur loading, which makes the initial porosity smaller than the pristine porosity by sulfur loading (expressed as volume of impregnated sulfur per unit volume of the electrode). Figure 2a−d show visual representations of the microstructure at different precipitation volume fractions. The microstructure growth is carried out by depositing a uniform thickness (δf) precipitate layer (homogeneous precipitation). As thickness of precipitate layer is increased, available pore space decreases and efficacy of different transport processes in the cathode also changes. Three microstructural properties appear in cell performance and impedance modeling (which quantify different transport modes): 1. Interfacial area: area available for electrochemical reactions per unit electrode volume 2. Pore-phase tortuosity: refers to pore-phase-transport resistance experienced by electrolyte phase species 3. Effective electronic conductivity: accounts for long-range electron conduction in the electrode solid phase Since carbon has very high bulk conductivity, corresponding long-range conduction is quite efficient and offers negligible impedance. Therefore, the structures are characterized in terms of interfacial area and pore-phase tortuosity only. First consider Figure 2i that plots evolution of interfacial area and tortuosity as a function of precipitation quantity for a mesoporous structure with 10 μm mean pore size and 60% pristine porosity. The solid lines represent trends from pore-scale direct numerical simulations for effective properties, while the dashed lines sketch changes as assumed in the literature (eqs 12 and 13). ⎛ ε ⎞3/2 a = a 0⎜ ⎟ ⎝ ε0 ⎠

(12)

τ = 1/ ε

(13)

In the pore-scale simulations, porosity reduces due to precipitation at the solid−pore interface. This maintains a similar pore network but reduces pore dimensions. Alternatively, eq 12 assumes a new mesoporous structure with smaller porosity. Thus, the area reduction is more drastic in the assumed eq 12. Note that the interfacial area is normalized using the area of the pristine structure while plotting Figure 2i. On the other hand, the tortuosity values are severely under predicted by the often cited Bruggeman relation. This makes sense since the Bruggeman relation is only valid for porous structure with nonoverlapping spherical particles, while the mesoporous structure represents a completely different microstructural arrangement. Such a considerable difference in traditionally assumed relations and actual pore-scale calculations, emphasizes the requirement for accurate microstructural characterization in order to comprehend electrochemical response (i.e., cell performance, impedance, cyclic voltammetry etc.) of Li−S cells. Such qualitatively different behaviors arise due to fundamental structural differences between the mesoporous and assumed microstructure for the traditional relations. Figure 2j reports the evolution of pore-size distribution in cathode upon precipitation. The pristine cathode has a lognormal pore-size distribution centered around 10 μm (here pore size is expressed in terms of pore diameter). As precipitation takes place, the mean pore size decreases monotonically. Simultaneously, the standard deviation also consistently increases as more and more deposition takes place. Note that the pore-size distribution is qualitatively very similar at different precipitation volumes. Tortuosity increase is positively correlated to porosity reduction. For pore-phase transport inside a porous medium, diffusing species have to travel a longer distance and a more curved path depending on pore network arrangement and pore dimensions. Tortuosity quantifies this pore-phase transport resistance. The corresponding paths for species migration are often referred to as percolation pathways. Decrease in number of percolation pathways represents increase in tortuosity. Figures 2e−h graphically show the percolation pathways in x-direction for different porous structures corresponding to Figures 2a−d. The evolution of percolation pathways is quantified on Figure 2k. Here number of percolation pathways for mesoporous carbon cathode with different precipitation amounts is expressed. The number of pathways for the pristine structure are used to normalize the plot k. Percolation pathways also vary monotonically with precipitation volume. Thus, the Li−S cathodes are characterized in terms of effective microstructural properties as a function of precipitation volume. These characterization results are then passed onto the performance and impedance models.

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RESULTS AND DISCUSSION The impedance response of a porous electrode with electronically insulating film (precipitation) is influenced by interfacial area, film thickness, charge-transfer resistance, and ionic conductivity of the electrolyte phase. Figure 3 explores the effect of each of these variables on impedance response in order to obtain qualitative insights into impedance response at different measurement frequencies. The electrodes are with 60% pristine porosity (without film volume). Consider Figure 3a that explores the effect of interfacial area. Higher interfacial area leads to smaller interfacial impedance and in turn overall smaller electrode impedance. This does not change the nature

The comparison presented on Figure 2i shows that the assumed behavior (eqs 12 and 13) is very different from that computed on the basis of accurate microstructure. For example, the computed evolution of interfacial area demonstrates a smaller area reduction upon precipitation compared to eq 12. 21210


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the film impedance stays relatively invariant upon changes in rct. Figure 3d demonstrates the effect of film thickness on impedance response. With higher film thicknesses, the film impedance increases according to eq 8. This manifests as larger size first semicircle (at higher frequencies). Since the second semicircle is dominantly influenced by charge-transfer resistance, and the charge-transfer resistance is identical for the four profiles on Figure 3d, it is fairly uninfluenced by changes in film thickness. Notice that for small values of charge-transfer resistance or large values of film thickness, the second semicircle (charge-transfer signature) gradually merges with the film impedance and it becomes difficult to visually distinguish the two, e.g., Figure 3c response for rct/2 and rct/10. With the impedance spectra understanding developed by studying Figure 3, the impedance evolution in a Li−S cathode can be interpreted. The electrochemical state of Li−S cathode is obtained from performance model at different depths of discharge (with theoretical capacity, 1675 mAh/g of sulfur, meaning 100% depth of discharge). The electrochemical state of the cathode is quantified in terms of: 1. Potentials of electrolyte and electrode phase 2. Concentrations of different polysulfide species 3. Microstructural attributes, i.e., interfacial area, porosity, and tortuosity The cell is discharged at C/20 rate (1675 mA/g corresponds to 1C current). Such a slow rate of discharge is selected in order to study cell’s thermodynamic operation. During the course of the discharge simulation, the electrochemical state (characterized as described before) is recorded, and subsequently, impedance calculations are performed at these different states in order to correlate impedance spectra with electrochemical state. The corresponding results are presented hereafter. Impedance Evolution during Discharge. As described earlier, the discharge of a Li−S cell starts with a cathode containing solid sulfur. As discharge progresses, this sulfur dissolves and undergoes electrochemical reduction to successively lower order polysulfide ions. Subsequently, lithium sulfide (Li2S) precipitates out and lodges in the pores of the cathode. Tracking the porosity evolution of the Li−S cathode, during the early stages of discharge, porosity increases on account of sulfur dissolution and eventually starts to decrease upon Li2S precipitation. On a stoichiometric basis, conversion of S8 to Li2S leads to 80% volume increase. Consequently, the final porosity (upon complete Li2S precipitation) is lower than the initial porosity. These trends are apparent in Figure 4a, where cell voltage evolution and porosity variations are plotted together, for a cathode with 60% pristine porosity and 20% sulfur loading (i.e., 40% initial porosity). The corresponding temporal variations in active (interfacial) area and tortuosity are depicted on Figure 4b, while the electrode microstructures are visualized on Figure 4d−g. Figure 4c plots the evolution of pore-size distribution during cell operation, where mean pore size increases upon sulfur dissolution (the distribution shifts toward right), and upon Li2S precipitation mean pore size again reduces. Note that these variations refer to precipitate phase forming a uniform deposit film. Following the variation in electrochemical state of Li−S cathode during discharge, different forms of resistance (i.e., charge-transfer, rct, and film resistance, rf) buildups are computed. Given the complexity of reaction mechanism on the cathode side, computation of charge-transfer resistance

Figure 3. Effect of different physical variables on impedance response for a cathode with 60% pristine porosity: (a) interfacial area, (b) ionic conductivity, (c) charge-transfer resistance, and (d) precipitation film thickness.

of impedance profile (i.e., features on the plot), but only leads to quantitative variations. Complex impedance points are also identified on the Argand plane at two different frequencies: 150 Hz, a small frequency with dominant charge-transfer effects, and 400 kHz with, a high frequency with major contribution from film impedance. Comparison of impedance values at similar frequencies also assist a quantitative understanding of changes in interfacial area. Also note that the contact resistance stemming from imperfect contact at carbon−precipitate interface as well as cell components accounts for nonzero resistance at the highest frequency point. Figure 3b explores the effect of ionic conductivity. Higher ionic conductivity makes the electrolyte phase transport more efficient and gives rise to smaller impedance. This monotonic dependence can be clearly identified on Figure 3b. There is an interesting trend with respect to conductivity variations. Beyond a certain value of conductivity, i.e., once the ionic conduction is sufficiently efficient compared to other impedance modes such as charge transfer, further increase in conductivity would have little effect on impedance spectra. Figure 3b demonstrates such a converging response. Increasing ionic conductivity from 5.0 × 10−3 S/m to 7.5 × 10−3 S/m to 10−2 S/m brings gradually smaller improvements to the impedance profile. Similar attributes are also observed on Figure 3a, where increasing active area leads to smaller interfacial resistance, but the profile does not become insensitive as observed in Figure 3b. Figure 3c studies the effect of charge-transfer resistance on impedance profile. Note that the second semicircle at lower frequency range corresponds to charge-transfer contribution. The radius of this semicircle is positively correlated to chargetransfer resistance. Therefore, increasing charge-transfer resistance leads to higher impedance and larger size second semicircle. Notice that the first semicircle that characterizes 21211


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Figure 4. Study of impedance evolution for a mesoporous carbon cathode with 60% pristine porosity, 20% sulfur loading, and 10 μm mean pore size: (a) cell voltage and porosity variations upon discharge, (b) evolution of interfacial area and pore-phase tortuosity, (c) temporal changes in cathode pore network (presented as changes in pore-size distribution). (d−g) Different microstructural states, (h) changes in charge-transfer resistance and film resistance during discharge operation, (i) impedance evolution during sulfur dissolution−early discharge stages, and (j) impedance evolution during Li2S precipitation.

requires some elaboration. As sketched on Figure 1a, electrochemical reduction of sulfur, S8(l) to S2− is assumed to take place in five steps.29,46 Each of these reactions are dominant over different durations of discharge and their individual charge-transfer resistances evolve over the course of electrochemical operation in response to changes in respective ionic concentrations. As summation of individual reaction currents constitute the total faradic current, total chargetransfer resistance is expressed in terms of individual reaction charge-transfer resistances as follows: 1 1 1 1 1 1 = + + + + rct rct,1 rct,2 rct,3 rct,4 rct,5 reaction 1: reaction 2: reaction 3: reaction 4: reaction 5:

1 2− 1 S8 ⇌ S8(l) + e− 2 2 3 2S62 − ⇌ S82 − + e− 2 3 2− S4 ⇌ S62 − + e− 2 1 S22 − ⇌ S42 − + e− 2 1 S2 − ⇌ S22 − + e− 2

The reaction subscripts on charge-transfer resistances in eq 14 correspond to reaction scheme numbered in eq 15. As explained in eq 5, the charge-transfer resistance is inversely dependent on reactant concentrations (raised to appropriate powers). Therefore, if a particular reaction is not active on account of not having enough reactants, its charge-transfer resistance becomes very large and equivalently its contribution to eq 14 is negligible, and it does not appear on total chargetransfer resistance. An interesting situation arises for reaction 5 that accounts for electrochemical reduction of S22− to S2−. Due to very low solubility of Li2S in organic electrolyte, the concentration of S2− ions is always considerably less, which in turn makes the corresponding charge-transfer resistance very large. Note that since all the reactions act in parallel, the overall resistance is actually smaller than the smallest of five resistances, and any one of them becoming very large (in the event of small reactant concentrations) does not increase overall resistance drastically. Putting it in other words, when multiple reactions are simultaneously active, reaction with lower charge-transfer resistance contributes more current (since current flow prefers the lowest resistance path). Figure 4h plots the variations in total charge-transfer resistance and film resistance during the course of discharge. At the onset of discharge, an insulating film of solid sulfur is present, porosity is smaller than the pristine porosity, corresponding interfacial area is smaller, and tortuosity is larger (compared to the pristine structure). Additionally, reactant concentrations are very low since only dissolved sulfur is

(14)

(15) 21212


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Figure 5. Effect of microstructural modifications on impedance response−study of mean pore size: (a) different electrode microstructural states for three different cathodes with 5, 10, and 15 μm mean pore size, (b) interfacial area of the pristine electrode as a function of mean pore size, (c) comparison of charge-transfer resistance evolution, (d) comparison of film resistances for the three different electrodes, (e) electrode impedance during early discharge stages (1% DOD), and (f) electrode impedance toward the end of discharge (95% DOD).

discharge should be related to increased cell impedance (Figures 4h,j). Influence of Mean Pore Size on Impedance. In addition to being a conversion chemistry, Li−S cells differ from traditional Li-ion given the presence of substantial microstructural changes (even during a single discharge or charge operation). Higher values of impedance correlate to more losses in the cell. Figure 5 investigates the relative importance of microstructural modifications. Specifically, three different cathodes are investigated to assess their impedance response. Each of these have the same pristine porosity (60%) and identical initial sulfur loading (20%), but different pore-size distribution. The (pristine) microstructures have been generated with log-normal distributions having different mean pore sizes (same deviation) 5, 10, and 15 μm. As mentioned earlier, the sulfur loading is carried out in the form of a uniform film at the carbon−void interface. The film thickness is gradually increased until it meets the prescribed (volumetric) sulfur loading. Given different pore dimensions, corresponding interfacial area are different, which in turn leads to different film thicknesses for the same amount of sulfur loading. Specifically, smaller mean pore size has smaller sulfur film thickness as it has higher interfacial area. Figure 5a shows corresponding cathode microstructures at three different depths of discharge. The selected DOD values (1%, 20% and 95%) are such that they highlight three qualitatively distinct microstructural states: 1% DOD correlates to cathode with present solid sulfur, 20% DOD

present. All these combined together lead to considerable charge-transfer resistance as well as film impedance, and consecutively both the semicircles are present on the impedance plot Figure 4i. As the discharge progresses, various transport processes become more efficient on account of increased porosity and higher reactant concentrations. Jointly, these lead to reduction in impedance values. Moreover, the film impedance semicircle also slowly disappears. At about 20% depth of discharge, which corresponds to a point between the first and the second discharge plateau, the cathode has recovered its pristine state due to complete dissolution of the solid sulfur, and the cathode impedance becomes minimum with only the charge-transfer semicircle left (no solid film present). Thus, both total charge-transfer resistance, rct, and film resistance, rf, become smaller around 20% state of charge, given the favorable electrochemical state of the Li−S cathode. Later on, as Li2S precipitation takes place, ionic concentration in the electrolyte phase starts declining as S2− starts to form Li2S. The cathode porosity also simultaneously decreases due to precipitation, which in turn leads to reduced active area, increased tortuosity and higher film thickness. Since, Li2S occupies more space compared to stoichiometrically equivalent sulfur, corresponding deposits have greater thickness and subsequently higher film impedance at the end of discharge compared to the start of discharge (Figures 4i,j). These observations also meet the physical intuition that the end of 21213


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microstructural effects on impedance, EIS diagnosis is carried out on three cathodes with same pristine porosity (60%) and sulfur loading (20% by vol) but different pore-size distributions (mean diameter 5, 10, and 15 μm). The electrode with smaller pores demonstrates smaller impedance values since charge transfer is more efficient (higher active area) as well as precipitate thickness is smaller (for the same precipitate volume). Thus, impedance behavior of a Li−S cell is correlated with electrochemical and microstructural state of the cathode, which in turn allows probing the microstructural state from the observed impedance characteristics.

is pristine cathode, and 95% DOD is cathode with appreciable Li2S precipitated. Analyzing across cathodes with different pore sizes, the number of pores in same electrode volume decreases going from 5 to 15 μm. This gives rise to smaller interfacial area for a cathode with larger pores (Figure 5b). Doubling the pore size (from 5 to 10 μm) reduces the area to (almost) half. To better understand the dependence on pore size, the interfacial area (m2/m3) is cast into a nondimensional form by multiplying with mean pore size (diameter). The corresponding trend is also presented on Figure 5b, superimposed on top of the interfacial area bar chart. The dimensionless area, a·Dmean increases with mean diameter, suggesting that the interfacial area scales with mean pore diameter as a ∝ D−p mean, with exponent p ∈ (0,1). Plots (c) and (d) in Figure 5 represent the changes in chargetransfer resistance and film thickness resistance, respectively, during discharge for these different cathode structures. The cathode with the smallest pore size (5 μm) has the smallest resistance throughout discharge operation. As discussed in Figure 5b, a smaller-pore-size electrode has a higher interfacial area, and equivalently for the same precipitation volume, leads to smaller precipitate thicknesses. This explains the trends in film resistance as it scales linearly with thickness of deposits (Figure 5d). The area-specific charge-transfer resistance (plot c) depends on concentrations. For an electrode with higher interfacial area, overpotential is smaller, and equivalently, the concentration evolution is more favorable, leading to smaller charge-transfer resistance (smaller concentration overpotential). Both charge-transfer resistance and film impedance trends jointly influence the overall impedance behavior (Figures 5e,f). Smaller pore size has higher active area, smaller precipitation thickness and smaller impedance values. The smallest-pore-size cathode (5 μm) has consistently smaller impedance at different depths of discharge. Electrode impedances are compared for two cathode microstructure states: Figure 5e 1% DOD, which represents cathode with present sulfur and Figure 5f 95% DOD with Li2S as the solid phase. As precipitate thickness toward the end of discharge (Figure 5f) is higher than that during the start of discharge (Figure 5e), the film impedance semicircle is larger on Figure 5f.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Partha P. Mukherjee: 0000-0001-7900-7261 Present Address #

(A.M. and P.P.M.) School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, U.S.A. Author Contributions †

(C.-F.C. and A.M.) These authors contributed equally to this work. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The information, data, or work presented herein was funded in part by the Office of Energy Efficiency and Renewable Energy (EERE), U.S. Department of Energy, under Award DEEE0006832.

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NOMENCLATURE symbol description value (units) Li−S Lithium sulfur EIS Electrochemical Impedance Spectroscopy OCP Open-circuit potential (V) U Open circuit potential (U0 refers to corresponding value at 1 mol/m3 reactant concentrations) (V) R Universal gas constant (J/mol·K) T Temperature (K) F Faraday’s constant 96487 (C/mol) C Reactant concentration (mol/m3) i Electrochemical reaction current density (A/m2) 0 k Reaction rate constant (A/m2/(mol/m3)7/4) rct Charge-transfer resistance (Ωm2) cdl Double layer capacitance (F/m2) rc Contact resistance (Ωm2) ρf Resistivity of precipitate film 7 × 104 (Ωm)a δf Thickness of precipitate film (m) rf Film resistance (Ωm2) cf Film capacitance (F/m2) εf Permitting of precipitate film 31 × 10−11 (F/m)a zf Film impedance (Ωm2) ω Angular frequency (rad·s−1) σeff Effective electronic conductivity (S/m) Effective ionic conductivity (S/m) κeff Di Species diffusivity (m2/s) zinterface Interfacial impedance(Ωm2) z Total impedance of a porous Li−S cathode (Ωm2) L Cathode thickness 60 (μm) SEM Scanning Electron Microscopy

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CONCLUSIONS Electrochemical impedance spectroscopy is a useful diagnostic tool to probe different forms of electrode impedances. The present work proposes a physics-based model to understand impedance response of a Li−S cathode. This model is used to study impedance evolution during the course of discharge operation. The Li−S cathode undergoes substantial microstructural evolution during operation as solid sulfur first dissolves (earlier stages of discharge) and later precipitates as Li2S. This leads to distinct electrode transport characteristics at different depths of discharge and equivalently different electrode impedance. The impedance profile for a Li−S cathode consists of two semicircles representing different forms of impedance behavior. The lower frequency (right most) semicircle quantifies the charge-transfer resistance, while the higher frequency semicircle characterizes impedance of the precipitate film. As cathode operation has multiple electrochemical reactions taking place, the charge-transfer resistance also evolves during operation, with it being highest at the onset of discharge and toward end of discharge (as the reactant concentrations become very small). To further explore the 21214


Article

The Journal of Physical Chemistry C a τ DOD

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Interfacial area(m2/m3) Tortuosity(m/m) Depth of discharge (= discharge time/theoretical discharge time) aassumed values

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Probing Impedance and Microstructure Evolution in LithiumâSulfur

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