Multistep Reaction Mechanisms in Nonaqueous LithiumâOxygen

Article pubs.acs.org/JPCC

Multistep Reaction Mechanisms in Nonaqueous Lithium−Oxygen Batteries with Redox Mediator: A Model-Based Study Daniel Grübl,† Benjamin Bergner,‡,§ Daniel Schröder,‡ Jürgen Janek,‡ and Wolfgang G. Bessler*,† †

Institute of Energy Systems Technology (INES), Offenburg University of Applied Sciences, 77652 Offenburg, Germany Physikalisch-Chemisches Institut, Justus-Liebig-Universität Giessen, 35392 Giessen, Germany

‡

ABSTRACT: Lithium−oxygen cells with nonaqueous electrolyte show high overpotentials during charge, indicating asymmetric charge/discharge reaction mechanisms. We present a kinetic modeling and simulation study of the lithium−oxygen cell cycling behavior. The model includes a multistep reaction mechanism of the cell reaction (2Li + O2 ⇄ Li2O2) forming lithium peroxide by precipitation, coupled to a 1D porous-electrode transport model. We apply the model to study the asymmetric discharge/charge characteristics and analyze the influence of a redox mediator dissolved homogeneously in the liquid electrolyte. Model predictions are compared to experimental galvanostatic cycling data of cells without and with 2,2,6,6-tetramethylpiperidinyloxyl (TEMPO) as redox mediator. The predicted discharge behavior shows good agreement with the experimental results. A spatiotemporal analysis of species concentrations reveals inhomogeneous distributions of dissolved oxygen and reaction products within the cathode during discharge. The experimentally observed charge overpotentials as well as their reduction by using a redox mediator can be qualitatively reproduced with a partially irreversible reaction mechanism. However, the proposed models fail to reproduce the particular shape of the experimental charge curve with continuously increasing charge overpotential, which implies that part of the reaction mechanism is still open for investigation in future work. electrode surface (Figure 1c),16 the core−shell structure of lithium peroxide with a potential superoxide-like surface (Figure 1d),17−20 the presence of a nonstoichiometric phase Li2‑xO2 and its topotactic delithiation (Figure 1e),21−23 or the morphology of the discharge product (Figure 1f).24,25 Furthermore, the increasing charge overpotential might indicate that different reactions take place in parallel. To reduce the charge overpotentials in Li/O2 batteries, the addition of a soluble redox mediator to the liquid electrolyte has been proposed. A redox mediator can be seen as an electron (hole) transporter within the electrolyte, shuttling electrons between two charge-transfer reaction zones, thus also connecting electrically insulating discharge products with the cathode. Different redox mediators have been successfully applied, such as tetrathiafulvalene (TTF),26 lithium iodide (LiI),27 lithium bromide (LiBr),28,29 iron phthalocyanine (FePc),30 and nitroxides31 like 2,2,6,6-tetramethylpiperidinyloxyl (TEMPO).12 Recent modeling studies on Li/O2 cells focus mainly on the discharge mechanism, for example, the rate-dependent formation of the Li2O2,32 precipitation mechanisms,33 influence of selected parameters on the cathode design,34 development of cell design concepts,35 influence of cathode microstructure and pore size distribution on discharge capacity and power,36

1. INTRODUCTION Lithium−oxygen (Li/O2) batteries are highly interesting due to their high theoretical specific energy (3458 Wh/kg including only the active materials Li and O2, and assuming lithium peroxide (Li2O2) as primary discharge product1). However, state of the art laboratory cells suffer from poor cycle stability, low power density, and especially poor reversibility, implying asymmetric discharge/ charge overpotentials and side reactions.1−3 Even after intense research was conducted in the past, many open questions remain, including the influence of electrolyte properties such as LiO2 solubility in lithium electrolytes,4 stability,5−7 water content8 or role of suitable additives,9 stability of the cathode material (mostly carbon based),10 slow oxygen reaction kinetics, and, especially, the oxidation of Li2O2 during charge leading to asymmetric discharge/ charge curves.11 The charge overpotential is not only higher than the discharge overpotential but also continuously rises upon progressing state of charge and/or shows more than the one expected plateau region.1,12 This results in a low round-trip efficiency and can increase the rate of potential-driven side reactions (e.g., electrolyte decomposition).13 The origin of the high and/or increasing charge overpotential has not yet been understood. In several studies different reaction mechanisms have been proposed that may explain the observed charge overpotential, as schematically shown in Figure 1. This includes, among others, the particle size distribution (Figure 1a),14,15 the high electric resistance of lithium peroxide (Figure 1b),1 the formation of an interfacial lithium carbonate layer on the © 2016 American Chemical Society

Received: August 4, 2016 Revised: September 20, 2016 Published: September 26, 2016 24623

DOI: 10.1021/acs.jpcc.6b07886 J. Phys. Chem. C 2016, 120, 24623−24636

Article

The Journal of Physical Chemistry C

The computational domain consists of four layers, (1) a closed gas reservoir filled with oxygen, (2) a flooded porous carbon cathode, (3) a glass-fiber separator, and (4) a metal anode. Each layer is assumed to consist of several (solid, liquid, or gaseous) bulk phases that are described in terms of their respective volume fraction using a homogenization approach (porous electrode theory).46 The model is parametrized according to experimental cells from Bergner et al.,12 as described in section 3. A 1 M solution of LiTFSI in diglyme is used as electrolyte, which also contains other species (O2, O2−, TEMPO, TEMPO+). The model geometry and transport parameters are summarized in Table 1, and the compositions of each layer and phase are given in Table 2. Table 1. Geometry and Transport Parameters parameter

Figure 1. Possible reasons for high charge overpotentials (a−f)45 and the investigated approach in this work (g,h).

analytical prediction of the discharge voltage characteristics,37 or a reaction pathway analysis of surface and solution-mediated reaction.38 The charge mechanism has, to the best of our knowledge, only been the subject of very few modeling and simulation studies39,40 lacking detailed investigations on the reaction mechanism. In this work, we present a dynamic one-dimensional model with detailed electrochemical kinetics, including a solutionmediated multistep discharge mechanism and a redox-mediated multistep charge mechanism. First results of this approach have already been presented.41 The herein presented investigations focus particularly on the irreversibility of the mechanisms. We assume a film-like growth4,32 (constant thickness) of reaction products on the electrode surface during discharge (Figure 1g) and electrostripping during charge42−44 (Figure 1h). The modeling and simulation framework is presented in section 2. Discharge and charge experiments as well as cyclic voltammetry are introduced in section 3. Electrochemical reaction mechanisms are derived in section 4, including TEMPO as redox mediator. Simulation results are shown and discussed in section 5, including a spatiotemporal analysis of state variables.

value

ref

volume-to-area ratio of gas reservoira thickness of cathode thickness of separator

6 cm3/cm2

measured12

50 μm 260 μm

thickness of anode tortuosity of cathode τCA tortuosity of separator τSEP diffusion coefficient Diglyme diffusion coefficient O2(solv) diffusion coefficient O2−(solv) diffusion coefficient Li+(solv) diffusion coefficient TFSI−(solv) diffusion coefficient TEMPO(solv) diffusion coefficient TEMPO+(solv)

1 μm 1.5 1.2 2 × 10−10 m2/s 4.4 × 10−9 m2/s

measured12 measured12 including assumed compression (factor 2) assumed assumed assumed measured52 measured12

1.4 × 10−10 m2/s

assumed

1 × 10−10 m2/s

measured52

1.1 × 10−10 m2/s

measured52

4 × 10−10 m2/s

measured12

4 × 10−10 m2/s

measured12

a

The volume-to-area ratio of the gas reservoir corresponds to an absolute reservoir volume of 4.7 cm3 and a cell surface area of 0.785 cm2.

2.2. Model Assumptions. The modeling framework and simulation methodology have been presented previously,46−48 but for the sake of completeness model equations are summarized in the Appendix. The model is based on the following assumptions: • Transport of species within the multicomponent electrolyte is modeled in one dimension y through the thickness of the electrode pair assuming dilute solution theory (diffusion and migration of noninteracting species). • Convective mass transport of the electrolyte is neglected. In order to ensure mass conservation, decreasing porosity due to precipitation of solid products is assumed to lead to a compression of the liquid electrolyte.46 • The (electro-)chemical reaction kinetics are described based on transition state theory and rate equations. The reactions are assumed to take place at the interfaces between the various solid, liquid, and gaseous phases inside the cell. The solid products (LiO2 and Li2O2) are only formed within the cathode layer. Additional side reactions (e.g., lithium carbonate formation, redox-shuttle reaction, and direct reduction of O2(solv) at the lithium anode) are neglected. • The electrical resistance of the reaction products is neglected; double layer capacitances are neglected; and the model assumes isothermal conditions.

2. MODELING AND SIMULATION APPROACH 2.1. Modeling Domain. We use a one-dimensional multiphysics continuum model shown schematically in Figure 2.

Figure 2. Computational domain of the one-dimensional continuum model (not to scale, see geometric data in Table 1). 24624


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The Journal of Physical Chemistry C Table 2. Properties and Composition of the Modeled Layers (cf. Figure 2) density ρ (kg/m3)

phase

gas reservoir cathode

gas carbon electrolyte

1 0.75412 0.24612

1.353 226054 943.453

LiO2 Li2O2 separator electrolyte lithium electrolyte

1 × 10−10 1 × 10−10 0.46b 0.54b 0.9a 0.1a

2040a 231053

separator anode a

initial volume fraction ε

layer

species [initial mole fraction] O2(gas) (1) Carbon (1) diglyme (0.840), Li+(solv) (0.172), TFSI−(solv) (0.172), O2(solv) (1.119 × 10−3), O2−(solv) (1 × 10−15), TEMPO(solv) (1.722 × 10−3), TEMPO+(solv) (8 × 10−10) LiO2(bulk) (1) Li2O2(bulk) (1) glass microfiber (1) see cathode layer Li(bulk) (1) see cathode layer

943.453 53453 943.453

Assumed. bCalculated from ref 12 for compressed separator (compression factor 2).

Table 3. Interfacial Chemical Reactions and Rate Coefficients (cf. Appendix)a interface

volume-specific surface area AV or three-phase boundary length LVTPB

reaction number

2

reaction

pre-exponential factor k0f

gas/electrolyte

3.09 × 109

m m3

R1

O2(gas) ⇄ O2(solv)

5 × 109 m/sb

carbon/electrolyte

1.54 × 108

εLiO2 εLi O m2 − − 2 2 d d m3

R2 R6

O2(solv) + e− ⇄ O2−(solv) TEMPO+(solv) + e− ⇄ TEMPO(solv)

4 × 10−15 m/sc 9 × 10−11 m/sd

R3

Li+(solv) + O2−(solv) ⇄ LiO2(bulk)

1.1 × 10−7 (1.1 × 10−3e) m4/(mol·s)c

R4

2LiO2(bulk) ⇄ Li2O2(bulk) + O2(solv)

5 × 10−17 (1.2 × 10−11e) m4/(mol·s)c

R5

Li2O2(bulk) ⇄ 2Li+(solv) + O2(solv) + 2e−

3.9 × 10−46 m4/(mol·s)c

R7

Li2O2(bulk) + 2TEMPO+(solv) ⇄ 2Li+(solv) + O2(solv) + 2TEMPO(solv)

1.3 × 10−13 m7/(mol2·s)c

R0

Li(bulk) ⇄ Li+(solv) + e−

1 × 1020 m/sb

LiO2/electrolyte LiO2/Li2O2 carbon/Li2O2/ electrolyte TPB Li2O2/electrolyte Li/electrolyte

εLiO2 d εLiO2 d

4· εLi2O2NV /d εLi2O2 d

1 × 106

m2 m3

a Charge-transfer symmetry factors αf are set to 0.5, and activation energies Eact,f are neglected. bSet fast. cFitted to experimental discharge/charge curves.12 dFitted to experimental cyclic voltammetry of redox-mediator reaction.12 eReversible reaction mechanism (cf. section 5.1).

TEMPO were applied as electrolytes. Oxygen was supplied from a closed gas tank atop of the cathode. Further information on the setup, the battery components, and the processing of the materials can be found in Bergner et al.12,31

The (electro-)chemical reaction mechanisms and corresponding thermodynamic and kinetic data will be presented in section 4. 2.3. Simulation Methodology. Finite volume discretization is used within the layers using a resolution of one grid point in the gas reservoir, 18 in the cathode (various thicknesses), 11 in the separator (various thicknesses), and one in the anode. Multiphase management and transport are solved in the in-house code DENIS.46 The multistep electrochemistry is evaluated using Cantera.49 For all bulk phases we use Cantera’s ConstDensityThermo (“incompressible_solid”) model, except for the gas phase for which we use the IdealGas model. For all interfacial reactions the InterfaceKinetics model is used. The numerical solution is carried out using LIMEX.50,51

4. REACTION MECHANISMS 4.1. Overview. In this study we investigate and compare three different multistep reaction mechanisms at the cathode: (A) fully reversible discharge/charge reaction mechanism, (B) partially irreversible reaction mechanism, and (C) partially irreversible reaction mechanism with redox mediator. These three main mechanisms are broken down in a total of up to eight elementary-step reactions, including electrochemical, thermochemical, and phase-change reactions. All reactions are assumed to take place at interfaces between the various solid, liquid, and gaseous phases inside the cell. The reactions and their interfaces are summarized in Table 3. This table also includes the specific interfacial areas that are generally assumed to depend on the volume fractions of the adjacent phases, as derived in the Appendix, thereby representing morphology changes, as discussed in the following subsections. 4.2. Lithium Metal Electrode. At the negative electrode, the identical lithium oxidation reaction is used for all mechanisms: Metallic lithium reacts to form lithium ions and electrons at the lithium/electrolyte interface

3. EXPERIMENTAL SECTION Cyclic voltammetry measurements of the TEMPO/TEMPO+ redox couple were conducted in a three-electrode setup using a glassy carbon working electrode, a lithium reference electrode, a platinum counter electrode, and 1 M LiTFSI/diglyme with 10 mM TEMPO as electrolyte. Galvanostatic cycling was performed in Swagelok type cells comprising a lithium anode, a porous separator (Whatman, GF/A) soaked with liquid electrolyte (60 μL), a lithium reference electrode, and a porous carbon cathode (87.5% Ketjen Black and 12.5% PTFE). Either pure 1 M LiTFSI/diglyme or 1 M LiTFSI/diglyme with 10 mM 24625


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The Journal of Physical Chemistry C Li(bulk) ⇄ Li+(solv) + e−

This reaction is implemented as reversible reaction (reaction R4a) for the fully reversible reaction mechanism “A” (cf. section 4.1)

(Li/electrolyte interface) (R0)

The kinetics for this reaction are assumed to occur very fast; therefore, this electrode represents the reference electrode of the experimental setup. 4.3. Discharge Reaction Mechanism (Oxygen Reduction Reaction). The oxygen reduction reaction can be formulated globally as

2LiO2 (bulk) ⇄ Li 2O2 (bulk) + O2 (solv) (LiO2 /Li 2O2 interface)

(R4a)

and as irreversible reaction (equation R4b) in the partially irreversible reaction mechanisms “B” and “C” (cf. section 4.1)

2Li+(solv) + 2e− + O2 (gas) ⇄ Li 2O2 (bulk)

2LiO2 (bulk) → Li 2O2 (bulk) + O2 (solv)

Here we investigate a multistep reaction mechanism as schematically shown in Figure 3a. During discharge, gaseous oxygen

(LiO2 /Li 2O2 interface)

(R4b)

Alternatively to this disproportionation step, a further reduction of LiO2 on the electrode surface has been proposed56,57 according to LiO2(bulk) + Li+(solv) + e− ⇆ Li2O2(bulk), especially at high overpotentials.58,59 Furthermore, it has been reported that both reactions might take place in parallel.4 However, in this study, chemical disproportionation is assumed as dominant reaction step. As will be shown below, this mechanism provides a good reproduction of experimentally observed discharge behavior; therefore, the second reduction step was not further studied. We assume that the solid products, LiO2 and Li2O2, form a film4,32 with a constant thickness d on the carbon surface (cf. Figure 1g), thus continuously reducing the available free carbon surface. This assumption results in a volume-specific carbon surface area AVcarbon according to the expression ⎛ εLi 2O2 +εLiO2 ⎞ V ⎟ Acarbon = A 0V ⎜1 − dA 0V ⎠ ⎝

where, AV0 represents the initial volume-specific surface area of the carbon electrode and εLiO2 and εLiO2 the volume fractions of lithium superoxide and peroxide, respectively. The reduced surface area slows down and eventually deactivates the chargetransfer reaction (equation R2), leading to the end of discharge. In the present study, the film thickness d is adjusted to match the experimental end of discharge, which can be interpreted as change in film morphology. The assumed film morphology4,32 is alternative to particle morphologies that have also been observed.4,9,18,60 The transition between film and particle morphologies was described before as a function of local current density.32 Note that a film can also be interpreted as continuous coating by small particles with constant thickness (cf. Appendix). As the assumed filmgrowth mechanism provides a good reproduction of experimentally observed discharge behavior (cf. below), we have not explicitly studied particle-growth mechanisms. 4.4. Charge Reaction Mechanism (Oxygen Evolution Reaction). For the fully reversible mechanism “A” (cf. section 4.1), reactions R1−R4 are assumed reversible, thus enabling the complete dissolution of all lithium oxides upon charge without having to assume additional reactions. Results of this mechanism will be presented and discussed in section 5.1. The charge process in the partially irreversible reaction mechanisms “B” and “C” (cf. section 4.1) is shown in Figure 3b. Results of these reaction mechanisms will be presented and discussed in sections 5.2 and 5.3, respectively. Here, an additional reaction for the dissolution of lithium peroxide needs to be added due to its irreversible formation. We assume an electrostripping reaction of lithium peroxide (Figure 1h) as proposed before42−44

Figure 3. Schematic of investigated reaction mechanisms at the Li/O2 positive electrode: (a) Multistep discharge reaction mechanism (reactions R1−R4). (b) Charge reaction mechanism: electrostripping of Li2O2 (reaction R5) and redox-mediated dissolution of Li2O2 (reactions R6 and R7).

O2(gas) dissolves at the interface between the closed gas reservoir and the cathode O2 (gas) ⇄ O2 (solv)

(gas/electrolyte interface)

(R1)

Next, a charge-transfer reaction forming soluble superoxide ions O2−(solv) takes place at the carbon surface O2 (solv) + e− ⇄ O2−(solv) (carbon/electrolyte interface)

(R2)

As soon as the solubility limit of lithium superoxide LiO2 within the electrolyte is reached, it starts to precipitate Li+(solv) + O2−(solv) ⇄ LiO2 (bulk) (electrolyte/LiO2 interface)

(R3)

The presence of solid LiO2 on the electrode surface has been proposed before42−44 and observed in ex situ measurements;20 however, it has also been interpreted as adsorbed surface species LiO2*.8 As the last step of the global reaction, we assume the following chemical disproportionation of LiO2 forming Li2O2.42,44,55 24626


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The Journal of Physical Chemistry C Table 4. Thermodynamic Properties of Species Present at 298.15 K species

molar enthalpy (kJ/mol)

molar entropy (J/(mol·K))

ref

O2(gas) O2(solv) O2−(solv) Li+(solv) LiO2(bulk) Li2O2(bulk) Li(bulk) TEMPO(solv) TEMPO+(solv)

1.63 × 10−5 −43.9 −22.8 −278 −340 −649 0 −79.6 0

205 0 0 13.4 0 0 29.1 0 0

NIST thermochemical database63 O2 solubility measurements12 fitted to a solubility of LiO2 of 40 μmol/L4 Atkins54 fitted to experimental OCV12 Bender et al.61 Atkins54 experimental TEMPO/TEMPO+ redox potential12 reference value

The kinetics of the redox mediator reaction (reaction R6) were parametrized by comparing simulated and experimental cyclic voltammetry of the TEMPO/TEMPO+ redox couple with a scan rate of 50 mV/s between 2.0 and 4.55 V. Results are shown in Figure 4. They demonstrate good agreement between

Li 2O2 (bulk) ⇄ Li+(solv) + O2 (solv) + 2e− (Li 2O2 /carbon/electrolyte TPB)

(R5)

Reaction R5 takes place at the three-phase boundary (TPB) between lithium peroxide, carbon surface and electrolyte. The length of the TPB is assumed to depend on the volume fraction of Li2O2. As result, the reaction rate is decreasing upon charging because the volume fraction of Li2O2 decreases. Note that the intermediate LiO2 formed during discharge can be dissolved via the reversible reactions R1−R3. In addition, a redox-mediated charge mechanism (“C”) is investigated. Two additional redox reactions for the TEMPO+/ TEMPO redox couple are assumed, one on the carbon surface and one on the Li2O2 surface. Both are also depicted in Figure 3b. The redox reaction on the carbon surface in the cathode is given by TEMPO+(solv) + e− ⇄ TEMPO(solv) (carbon/electrolyte interface)

(R6)

Figure 4. Cyclic voltammogram of dissolved redox mediator (TEMPO): simulation and experimental data12 (voltage, 2−4.55 V; scan rate, 50 mV/s; initial concentration of TEMPO, 10 mM).

The second reaction is the oxidative dissolution of Li2O2 by TEMPO+(solv) according to

simulated cyclic voltammetry and experiments, while minor deviations could be explained by the nonideality of the glassy carbon electrode in the experiment. Reactions R0 and R1 were set fast and not rate-determining. As isothermal conditions were assumed, all activation energies were set to zero. Resulting kinetic data is listed in Table 3. We assume a saturated initial concentration of O2(solv) (6.5 mmol/L12 at a gas reservoir pressure of 101 325 Pa for the investigated electrolyte). The solubility of LiO2 has been shown to play a critical role influencing the discharge reaction mechanism.4 It depends strongly on the donor number of the solvent used in the electrolyte. Lacking data for diglyme, we use the proposed solubility of 40 μmol/L in monoglyme.4

Li 2O2 (bulk) + 2TEMPO+(solv) ⇄ 2Li+(solv) + O2 (solv) + 2TEMPO(solv)

(Li 2O2 /electrolyte interface) (R7)

which takes place on the complete surface of Li2O2, in contrast to the electrostripping reaction of Li2O2 (reaction R5) at the three-phase boundary. 4.5. Thermodynamic and Kinetic Parameters. In order to establish a self-consistent kinetic model, a complete set of thermodynamic data of the species is required. Data is given in Table 4. Values were taken from literature where available. Otherwise, reasonable assumptions were made or values from similar species were taken. Molar Gibbs free energies of the lithium oxides determine the predicted open-circuit voltage. Using literature values for lithium superoxide LiO2 of 307.161 or 319.8 kJ/mol,62 we were unable to reproduce the experimentally observed open-circuit voltage. Therefore, this parameter was taken as additional free parameter, for which a value of 340 kJ/mol was observed to show good agreement with experiments. Kinetic data were obtained by adjusting the pre-exponential factors of reactions R2−R5 and R7 such that simulated discharge/charge curves showed best fit to cycling experiments. Only the kinetics of reaction R3 had to be readjusted when switching between the reversible to the partially irreversible reaction mechanism.

5. RESULTS AND DISCUSSION 5.1. Reversible Reaction Mechanism. We start by investigating a fully reversible four-step reaction mechanism (mechanism “A”) (reactions R1−R4 in section 4.3 and Figure 3). Figure 5 shows simulation results for one galvanostatic cycle at a current density of 0.1 mA/cm2. (1 h rest, CC discharge, 2.35 V cutoff voltage, no rest, CC charge, 4.5 V cutoff voltage). This reaction mechanism predicts symmetric charge/discharge overpotentials (Figure 5a). The simulated discharge curve is in good agreement with the experimental data, deviating only at the end of discharge from the experimental curve. The cell voltage is constant over a wide capacity range, as would be expected for a phase-change reaction, before it drops at the end of discharge. 24627


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discharge/charge behavior is expected for a fully reversible reaction mechanism and is confirmed here by the kinetic model. The concentration of dissolved molecular oxygen (O2(solv), Figure 5b) shows a drop at the begin of the discharge and further decreases during discharge due to (a) its consumption in the cathode reaction, which leads to a decrease of the pressure in the closed gas reservoir resulting in a lower solubility limit of O2(solv) (cf. section 5.5), and (b) a decreasing effective diffusion coefficient due to lower available pore volume in the cathode, which is filled with the reaction products LiO2 and Li 2 O 2 (cf. section 5.5). During charge, the O 2 (solv) concentration increases slightly as expected due to its formation. The concentration of dissolved superoxide ions O2−(solv) shows distinct supersaturation at the beginning of discharge before lithium superoxide starts to precipitate. During the following growth of LiO2, the O2−(solv) concentration remains constant. At the beginning of charge it drops to a lower concentration before it is oxidized completely at the end of charge. The volume fraction of the reaction end product lithium peroxide (Li2O2) is increasing during discharge and decreasing during charge (Figure 5c). The volume fraction of the intermediate lithium superoxide (LiO2) remains very small during the complete cycle due to its further disproportionation. It decreases slightly during the discharge phase and increases slightly during the charging phase. The ratio of LiO2/Li2O2 can be adjusted by the rate of the chemical disproportionation reaction R4. 5.2. Partially Irreversible Reaction Mechanism. As shown in the previous subsection, the reversible reaction mechanism (“A”) does not provide asymmetric discharge/charge overpotentials. Thus, we analyze a partially irreversible reaction mechanism (“B”) as presented in section 4 (cf. reactions R1−R3, R4b and R5 in Figure 3). The results of this model are shown in Figure 6 for one galvanostatic cycle at a current density of 0.1 mA/cm2 (same cycling protocol as before). This simulation predicts asymmetric discharge/charge overpotentials (see Figure 6a). The discharge curve is not influenced by the change of the reaction pathway (cf. Figure 5a), but the charge curve now shows a plateau at around 4.15 V. A plateau-like behavior is expected due to the phase-change nature of the electrostripping reaction R5, and the plateau voltage has been adjusted by changing the pre-exponential factor of reaction R5 in order to match the small experimental plateau around 40 h. A closer look reveals that the charge voltage is not constant, but slightly increases during charge (1.5 mV/h). This is due to the decreasing length of the three-phase boundary between carbon, Li2O2 and electrolyte, where reaction R5 takes place (cf. section 4.4). The steep voltage increase toward end of charge is caused by the depletion of reactants, specifically Li2O2(bulk). Still, the simulated slope does not represent the one observed experimentally with its continuous increase of charge voltage. The concentration of dissolved molecular oxygen (O2(solv), Figure 6b) shows a similar trend as for the reversible reaction mechanism (Figure 5b). The concentration of O2−(solv) again shows supersaturation at the beginning of discharge before the lithium superoxide starts to precipitate. During the following growth of LiO2, the O2−(solv) concentration remains constant until it is oxidized at the beginning of the charge phase. The slope of the decrease is exponential, which can be seen at the linear slope in the logarithmic scale. Figure 6c shows that the volume fraction of the intermediate LiO2 rises at the beginning of discharge and remains at a very low constant value during the complete discharge. At the

Figure 5. Reversible reaction mechanism (without redox mediator): galvanostatic cycling at 0.1 mA/cm2. (a) Cell voltage as a function of time compared to experimental data, (b) spatially averaged concentrations of dissolved oxygen and superoxide ions as a function of time, and (c) spatially averaged volume fractions of Li2O2 and LiO2 within the cathode as a function of time.

The voltage drop is due to the blocking of the electrode surface with lithium oxides, leading to a decrease of carbon/electrolyte interfacial area (cf. section 4.3) from its initial value of 1.54 × 108 m2/m3 to a value of 1.7 × 105 m2/m3 at end of discharge, thus to only 0.01% of the initial free surface area. As a result, the reaction rate slows down leading to a high discharge overpotential. The film thickness d of lithium oxides on the carbon surface was adjusted to match the experimental discharge time, resulting in a value of d = 1.07 nm, which is in reasonable agreement to a suggested thickness of d = 4−5 nm as an average thickness of Li2O2 for cell death using a glassy carbon electrode.64 The simulated charge overpotential, however, is not only significantly lower than observed experimentally, it is even decreasing during the complete charge before it starts to increase at the end-of-charge phase. The decrease of the overpotential can be explained by the increasing carbon/electrolyte surface area during charge, which is symmetrical to the discharge process.65 The steep voltage increase toward end of charge is caused by the depletion of reactants, specifically, O2−(solv). A symmetrical 24628


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(with redox mediator). To match the experimental capacity with the model, the assumed film thickness d of the lithium oxides (cf. section 4.3) was reduced from d = 1.07 nm to d = 0.75 nm. Therefore, the drop in discharge capacity due to the addition of the redox mediator is interpreted as a change of film morphology. As changing the LiO2 solubility was also observed to lead to a different morphology,4 a direct influence of the presence of redox mediator on film morphology might be possible. It is beyond the scope of this model to further clarify this point. The simulations are able to reproduce the influence of the redox mediator on the charge overpotential by showing a charging plateau at around 3.65 V (4.15 V without redox mediator as shown in section 5.2). Hence, the charging voltage is completely governed by the redox potential of the couple TEMPO+/TEMPO (see Figure 4). Within the charge plateau, the overpotential slightly increases due to the decreasing surface of Li2O2. The concentrations of O2(solv) and O2−(solv) (Figure 7b) as well as the volume fractions of Li2O2 and LiO2 (Figure 7c) show the same qualitative behavior compared to the results without redox mediator (cf. Figure 6). In Figure 7d concentrations of the redox mediator species, dissolved TEMPO and TEMPO+, are shown. The initial values of both species stay almost constant during discharge, but during charge, the concentration of TEMPO+ increases and the concentration of TEMPO decreases. As the potential during discharge is far below the redox potential of 3.74 V vs Li/Li+, the formation of TEMPO in reaction R6 is favorizied. The plateau-like concentrations of these species during a large part of charge are due to the competing electrochemical formation of TEMPO+ (reverse reaction R6) and the chemical consumption of TEMPO+ (reaction R7). At the end of charge, when Li2O2 is completely dissolved, TEMPO+ is not consumed any more; therefore, the TEMPO concentration drops toward zero, while the TEMPO+ strongly increases at the end of charge. This has been experimentally proven by in situ pressure monitoring of Li−O2 cells with TEMPO as well.66 5.4. Influence of Current Density. The model with irreversible reaction mechanism without redox mediator (“B”) is evaluated at two discharge current densities, 0.1 and 0.5 mA/cm2. The cycling behavior for i = 0.1 mA/cm2 is shown above (section 5.2 and Figure 6). Figure 8 shows results for one galvanostatic cycle at a current density of 0.5 mA/cm2 (0.1 h rest, otherwise same cycling protocol as above). The model predicts the discharge voltage characteristics of the experimental cell very well without readjusting kinetic parameters. Again, the experimental charge curve cannot be reproduced correctly with this mechanism. At the beginning of the charge, the simulations indicate a short plateau at around 3.2 V, which can be assigned to the dissolution reaction of lithium superoxide (reaction R3 in section 4.4). The main charge plateau has an elevated potential of 4.25 V compared to 4.15 V at the lower current density of 0.1 mA/cm2. As expected, a higher charge current density leads to higher charge overpotentials. The experimentally observed cell capacity drops significantly at higher current densities (0.53 mAh at 0.5 mA/cm2 compared to 1.74 mAh at 0.1 mA/cm2). This capacity-rate effect is wellknown for Li/O2 batteries.46,67 In order to accommodate the capacity loss with the model, we decreased the film thickness d of the lithium oxides from 1.07 nm (at 0.1 mA/cm2) to 0.33 nm (at 0.5 mA/cm2). Again, as in the case of the redox mediator, the change in discharge capacity is interpreted as film morphology

Figure 6. Irreversible reaction mechanism (without redox mediator): galvanostatic cycling at 0.1 mA/cm2. (a) Cell voltage as a function of time compared to experimental data, (b) spatially averaged concentrations of dissolved oxygen and superoxide ions as a function of time, and (c) spatially averaged volume fractions of Li2O2 and LiO2 within the cathode as a function of time.

beginning of charge it dissolves completely. The volume fraction of Li2O2 increases during discharge and decreases during charge, very similar to the reversible mechanism. 5.3. Influence of Redox Mediator. To analyze the influence of the redox mediator TEMPO on the cycling behavior of the Li/O2 cell, the two dissolved species TEMPO (2,2,6,6-tetramethylpiperidinyloxyl) and TEMPO+ were integrated into the mechanism. The full mechanism (“C”) consists of the partially irreversible mechanism (reactions R1−R3, R4b and R5) and the two redox mediator reactions R6 and R7, as described in section 4.4. According to the experimental conditions, an initial concentration of 10 mM TEMPO in the electrolyte was chosen. The results of this model are shown in Figure 7 for one galvanostatic cycle at a current density of 0.1 mA/cm2 (same protocol as before). The discharge overpotential is not influenced by the redox mediator (see Figure 7a). However, the experimentally observed discharge capacity drops significantly from 1.74 mAh (without redox mediator) to 1.22 mAh 24629


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Figure 8. Irreversible reaction mechanism (without redox mediator): galvanostatic cycling at 0.5 mA/cm2: cell voltage as a function of time compared to experimental data.12

5.5. Spatial Distributions. The use of our spatially resolved model allows insight into the spatial distribution of relevant state variables. Here, we focus on concentration profiles of dissolved oxygen and the volume fraction profiles of solid lithium peroxide Li2O2 during discharge. Figure 9 shows spatial profiles at two different current densities (0.1 and 0.5 mA/cm2) for 11 equidistant time steps during one complete discharge. The spatial distribution of dissolved oxygen O2(solv) concentration starts homogeneously at its initial condition (solubility limit of 6.5 mmol/L12) (Figure 9, left). During discharge a concentration gradient along the cathode thickness is observed, which depends on current density. A higher current density leads to a larger concentration gradient due to the higher local consumption of the species at the reaction surface. The concentration within the separator is homogeneous. At the interface between the cathode and gas reservoir the saturation concentration is maintained but decreases during discharge due to decreasing oxygen pressure in the closed gas reservoir. The drop is lower at the higher current density because the discharge capacity is lower (see section 5.4), which linearly influences the oxygen pressure due to the total amount of consumed oxygen.48,68 The model predicts that, even for a current density of 0.5 mA/cm2, no complete depletion of dissolved oxygen is observed in any region within a 50 μm thick cathode during discharge, which would lead to a deactivation of relevant reaction zones starting close to the cathode/separator interface. The precipitation of solid lithium peroxide Li2O2 (see Figure 9, right) is predicted to occur preferentially at the gas reservoir/ cathode interface leading to a slightly inhomogeneous distribution within the cathode. This behavior can be explained by the locally higher concentration of dissolved oxygen at the interface between gas reservoir and cathode, which leads to locally higher reaction rate of reaction R2 and the following reaction steps forming Li2O2. This trend is sensitive to the discharge current density, and more distinct at 0.5 mA/cm2 compared to 0.1 mA/cm2. A similar tendency has been recently shown experimentally for Li/O269 and Na/O270 cells. Interestingly, the gradient levels off toward end of discharge. This is due to the assumed film formation of lithium oxides, blocking the carbon/electrolyte interface: the regions of the electrode with initially preferred precipitation are deactivated by the film formation earlier, resulting in a more homogeneous distribution at the end of discharge.

Figure 7. Irreversible reaction mechanism with redox mediator: galvanostatic cycling at 0.1 mA/cm2. (a) Cell voltage as a function of time compared to experimental data, (b) spatially averaged concentrations of dissolved oxygen and superoxide ions as a function of time, (c) spatially averaged volume fractions of Li2O2 and LiO2 within the cathode as a function of time, and (d) spatially averaged concentrations of redox mediator species (TEMPO and TEMPO+) as a function of time.

change. The morphology of lithium peroxide has been shown to be strongly dependent on the discharge current density4,24,25,32,67 leading to more film-like precipitation at high current densities and particles with toroidic shape at very low discharge current densities.60

6. CONCLUSIONS Modeling of multistep electrochemistry allows detailed insights into the complex mechanisms and performance potentials of 24630


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Figure 9. Spatiotemporal analysis during discharge for a current density of 0.1 mA/cm2 (top) and 0.5 mA/cm2 (bottom): dissolved oxygen concentration (left) in the electrolyte in the cathode (CA) and separator (SEP), and lithium peroxide volume fraction (right) in the cathode. All profiles are plotted as a function of the distance from cathode/gas interface. Each panel shows 11 spatial profiles taken at equidistant time steps between beginning and end of discharge.

lithium−oxygen batteries. We have presented and analyzed three different multistep mechanisms of the 2Li + O2 ⇆ Li2O2 cell reaction, (A) a reversible 5-step mechanism, (B) a partially irreversible 6-step mechanism, and (C) a partially irreversible 8-step mechanism that includes reactions of a redox mediator. Model predictions were compared to experimental galvanostatic cycling data of Swagelok cells without and with TEMPO as redox mediator at two different current densities. As result, all mechanisms are able to predict the discharge behavior in good agreement to the experimental results. The experimentally observed high charge overpotentials can be qualitatively reproduced with the irreversible reaction mechanisms. The reduction of the charge overpotential by using a redox mediator was successfully reproduced. However, the particular shape of the experimental charge curve with continuously increasing charge overpotential could not be reproduced with the investigated mechanisms. Predicted spatiotemporal distributions of state variables reveal inhomogeneous oxygen concentrations as well as reaction product precipitations within the cathode. The combined modeling and experimental study allows the following conclusions on the mechanism of nonaqueous lithium−oxygen batteries: • The asymmetry of discharge/charge behavior is due to a partially irreversible reaction mechanism. Fully reversible reaction mechanisms are unable to reproduce the asymmetry. Here, we assume the 2LiO2 → Li2O2 + O2 disproportionation reaction to be the irreversible discharge step. • The model assumes film-like growth of lithium oxides (LiO2 and Li2O2) on the electrode surface. End of discharge is due to a blocking of the carbon/electrolyte interface. Under this assumption, the observed discharge capacity can be well-reproduced by assuming realistic nanometer-range film thicknesses. • The use of a redox mediator reduces the charge voltage indirectly by transferring electrons from the carbon surface

Figure A1. Scheme of a Li2O2 particle on the carbon electrode surface depicting reaction zones: the top surface ALi2O2,top and the three-phase boundary LTPB, the bottom surface ALi2O2,bottom covering the carbon surface Acarbon, and film thickness d.

to the Li2O2 surface. The oxidation of the redox mediator is sufficiently fast and therefore not influenced by the partial blocking of the carbon/electrolyte surface after end of discharge. • The end of charge is due to depletion of reactants: dissolved superoxide ions O2− in case of the reversible reaction mechanism and solid lithium peroxide Li2O2 in case of the partially reversible reaction mechanism. • Changes of discharge capacity due to the introduction of a redox mediator or due to increase of current density can be interpreted as morphological changes of the reaction products. Here, we assume formation of lithium oxide films of different thicknesses. It should be noted that alternative discharge mechanisms have been proposed or shown before, including particle-like growth, two-step electrochemical oxygen reduction, etc. This study does not disprove these mechanisms, but shows that the present assumptions (film growth, LiO2 disproportionation) are sufficient to reproduce our experimental discharge behavior. Ongoing studies focus on the further development of the charge mechanism, including a surface film model hosting nonstoichiometric phases like Li2‑xO2, which might allow to more accurately predict the increasing charge overpotential. 24631


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The Journal of Physical Chemistry C Table A.1. Summary of Model Equations;48 See Table A.2 for a List of Symbols physicochemical process

model equation

Gas Reservoir

∂ρgas

continuity

∂t

∂(ρgas Yi )

species conservation

Ns

A res Vres

=

∑ Jicat | res Mi i=1

A = res Jicat | res Mi Vres

∂t

Ns

pgas = ρgas RT ∑ Yi /Mi

ideal gas law

i=1

Mass and Charge Transport in Electrodes and Separator ∂(ϵelytci)

species conservation

=−

∂t

species fluxes (Nernst−Planck equation)

Ji = − Dieff

charge conservation (charge neutrality)

V C DL

Dimigr =

effective transport coefficients (porous-electrode theory)

Dieff =

∂t

i=1

∂Ji ∂y

− iFV

2 τelyt

Di and Dimigr,eff =

ϵelyt 2 τelyt

Dimigr with τelyt =

le 71 l

NS, j

=

∑ si̇VMi i=1

Nr

∑ νirnA nV

si̇ V =

species source terms

Ns

∑ ziF

ziF ciDi RT

ϵelyt

∂(ρj ϵj)

multiphase management

+ si̇ V + si̇ V,DL

∂ϕelyt ∂ci − Dimigr,eff ∂y ∂y

∂(Δϕ) = ∂t

migration coefficient (dilute solution theory)

∂Ji ∂y

n=1

Cell Current and Voltage cell voltage

Ecell = ϕelde,ca − ϕelde,an

potential step (anode and cathode)

Δϕ = ϕelde − ϕelyt

Faradaic current density

iFV = Fsė V− =

Nr

∑ Fνe−rnA nV n=1

V d(Δϕ) = C DL dt

Current density due to electrical double layer (anode and cathode)

V iDL

total current density (anode and cathode)

icell =

Lelectrode

∫

V (iFV + iDL ) dy

y=0

(Electro-)Chemistry NP

i=1

i=1

⎛ Eact,f ⎞ ⎛ α zF ⎞ k f0·exp⎜ − ⎟ ·exp⎜ − f Δϕ⎟

forward rate constant

kf =

reverse rate constant (thermodynamic consistency)

⎛ ΔG 0 ⎞ ⎛ (1 − αf )zF ⎞ −∑ν ⎛ Eact,f ⎞ i Δϕ⎟ctot k r = k f0· exp⎜ − ⎟ ·exp⎜ ⎟ · exp⎜ ⎝ RT ⎠ ⎝ ⎠ RT ⎝ RT ⎠

⎝

RT ⎠

⎝

RT

⎠

depending on the time-dependent volume fractions of adjacent bulk phases as described in the following. Lithium oxide particles as part of the film-like coating on the electrode surface are assumed as equilateral rectangular-shaped with the edge length aLi2O2, as schematically shown in Figure A1 for a Li2O2 particle on the carbon surface. We assumed that reaction R3 takes place at the top surface area ALi2O2,top, while the lateral surfaces are neglected due the low film thickness of d (d ≲ 1 nm). The absolute volume of the Li2O2 phase VLi2O2 can be given either by the product of the volume of one Li2O2 particle Vparticle times the volume-specific particle density NV and the total electrode volume Vtotal

The model framework presented in this work can also allow to systematically study further redox mediators in Li/O2 cells. Suitable candidates to reduce the high charging overpotentials might be screened before extensive experimental studies are carried out.

■

NR

r = k f ∏ ci|νi| − k r ∏ ci|νi|

reaction rate (mass-action kinetics)

APPENDIX: VOLUME-SPECIFIC SURFACE AREAS AND THREE-PHASE BOUNDARY

Reactions in the presented model approach take place at interfaces or three-phase boundaries, which are characterized by their volume-specific surface area or volume-specific length (cf. section 2.3 and Table 3). We scale these reaction zones 24632


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The Journal of Physical Chemistry C Table A.2. List of Symbols symbol

unit

AV0 Acarbon ALi2O2,bottom

m2/m3 m2 m2

ALi2O2,top

m2

Ares AV CVDL

m2 m2/m3 F/m3

Di Deff i

m2/s m2/s

Dmigr,eff i

mol/(V·m·s)

Dmigr i Eact,f

mol/(V·m·s) kJ/mol

Ecell E0 Ji Jcat|res i

V V mol/(m2·s) mol/(m2·s)

LTPB,particle

m

LVTPB

m/m3

NP

1

Nr NR

1 1

NS,j NS NV Vparticle Vres Vtotal VLi2O2

meaning

symbol

unit

meaning

initial volume-specific surface area surface area of carbon bottom surface area of Li2O2 particle top surface area of Li2O2 particle

iVDL

A/m3

iFV kf kf0

pgas ṡiV ṡi,DLV zi F Lelectrode M R T Y i j l le n r t y

1 As/mol m kg/mol J/(mol·K) K 1 1 1 m m 1 mol/(m2·s) s m

1 1 1/m3 m3 m3 m3 m3

electrode surface area volume-specific surface area volume-specific double layer capacity diffusion coefficient of species i effective diffusion coefficient of species i effective migration coefficient of species i migration coefficient activation energy of forward reaction cell voltage equilibrium cell voltage molar flux of species i molar flux of species i through cathode/gas reservoir interface length of three-phase boundary of one particle volume-specific length of threephase boundary number of products participating in reaction number of reactions number of reactants participating in reaction number of species in phase j number of species volume-specific particle density volume of particle volume of gas reservoir total electrode volume absolute volume of Li2O2 phase

A/m3 mol/(m2·s)·(mol/m3)N ∑i =R1|νi| mol/(m2·s)·(mol/m3)N ∑i =R1|νi| mol/(m2·s)·(mol/m3)N ∑i =P1|νi| Pa mol/(m3·s) mol/(m3·s)

z αf

1 1

νi

1

aLi2O2

m

edge length of Li2O2 particle

ci

mol/m3 mol/m3

kg/m3 1 1 V V kJ/mol

ctot

m A/m2

Δϕ

V

d icell

concentration of species i in a bulk phase total concentration of the bulk phase film thickness area-specific cell current

ρj τelyt ϵj ϕelde ϕelyt ΔG0

kr

VLi 2O2 = VparticleNVVtotal

A LiV2O2 = a Li 2O2 2NV =

or by the product of the volume fraction of the Li2O2 phase εLi2O2 and the total volume Vtotal

pre-exponential factor of forward reaction reaction rate constant of reverse reaction pressure of gas in reservoir volumetric species source term volumetric species source term due to double layer charge/discharge charge number of species i faraday constant thickness of electrode molar mass ideal gas constant temperature mass fraction index of species index of bulk phases length of porous medium effective length of pores index of chemical reactions interfacial reaction rate time spatial position in dimension of electrode-pair thickness number of transferred electrons symmetry factor of forward reaction stoichiometric coefficient of species i density of bulk phase j tortuosity of electrolyte volume fraction of bulk phase j electric potential of the electrode electric potential of the electrolyte standard Gibbs free reaction energy electric potential difference between electrode and electrolyte

εLi 2O2 d

This derivation is also valid for the LiO2 phase and for the calculation of Aj,bottom of phase j, which is assumed to cover the carbon surface (cf. section 4.3). The volume-specific three-phase boundary length LVTPB is the total length of all particle edges per volume, defined as

VLi 2O2 = εLi 2O2Vtotal

Furthermore, the volume of one Li2O2 particle is given by Vparticle = a Li 2O2 2d

V L TPB = L TPB,particleNV = 4a Li 2O2NV

V

Taken together, N can be expressed as εLi 2O2 NV = a Li 2O2 2d

volume-specific current due to double layer volume-specific Faradaic current reaction rate constant of forward reaction

Substituting aLi2O2 from eq 1 yields (1)

V L TPB

The volume-specific surface area AVLi2O2 of Li2O2 is calculated as the top surface area of each Li2O2 particle times the volumespecific particle density resulting in

=4

εLi 2O2NV d

We assume volume-specific particle density of 4.466 × 1020 1/m3.72 24633


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The Journal of Physical Chemistry C In summary, we have derived expressions for AV and LVTPB as a function of ε and the independent parameters d and NV (see Table 3).

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AUTHOR INFORMATION

Corresponding Author

*E-mail: wolfgang.bessler@hs-offenburg.de. Tel: +49 781 2054653. Present Address §

Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the German Ministry of Education and Research (BMBF) in the framework of the “LiO2Mech” project (01DM14002) is gratefully acknowledged. D.G. thanks Manik Mayur (Offenburg University of Applied Sciences), and Robert J. Kee and Steven DeCaluwe (Colorado School of Mines) for detailed discussions. J.J. acknowledges support by the BASF International Network for Electrochemistry and Batteries, as also by the BMBF (Federal Ministry of Education and Research) within the project BenchBatt (03XP0047A).

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