Modeling, Simulation, and Experimental Verification of Impedance Spectra in Li-Air Batteries Mohit Rakesh Mehta

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2015 Modeling, Simulation, and Experimental Verification of Impedance Spectra in Li-Air Batteries Mohit Rakesh Mehta

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COLLEGE OF ENGINEERING

MODELING, SIMULATION, AND EXPERIMENTAL VERIFICATION

OF IMPEDANCE SPECTRA IN LI-AIR BATTERIES

MOHIT RAKESH MEHTA

A Dissertation submitted to the Department of Electrical and Computer Engineering in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

2015

Mohit Rakesh Mehta defended this dissertation on August 26, 2015. The members of the supervisory committee were:

Petru Andrei Professor Directing Dissertation

Joseph B. Schlenoﬀ University Representative

Jim P. Zheng Committee Member

Pedro Moss Committee Member

Hui Li Committee Member

The Graduate School has veriﬁed and approved the above-named committee members, and certiﬁes that the dissertation has been approved in accordance with university requirements.

ii Dedicated to My Beloved Parents

iii ACKNOWLEDGMENTS

I am ﬁrst and foremost grateful to my advisor, Petru Andrei, for all his guidance, support and most importantly his patience. I would also like to thank Prof. Jim Zheng for allowing me to perform experiments on Li-air batteries with his students. I would also like to thank Dr. Xujie Chen for building Li-air batteries and performing impedance measurements in order to validate the idea of measuring impedance spectra during discharge. Dr. Xujie Chen also was of great help in guiding and assisting me in performing the EIS experiments. I would also take this opportunity to thank all my friends for their support and constant encouragement.

iv TABLE OF CONTENTS

ListofTables...... vii ListofFigures ...... viii ListofSymbols...... xi Abstract...... xiii

1 Introduction 1 1.1 Electrochemical impedance spectroscopy ...... 1 1.1.1 Common EIS measuring techniques ...... 2 1.2 Modeling of EIS spectra in electrochemical systems ...... 3 1.2.1 Phenomenological modeling ...... 4 1.2.2 Equivalentcircuitmodeling ...... 8 1.3 Li-air batteries (LABs) ...... 8 1.4 Organization of thesis ...... 10

2 EIS of Li-air batteries under d.c. discharge 11 2.1 Introduction...... 11 2.2 Assumptions ...... 12 2.3 Modeling ...... 12 2.3.1 The case of high d.c. discharge currents ...... 13 2.3.2 The case of low d.c. discharge currents ...... 19 2.4 Simulationresults ...... 23

3 EIS of Li-air batteries under d.c. charge and discharge 32 3.1 Introduction...... 32 3.2 Steady-state analysis ...... 33 3.2.1 Oxygen concentration at steady-state ...... 33 3.2.2 Faradaic current at steady-state ...... 35 3.2.3 Steady-state Faradaic current during discharge ...... 36 3.3 Small-signal analysis ...... 36 3.3.1 The Faradaic impedance during discharge ...... 43

4 Experimental verification and parameter extraction 44 4.1 Introduction...... 44 4.2 Parameter extraction at low cathode specific areas ...... 44 4.3 Parameter extraction at high cathode specific areas ...... 47 4.4 Experimentalverification ...... 47 4.5 Equivalentcircuitmodel(ECM) ...... 50

v 5 Eﬀect of ﬁnite oxygen dissolution on the impedance spectra 54 5.1 Introduction...... 54 5.2 Modelfordissolutionkinetics ...... 55 5.2.1 Steady-state analysis ...... 56 5.3 Impedancespectraunderd.c. discharge ...... 58 5.4 The case of high dissolution kinetics ...... 63

6 Finite element modeling of EIS 65 6.1 Introduction...... 65 6.2 Finiteelementmodel...... 65 6.3 Computation of impedance spectra ...... 68 6.4 Simulationresults ...... 71

7 Conclusions 79

References...... 81 BiographicalSketch ...... 89

vi LIST OF TABLES

2.1 List of simulation parameters for ﬁgures 2.4 and 2.5 ...... 25

2.2 List of simulation parameters for ﬁgures 2.6 and 2.7 ...... 26

2.3 List of simulation parameters for ﬁgures 2.8 and 2.9 ...... 28

2.4 List of simulation parameters for ﬁgures 2.10 and 2.11...... 29

2.5 List of simulation parameters for the comparison between the impedance spectra computed using Tafel kinetics (eqn. 2.38), Butler-Volmer kinetics with symmetric charge transfer (β = 0.5) (eqn. 2.49), and Butler-Volmer kinetics without the symmetric charge transfer limitation (eqn. 2.44) ...... 30

6.1 List of parameters used in the ﬁnite element simulations ...... 71

vii LIST OF FIGURES

1.1 Examples for representing the data of the impedance spectra: (a) A typical Nyquist plot for a Li-air battery with the real value of the impedance on the x-axis and the imaginary value of the impedance on the y-axis. (b) A typical Bode plot with the absolute value and the phase angle of the impedance are plotted for diﬀerent values of frequency corresponding to the Nyquist plot represented in (a)...... 2

1.2 Network for measuring the impedance spectra of a battery using the potentiostatic measuringtechnique...... 3

1.3 Network for measuring the impedance spectra of a battery using the galvanostatic measuringtechnique...... 3

1.4 The total value of the discharge current can be separated into two parts: charge transfercurrent andthedoublelayercurrent...... 6

1.5 The structure of a Grahame’s double layer with the inner and outer Helmholtz plane near the electrode surface and the diﬀuse layer near the bulk region. Here ϕm, ϕ1 and ϕ2 are the potentials at the metal surface, inner and outer Helmholtz plane respectively. 7

1.6 A schematic of a typical Li-air battery with oxygen entering the system from x = 0. . 9

2.1 Non-distributed (a) and distributed (porous) (b) cathode ...... 12

2.2 The range of values of the d.c. discharge current and the reaction rate constant over which the Tafel equation is assumed to be valid...... 20

2.3 The eﬀect of the initial oxygen concentration on the impedance plot for two values of the system length: l = 100 nm (a) and l = 10 µm (b). The other simulation parameters are, C = 4 µF/cm2, D = 7 10 6 cm2/s, i = 200 µA/cm2, k = 1.3 10 8 cm/s d o × − F 0 × − and n =3...... 23

2.4 The eﬀect of the reaction rate on the impedance plot for two values of the stoichiometric number of electrons: n = 2 (a) and n =3(b)...... 24

2.5 The effect of the stoichiometric number of electrons on the impedance plot for two values of the oxidant diffusion coefficient: D = 7 10 7 cm2/s (a) and D = 1.4 10 5 cm2/s o × − o × − (b)...... 24

2.6 The effect of the d.c. discharge current on the impedance plot for two values of the oxygen diffusion coefficient: D = 3.5 10 6 cm2/s (a) and D = 7 10 5 cm2/s (b). 25 o × − o × − 2.7 The effect of the oxygen diffusion coefficient on the impedance plot for two values of 2 2 the double layer capacitance: Cd = 4 µF/cm (a) and Cd = 40 µF/cm (b)...... 26

viii 2.8 The eﬀect of the system length on the impedance plot for two values of the stoichiometric number of electrons: n = 2 (a) and n =3(b)...... 27

2.9 The eﬀect of the double layer capacitance on the impedance plot for two values of the stoichiometric number of electrons: n = 2 (a) and n =3(b)...... 27

2.10 Comparison between the impedance spectra computed using Tafel kinetics (eqn. 2.38) and Butler-Volmer kinetics (eqn. 2.44) for diﬀerent values of d.c. discharge current density (i)...... 28

2.11 Comparison between the impedance spectra computed using the Tafel kinetics (eqn. 2.38) and Butler-Volmer kinetics (eqn. 2.44) for diﬀerent values of standard reaction rate constant...... 29

2.12 Comparison between the impedance spectra computed using Tafel kinetics (eqn. 2.38), Butler-Volmer kinetics with symmetric charge transfer (β = 0.5) (eqn. 2.49), and Butler-Volmer kinetics without the symmetric charge transfer limitation (eqn. 2.44) for diﬀerent values of the charge transfer coeﬃcient (β)...... 30

2.13 Comparison between the impedance spectra computed using Tafel kinetics (eqn. 2.38), Butler-Volmer kinetics with symmetric charge transfer (eqn. 2.49), and Butler-Volmer kinetics without the symmetric charge transfer limitation (eqn. 2.44) for diﬀerent values of the standard reaction rate constant k0 when β = 0.55...... 31

4.1 Possible low frequency impedance spectra of Li-air batteries with low speciﬁc area of the cathode. The dashed continuation line at high frequencies show that the Nyquist spectra might contain other semicircles, which are usually due to the anode and anode- separatorinterface...... 46

4.2 Possible low frequency impedance spectra of Li-air batteries with high speciﬁc area of the cathode. The dashed lines show that the Nyquist spectra might contain other semicircles at high frequencies, which are usually due to the anode and anode-separator interface...... 48

4.3 Comparison between the theoretical and experimental impedance spectra. R12 is the diameter of the second semicircle along the real axis and can be used to determine the effective value of the oxygen diffusion coefficient and the reaction rate in the cathode using equations 4.15, 4.16, and 4.18...... 49

4.4 Small-signal equivalent circuit of Li-air batteries. ZF denotes the Faradaic impedance, CD is the capacitance of the double layer, RΩ is the combined resistance of the electrolyte, Li-ions, and electrons in the cathode matrix...... 50

4.5 Nyquist plot of the Faradaic impedance at large discharge currents and cathode widths (i.e. l λ)...... 51 ≫ 4.6 Approximate small-signal equivalent circuit of Li-air batteries operating at large discharge currents and with large cathode width (l λ). The values of R , C , R, and ≫ Ω D

ix C can be expressed in terms of physical parameters using eqns. 2.26, 2.27, 4.10, and 4.28, respectively. This circuit should be used with care in practical applications and, instead, one should use the more general circuit from Fig. 4.4...... 52

5.1 Dissolution kinetics in a Lithium-air battery with oxygen dissolving at x = 0 . . . . . 55

6.1 Cross-section of a cathode with cylindrical pores with a single radius ...... 67

6.2 Comparison between the ﬁnite element simulation (symbols) and the analytical model (continuous) for diﬀerent values of the d.c. discharge current density...... 72

6.3 Distribution of the discharge product (a)-(d) for diﬀerent states of discharge: 20%, 50%, 90%, and 100% (for a discharge current density of 0.1 mA/cm2) and impedance spectra (e) simulated for diﬀerent values of the state-of-discharge...... 73

6.4 Distribution of the discharge product (a)-(d) for diﬀerent states of discharge: 20%, 50%, 90%, and 100% (for a discharge current density of 1 mA/cm2) and impedance spectra (e) simulated for diﬀerent values of the state-of-discharge...... 74

6.5 Cell voltage as a function of speciﬁc capacity for the two discharge currents used in the simulations presented in Fig. 6.3 and Fig. 6.4. The symbols represent the state of discharge at which the impedance spectra are simulated...... 75

6.6 (a)-(d) show the distribution proﬁle of the discharge product and (e) shows the impedance spectra for diﬀerent values of Li2O2 resistivity at 50% state-of-discharge. The simulations are carried out for d.c. discharge current density of 1 mA/cm2. . . . 76

6.7 Cell voltage as a function of speciﬁc capacity for diﬀerent values of resistivity of the discharge product. The dotted line and the black circle symbols denote the 50% state ofdischarge...... 77

6.8 Voltage discharge curves (a) and simulated EIS spectra (b) for diﬀerent values of oxygen concentration in the atmosphere (relative to 1 atmosphere)...... 78

x LIST OF SYMBOLS

Roman Symbols A cross-sectional area of the porous electrode (cm2) brugg bruggeman coefficient for oxygen diffusion bruggLi bruggeman coefficient for electrolyte diffusion bruggκ bruggeman coefficient for electrolyte conductivity 2 CD double layer capacitance (F/cm ) 3 ci concentration of species i (mol/cm ) 3 ci(x,t) concentration of species i as a function of time and space (mol/cm ) 3 ci(0,t) surface concentration of species i as a function of time (mol/cm ) ci∗ initial concentration of a species i (mol/cm) 2 Deff effective diffusion coefficient (cm /s) 2 Di bulk diffusivity of a species i (cm /s) d multiplying factor depending on the geometry of the pores E electrode potential (V) E0 open circuit potential (V) Eeq equilibrium electrode potential (V) e electronic charge (C) F Faraday’s constant (C/mol) 2 idis/chg d.c. value of the total current density while discharging/charging (A/cm ) 2 i0 exchange current density (A/cm ) 2 iF d.c. value of the Faradaic current density (A/cm ) 2 idl current density of the double layer (A/cm ) j √ 1 − Ji flux of species i (mol/cm2 s) k reaction rate constant (1/s) k0 standard reaction rate constant (cm/s) kf/r forward/reverse reaction rate constant (1/s) f/r k0 standard forward/reverse reaction rate constant (cm/s) kf/b forward/backward reaction rate for oxygen dissolution kinetics (cm/s) l cathode thickness (cm) Mdis molar mass of the discharge product (g/mol) n stoichiometric number of electrons consumed O oxidant po partial pressure of oxygen (atm) R (a) reductant (b) resistance defined in the equivalent circuit model (eqn. 4.10) R1 diameter of the first low frequency semicircle on the Nyquist plot R2 diameter of the second low frequency semicircle on the Nyquist plot R12 diameter of the combined low frequency semicircle on the Nyquist plot 3 Rc reaction rate at the cathode (A/cm )

xi rp average radius of the pores (cm) rp, 0 initial average radius of the pores (cm) t+ transference number v velocity of the volume element (cm/s) V cell voltage (V) V0 open cell voltage (V) VT thermal voltage (V) VΩ internal ohmic resistance, including the resistance of the separator, Li-ions, electrons, deposited layer, and contacts (Ω) x location inside the cathode (cm) y vector of state variables (eqn. 6.15) Y0 d.c. values of the state variables at each mesh point Z total impedance (Ω) ZF Faradaic impedance (Ω) zi charge on species i

Greek symbols β symmetry factor ǫ cathode porosity ǫ0 initial cathode porosity 3 δci small-signal concentration of species i (mol/cm ) δi total small-signal current density (mol/cm3) 2 δidl small-signal current density of the double layer (A/cm ) 2 δiF small-signal Faradaic current (A/cm ) δη small-signal over-potential (V) δy state vector of small-signal values of the state variables η over-potential (V) ω angular frequency (rad/s) φ electron potential (V) φLi electrostatic potential of the electrolyte (V) ϕm metal surface/electrode potential (V) ϕ1 inner Helmholtz plane potential (V) ϕ2 outer Helmholtz plane potential (V) ρdis resistivity of the discharge product (Ωcm) ρm,dis mass density of the discharge product (g/mol)

xii ABSTRACT

There has been a growing interest in electrochemical storage devices such as batteries, fuel cells and supercapacitors in recent years. This interest is due to our increasing dependence on portable electronic devices and due to the high demand for energy storage from the electric transport vehicles and electrical power grid industries. As, we transition towards cleaner renewable fuel sources such as solar and wind the need for energy storage devices in the power grid industry will continue to grow. Among the electrochemical storage devices, Li-air offers the highest theoretical specific capacity and energy density. However, these batteries suffer from a number of issues such as low cyclability, low practical energy densities, and low specific capacities. The deposition of lithium peroxide on the surface of the cathode is one of the main causes for the low practical specific capacity of lithium-air batteries with organic electrolyte. Electrochemical impedance spectroscopy (EIS) is a powerful analytical tool that has been used in the past to extract physical parameters such as chemical diffusion coefficient, effective diffusion coefficient, and Faradaic reaction rate and to determine the stability and the extent of degradation in an electrochemical device. In this dissertation, a physics-based analytical model is developed to study the EIS of Li-air batteries, in which the mass transport inside the cathode is limited by oxygen diffusion, during charge and discharge. The model takes into consideration the effects of double layer, Faradaic processes, and oxygen diffusion in the cathode, but neglects the effects of anode, separator, conductivity of the deposit layer, and Li-ion transport. The analytical model predicts that the effects of Faradaic processes and oxygen diffusion can be masked by the double layer capacitance. Therefore, the dissertation focuses on the two cases: 1) when the Faradaic and the diffusion processes are separate and can are easily observed as two different semicircles on the Nyquist plot and 2) when the Faradaic and the diffusion processes are shadowed by the double layer capacitance and is observed on the Nyquist plot as a single combined semicircle. In order to extract physical parameters such as the diffusion coefficient of oxygen and Faradaic reaction rate a simple mathematical expression is developed. The diffusion coefficient is determined by using the value of the resistances (real impedance intercept on the Nyquist plot) of both the semicircles for the first case and by using the value of the combined resistance for the second case. Once, the effective oxygen diffusion coefficient is estimated, it can be used to estimate the value

xiii of the reaction constant. This method of extracting physical parameters of the cathode in a Li- air battery can serve as a tool in identifying an effective electrolyte and a cathode material. In addition, it can also serve as a noninvasive technique to identify and also quantify the use of the catalyst to improve the reaction kinetics in an electrochemical system. Finally, finite element simulations are used to validate the analytical model and to study the effects of discharge products on the total impedance of Li-air batteries with organic electrolyte. The finite element simulations are based on the theory of concentrated solutions and the complex impedance spectra are computed by linearizing the partial differential equations that describe the mass and charge transport in Li-air batteries. These equations include the oxygen diffusion equation, the Li drift-diffusion equation, and the electron conduction equation. The reaction at the anode and cathode are described by Butler-Volmer kinetics. The finite element simulation show that the total impedance of a Li-air battery increases by more than 200% when the impedance spectra is measured near the end of the discharge cycle as compared to on a fresh battery. The impedance spectra and the deposition profile of lithium peroxide is significantly affected by the resistivity of the deposition layer. The finite element simulations also show that using electrolytes with high oxygen solubility and concentrated O2 gas at high pressures will reduce the total impedance of Li-air batteries.

xiv CHAPTER 1

INTRODUCTION

1.1 Electrochemical impedance spectroscopy

Electrochemical impedance spectroscopy (EIS) is a versatile measurement tool that measures the dynamic response of a system to a spectrum of sinusoidal a.c. perturbation signals, and the impedance of the system is calculated as the ratio of the small signal voltage over small-signal current. EIS can be performed on a battery during discharge (in-situ) as well as at different stages of discharge (ex-situ). One of the salient features of impedance spectroscopy is that it can measure the impedance response of a system non-destructively. Different physical processes, such as ion diffusion, intercalation diffusion, electrochemical reactions, and electrical double layer, have different time constants and can be identified easily using the impedance spectra easily. EIS is also an analytical tool that can provide information such as state-of-charge [1–5], state-of-health [1, 3], degradation of electrode materials [2, 6–8], electrode reaction [2, 9], morphological or geometrical changes inside the cathode [10], mass-transport processes [11], and intercalation [12] in a battery. EIS can also be used to extract quantitative information about the electrochemical system such as the values of diffusion coefficient [7, 13, 14] and reaction rates [14, 15]. In order to extract such information one needs to develop a theoretical model that describes the electrochemical system comprehensively by considering the material, geometrical and electrochemical properties, as well as the physical processes in a battery. The common modeling techniques include equivalent circuit modeling [8, 11, 16–19], transmission line modeling [20, 21], and phenomenological modeling [14,15,22,23]. Among these techniques, phenomenological models can provide important insights about the physical and electrochemical processes that take place during charge and discharge, thus, can help understanding the operation of the system most pre- cisely. The other two techniques are computationally much faster but have a number of drawbacks and limitations. In particular, they are usually based on curve fitting and sometimes have a pure mathematical and non-physical foundation. EIS measurements are usually represented using either

1 (a) 10 mHz (b) 30 300 300

0 ° ] ] Ω 200 Ω 200 |Z| [ |Z| -Z'' [ -Z'' -30 100 1 MHz 100 [ (Z) ] Arg

0 -60 0 100 200 300 10-2 0 102 104 106 Z' [Ω] Frequency [Hz]

Figure 1.1: Examples for representing the data of the impedance spectra: (a) A typical Nyquist plot for a Li-air battery with the real value of the impedance on the x-axis and the imaginary value of the impedance on the y-axis. (b) A typical Bode plot with the absolute value and the phase angle of the impedance are plotted for diﬀerent values of frequency corresponding to the Nyquist plot represented in (a).

Nyquist plot or Bode plot. A Nyquist diagram is a plot of real (abscissa) and imaginary (ordinate) impedance at different measurement frequencies. A typical Nyquist plot for a Li-air battery is shown in figure 1.1a, where the semicircle and the asymptotic line represent two different physical processes. The X-intercepts show the values of the different resistances of the Li-air battery. A typical Bode plot for the same impedance response is shown in figure 1.1b.

1.1.1 Common EIS measuring techniques

The a.c. response of a system can be measured using a number of techniques such as galvanostatic, potentiostatic, wavelet, Fourier transform, cyclic voltammetry at diﬀerent frequencies, and demodulation of a noisy signal into its frequency components [24]. The most common measuring methods that are used in literature and deployed by instrument manufacture companies are galvanostatic and potentiostatic. One of the reasons for their widespread acceptance is that they can be applied to a large number of electrochemical systems.

Potentiostatic. In a potentiostatic conﬁguration (ﬁgure 1.2), a superposition of a small- signal a.c. voltage (δv) and a d.c. voltage bias (v) is applied to the system under test and it’s response is measured. The impedance of the system is given as the small-signal voltage divided by the small-signal current (δv/δi). The applied small-signal voltage is usually limited to less than

2 Battery

δiac δvac

Figure 1.2: Network for measuring the impedance spectra of a battery using the potentiostatic measuring technique.

10 mV to meet the linearity condition for the impedance response (i.e. to be much smaller than the thermal voltage).

Galvanostatic. In a galvanostatic conﬁguration (ﬁgure 1.3), a small-signal a.c. current (δi) is superimposed on a d.c. current (i) and applied to the system being measured. The impedance of the system is given by the small-signal voltage divided by the small-signal current (δv/δi). The applied small-signal current is limited to relatively low magnitudes, which can be estimated by imposing the small-signal voltage response to be less than 10 mV.

1.2 Modeling of EIS spectra in electrochemical systems

EIS is widely used in electrochemical measurements because it has high sensitivity and has the ability to separate the effects of different processes. The ability to decouple different electrochemical

Battery

δvac δiac δiac

Figure 1.3: Network for measuring the impedance spectra of a battery using the galvanostatic measuring technique.

3 processes depends upon the accuracy and completeness of the model that is used to describe the system under study. The EIS models can be classiﬁed into physical and mathematical models. A physical model can be used to predict the response of the system by taking into consideration the underlying physical phenomena. On the other hand, a mathematical model can be used to ﬁt the experimental data to a phenomenological equation and often has a limited or no physical interpretation.

1.2.1 Phenomenological modeling

Transport and reaction kinetics in electrochemical systems.

Mass and charge transport equations. The movement of ions and other molecules in a liquid electrolyte can be described using the mass and charge transport equations. The concentration of the gas in the electrolyte is directly proportional to the partial pressure of the gas just outside the electrolyte, which can be expressed mathematically as (Henry’s law)

c = p k (1.1) × H where c is the concentration in the electrolyte, p is the partial pressure of the gas above the electrolyte, and kH is the proportionality constant and is known as Henry’s law constant. The Henry’s law constant depends on the electrolyte and the temperature of the solution. The movement of ions or molecules in a porous medium can described by the following Poisson- Nernst-Planck (PNP) equation

zi Ji = Deff,i ci Deff,ici φ + civ (1.2) − ∇ − VT ∇ where Ji is the flux vector of species i, is the gradient operator, D is the effective diffusion ∇ eff,i coefficient, c is the concentration gradient, φ is the electric field, V is the thermal voltage, v is ∇ i ∇ T the velocity vector, zi and ci are the charge and concentration of species i, respectively. The Poisson- Nernst-Planck equation contains three terms: diffusion, migration, and convection. The diffusion is due to the concentration gradient, the migration is due to the influence of the electric field, and the convection can be forced (laminar or turbulent flow, etc.) or natural (density gradients). The effective diffusion coefficient in the porous medium can be related to the diffusion coefficient in the bulk by the following expression [25–27] ǫ D = D 0 (1.3) eff,i i τ

4 where τ is the tortuosity. In the case of neutral species and in the absence of convection the PNP equation reduces to Fick’s ﬁrst law of diﬀusion

Ji = D c (1.4) − eﬀ,i∇ i By applying the continuity equation one obtains (Fick’s second law of diﬀusion)

∂c i = D 2c (1.5) ∂t eﬀ,i∇ i

Electrode kinetics. The charge transfer reaction taking place on the surface of the electrode governs the electrode kinetics.The charge transfer process can be modeled using the Butler-Volmer model, which is given by

−βn (1−β)n co(0,t) η cr(0,t) η i = i e VT e VT (1.6) F 0 · c − c o∗ r∗ where i0 is the exchange current, η denotes over-potential, n is the stoichiometric number of electrons consumed, VT is the thermal voltage, and β is the charge transfer coeﬃcient (it describes the shape of the energy barrier between the initial and ﬁnal states of the reaction, for symmetric barrier is β = 0.5). The exchange current is given by [28]

i0 = nF Ak0 (1.7) where F is the Faradaic constant, A is the area and k0 is the standard rate constant. The over- potential (η) is given by η = E E − eq

E is the actual electrode potential and Eeq is the equilibrium potential.

Eﬀect of double layer. In electrochemical systems the current can be divided into two parts: Faradaic and double layer current (ﬁgure 1.4). The Faradaic current is the current generated due to the electrochemical reaction taking place at the surface of the electrode and can be described by the Butler-Volmer equation (1.6). The double layer current is the current consumed to charge the layer formed at the solid/liquid or electrode/electrolyte interface.

5 i dl C i D i F Z F

Figure 1.4: The total value of the discharge current can be separated into two parts: charge transfer current and the double layer current.

Mathematically, the total current is the algebraic sum of the double layer current and Faradaic current and is given by dη i = i C (1.8) F − D dt

Here, i is the total current, iF is the Faradaic current, CD is the double layer capacitance, and η is the over-potential. The total impedance (Z) can be obtained by linearizing eqn. (1.8),

δi = δi δi = δi jωC δη (1.9) F − dl F − D where, δi is the small-signal total current, δiF is the small-signal Faradaic current, idl is the double layer current, j is the imaginary number (√ 1), ω is the angular frequency, and δη is the small- − signal over-potential. The above equation leads to the following expression for Z

δη Z = − δi δη = −δi jωC δη F − D δη/δi = F −1 jωC δη/δi − D F Z = F 1+ jωCDZF 1 = ZF kjωCD

The ﬁnal expression for the total impedance is

1 Z = ZF (1.10) kjωCD

6 Electrolyte φm φ1 φ2

Diffuse Metal Layer

x=0 x Surface IHP OHP

Figure 1.5: The structure of a Grahame’s double layer with the inner and outer Helmholtz plane near the electrode surface and the diﬀuse layer near the bulk region. Here ϕm, ϕ1 and ϕ2 are the potentials at the metal surface, inner and outer Helmholtz plane respectively.

where ZF is the Faradaic impedance. The model of the double layer has evolved over time. In 1885 Hermann von Helmholtz suggested a double layer model, which was similar to that of an ideal capacitor model. However, later in 1947, D.C Grahame’s experiments proved that the behavior of EDL was much more complicated than the ideal capacitor model. The experiments showed that the capacitance depends on the electrical potential and the concentration of the electrolyte. Gouy (1909) and Chapman (1913) independently suggested a modified model to overcome this shortcomings of the Helmholtz model. In 1924, Otto Stern proposed combining the Gouy-Chapman model with the Helmholtz model. Stern’s model accounted for the finite size of the ions and their distance from the electrode surface. D.C. Grahame (1947) modified the Stern’s model to include the variation in hydration. Grahame’s three layer model is shown in figure 1.5. The three layers are: 1) the inner Helmholtz plane (IHP), which is the locus of centers of all specifically adsorbed ions on the electrode surface, 2) the outer Helmholtz plane (OHP), which is limited by the closest distance the solvated ions can come to the electrode surface [29] and the ions interact with the surface using long range electrostatic forces, and 3) the diffuse layer, which

7 extends from the OHP to the bulk region, where the anions and cations are distributed according to Poisson and Boltzmann distribution [29,30].

1.2.2 Equivalent circuit modeling

Equivalent circuit modeling (ECM) is commonly used to describe electrochemical systems [31]. The simplicity of this technique lies in the fact that it uses elementary circuit elements such as resistors, inductors, and constant phase elements (CPE) to obtain the impedance response of the α battery. The constant phase element is mathematically given as ZCPE = 1/ (jω) Q, where α is the model parameter related to different time constants and Q is a capacitance-like term [32, 33]. However, there are many different physical interpretations for introducing factor α. This factor is generally attributed to the spatial distribution of physical properties such as structure, reactivity, dielectric constants, and resistivity [33] and to non-homogeneities in the system [34]. The values of the components in the circuit are found by fitting the experimental measurements of the EIS spectra [31] to the impedance network. Different sections of the circuit are attributed to different physical or electrochemical processes. An important requirement for building an equivalent circuit model is to have a priori knowledge of the physical processes taking place in the system. In the absence of this prior knowledge many different configurations can yield identical impedance response and thus making the predictions made using this model non-physical and non-practical. ECM modeling has already been used to provide additional information about Li-air batteries. For instance, ECM have been used to characterize the state-of-charge of Li-air batteries [35], analyze the performance of novel membrane materials [36, 37], and study Li-air batteries with aqueous electrolyte [38, 39].

1.3 Li-air batteries (LABs)

The modeling techniques discussed in the previous section can be applied to batteries with different chemistries. Among the metal-air batteries, Li-air batteries (LABs) with organic electrolyte have the highest speciﬁc energy density [40–42]. These batteries can store approximately 10 times more energy than conventional Li-ion batteries [43,44] and their theoretical energy density is com- parable to the theoretical energy density of gasoline [45]. The salient features of Li-air batteries

8 are: high open circuit voltage (2.96 V for Li2O2) and high speciﬁc energy (5200 Wh/kg inclusive of oxygen [46–48] and 11140 Wh/kg excluding oxygen [46, 49]). These highly desirable features of LABs is the primary driving force for companies, governments, and universities to invest their resources in their research and development. The main electrochemical reactions in an organic Li-air battery are

+ Li Li +e− (anode) (1.11) ←−→ + 2 Li +O +2e− Li O (deposit) (1.12) 2 ←−→ 2 2 + Li +O +e− LiO (deposit) (1.13) 2 −−→ 2

A typical Li-air battery consists of a closed anode and a porous cathode that is open to the atmosphere, which allows the oxygen from the atmosphere to enter battery, a Li+ conducting separator and an electrolyte that transports the Li-ions between the two electrodes (ﬁgure 1.6). During discharge, the lithium metal oxidizes to produce Li+ ions and electrons (eqn. 1.11); the Li-ions diﬀuse towards the cathode due of electrochemical potential gradient whereas the electrons

Load e- Anode Cathode Protective Layer

Li+

Conductive carbon Anode Solid O2 (air) (Li) separator Electrolyte

Resistive deposited Li2O2 layer

x l 0

Figure 1.6: A schematic of a typical Li-air battery with oxygen entering the system from x = 0.

9 flow through the external circuit. The oxygen from the atmosphere is reduced at the cathode to form lithium peroxide (eqn. 1.12) [19, 40, 43, 50–52] and lithium superoxide (eqn. 1.13) [44, 53]. These discharge products deposit on the cathode surface until they fill the pores and block the oxygen from reaching the cathode surface [54] or until the resistance of these batteries increases up to a point after which the electrochemical reaction can no longer be sustained [55, 56]. Despite high specific energy densities and high specific capacities the LABs have a number of limitations that has obstructed their large scale commercialization. Important limitations are low practical specific energy [57, 58], low power density [58–62], poor cycle life [58–70], poor charge efficiency [57, 59, 60, 66, 68, 69], and reduced rate capability [69].

1.4 Organization of thesis

In chapter 2, a physics based model is developed to compute the impedance spectra of Li-air batteries with organic electrolyte under high d.c discharge currents. Compact analytical equations are derived for the impedance of these batteries separately for the cases of high discharge current and low discharge currents. In chapter 3, a physics based model is developed for the impedance spectra in Li-air batteries during charge. In chapter 4, a model is developed to relate the Nyquist plots of Li-air batteries to the physical and electrochemical parameters of the cathode, and then the model is verified against experimental data published in the literature. In chapter 5, the initial model developed in chapter 2 is extended to include the effects of oxygen dissolution in the electrolyte. In chapter 6, finite element simulations are performed to validate the analytical model and to study various effects of electrochemical and physical parameters of the battery on the total impedance of Li-air batteries. Finally, conclusions are drawn in chapter 7.

10 CHAPTER 2

EIS OF LI-AIR BATTERIES UNDER D.C. DISCHARGE

2.1 Introduction

The power density of Li-air batteries is limited by oxygen diffusion and oxygen solubility in the electrolyte [40,54,71–74]. Limited availability of oxygen in the organic electrolyte can lead to non- uniform deposition of the reaction products and partial utilization of the electrode volume [14,75]. 2 The non-uniform deposition is observed at high discharge currents (idc > 0.5 mA/cm ), when the rate of consumption of oxygen is greater than the rate of oxygen dissolving in the electrolyte. In recent years, a number of authors have observed that the diffusion impedance on the Nyquist plot of Li-air batteries with organic electrolyte deviates from the traditional Warburg and Gerischer impedances, particularly, when the diffusion impedance is measured under d.c. discharge. The impedance response of Li-air batteries with organic electrolyte shows three semicircles under d.c. discharge: 1) one semicircle at high frequencies, which originates from the anode [55,76] (this semicircle will be neglected in our analysis since the focus is mostly on the middle and low frequencies), 2) one semicircle at middle-range frequencies (in the order of 1 Hz), which is due to the double layer formed on the metal/electrolyte interface in the cathode, and 3) another semicircle at low frequencies due to the charge transfer kinetics and oxygen diffusion. In this chapter, expressions for the Faradaic impedance for porous electrode under d.c. discharge under high discharge currents and low discharge currents are derived. The final equations for the Faradaic impedances are functions of the reaction rate, effective diffusion coefficient of oxygen, concentration profile of oxygen in the cathode, tortuosity of the cathode, discharge current, and the chemical pathway. This chapter is organized as follows: First, the assumptions under which the expression for the final Faradaic impedance is derived are stated. Then, expressions for the discharge current under steady-state conditions are derived, followed by the small-signal analysis under for large and small values of the d.c. discharge current; in the analysis, two separate cases

11 (a) (b) air/O2

Carbon - e- ox e Carbon e- electrolyte red ox red Carbon red ox Electrode Separator Carbon x=0 x→∞

Figure 2.1: Non-distributed (a) and distributed (porous) (b) cathode for β = 0.5 and β = 0.5 are considered. Finally, the expression for the total impedance of Li-air 6 batteries under high d.c. discharge currents is shown.

2.2 Assumptions

In order to derive the the analytical solution for the impedance spectra of LABs the following assumptions are made:

1. the spectra is measured on a fresh and fully charged battery (i.e. the porosity is uniform throughout the cathode)

2. there is a uniform deposition of the discharge product during the time the EIS measurements is performed,

3. the separator is impervious to oxygen (this is a reasonable assumption since the separator is often built to protect the lithium metal from oxygen).

4. the conductivity of the resistive is neglected

These assumptions are reasonable for impedance measurements performed on fresh batteries and when the d.c. discharge currents do not lead to non-uniform deposition.

2.3 Modeling

The anode of LABs consists of a metal plate, which can be modeled using the system shown in fig. 2.1a and the cathode consists of a conductive porous-material, which can be modeled by distributing the reaction rate throughout the cathode (a simplified version of the distributed cathode is shown in fig. 2.1b). The electrochemical reaction at the anode is much faster than the reaction in the distributed cathode.

12 2.3.1 The case of high d.c. discharge currents

The Faradaic current for distributed electrodes is sum of currents at accessible reaction sites and is given by l

iF = A Rc(x,t) dx (2.1) Z0 where iF is the Faradaic current, A is the cross-sectional area of the cathode, l is the length of the cathode, x is the position inside the cathode, t is the time, and Rc is the speciﬁc reaction rate at the cathode [50]. The reaction rate at large discharge currents can be described using

Rc(x,t)= nFkco(x,t) (2.2) where, n is the number of electrons transferred during charge transfer, F is the Faraday’s constant, k is the reaction rate constant, and co(x,t) is the distribution of oxygen concentration in the cathode. When η < 0 and η V (where V is the thermal voltage), the reaction rate can be described ≫ T T using the Tafel relation −nβ V η k = k0 a(ǫ) e T (2.3) where k0 is the standard reaction rate constant, a(ǫ) is the specific surface area of the cathode, η is the over-potential (the cell voltage), and β is the symmetry factor of the electrochemical reaction (sometimes it is also referred to as charge transfer coefficient and is denoted by α). The specific area is given using the following relation

a(ǫ)= ǫ/rp (2.4) where ǫ is the porosity of the cathode and rp is the average size of the pores in the cathode, which is given by 1 ǫ d r = r (2.5) p p, 0 ǫ 0 where ǫ0 is the initial porosity, rp, 0 is the initial average pore radius, and d is the coeﬃcient (2 for cylindrical pores and 3 for spherical pores). The pores are assumed to be cylindrical in shape, hence d = 2 throughout this chapter or unless speciﬁed. In order to derive closed-form equations for the impedance spectra, the porosity is assumed to vary slowly in time during discharge and this variation can be neglected during the time the a.c. impedance measurement is performed. Indeed,

13 in Li-air batteries with organic electrolyte the porosity decreases slowly in time when the reaction product deposits on the surface of the carbon. It takes a very short time to measure the impedance response of a battery as compared to the lifetime of the battery. In addition, it is convenient to assume that the porosity of the battery is uniformly constant throughout the cathode, which is usually the case when the battery is completely charged or is partially discharged at a very low discharge rate. The over-potential is related to cell voltage using

V = V + η V (2.6) 0 − Ω where V is the cell voltage, V0 is the open cell voltage, VΩ represents voltage drop across separator, anode, deposit layers, interfaces, electrolyte and metal contacts, and η is the over-potential. In general, the ohmic voltage drop VΩ depends on the value of the discharge current. For instance, if the deposit layer at the cathode is an insulator, the electrons have to tunnel through the deposit layer and the total current will depend nonlinearly on the over-potential as suggested by a number authors [58,77,78]. Quantum tunneling results in a strong nonlinear-dependence of VΩ as a function of the discharge current.

Substituting Rc from eqn. 2.2 into eqn. 2.1

iF = nAF k co (x,t) dx (2.7) Z0 The Faradaic current under steady-state conditions can be derived using the above expression.

The concentration of oxygen (co (x,t)) in the porous electrode can be derived using the modified Fick’s second law of diffusion. In Li-air batteries, during discharge, the oxygen and lithium ions are consumed and the reaction products such as lithium peroxide, lithium superoxide, and etc. are formed; these discharge products deposit inside the cathode and reduce the total surface area of the electrochemical reaction. The consumption of oxygen is taken into account by adding an additional term to Fick’s second law of diffusion, as described by the following equation ∂c (x,t) d2c (x,t) ǫ o = D o kc (x,t) (2.8) 0 ∂t eff dx2 − o replacing the value of k from eqn. 2.3

2 nβ ∂co (x,t) d co (x,t) η ǫ = D k a(ǫ)e− VT c (x,t) (2.9) 0 ∂t eﬀ dx2 − 0 o

14 where Deff is the effective diffusion coefficient of oxygen in the porous electrode. The effective diffusion coefficient is calculated using the Bruggeman relation

brugg Deff = ǫ Do (2.10) where, brugg is the Bruggeman constant and Do is the diffusion coefficient of the bulk. The value of brugg is taken to be 1.5 throughout the thesis (unless specified). In this section, the porosity of the cathode is considered to remain constant ǫ(x)= ǫ0, since the porosity does not change significantly during an EIS measurement. The modified diffusion equation 2.9 is subject to following initial and boundary conditions:

Initial condition

co (x, 0) = co∗

Boundary condition at x = 0

co (0,t)= co∗ (2.11a)

Boundary condition at x = l dc (x,t) o = 0 (2.11b) dx x=l

Steady-state analysis. Let us consider steady-state of the electrochemical system, i.e. time derivatives in eqn. 2.9 can be neglected, although, the perfect steady-state can never be reached, this approximation is valid if the discharge currents are not high. The oxygen diﬀusion equation at the steady state is 2 d co (x) kco (x) 2 = 0 (2.12) dx − Deﬀ The solution to the above equation is,

k k co (x)= C1 cosh x + C2 sinh x (2.13) sDeﬀ ! sDeﬀ !

The integration constants are obtained by applying to the boundary conditions 2.11a and 2.11b to eqn. 2.13

C1 = co∗ (2.14) k C2 = co∗ tanh l (2.15) − sDeﬀ !

15 The ﬁnal expression for the oxygen concentration inside the cathode under steady-state condition is obtained by solving eqns. 2.14, 2.15, and 2.13

k k k co (x)= co∗ cosh x tanh l sinh x (2.16) " sDeff ! − sDeff ! sDeff !# Using the above equation one can calculate the oxygen concentration at the two boundaries (air side and separator)

co (0) = co∗ (2.17a) k co (l)= co∗ sech l (2.17b) sDeﬀ ! The ﬁnal expression for the steady-state Faradaic current is obtained by substituting eqn. 2.16 into eqn. 2.7 and integrating

Deﬀ k iF = nAF kco∗ tanh l (2.18) r k sDeﬀ ! Under steady-state conditions, since, the current charging/discharging the double layer is zero, the total current is equal to the Faradaic current and, therefore, iF = i.

Small-signal analysis. The impedance spectra is computed by superposing a small-signal jωt perturbation of the Faradaic current (δiF e ) to the d.c. value of the discharge current (iF ) and computing the resulting variation in the cell voltage (δvejωt). Applying a small perturbation to jωt the discharge current results into an inﬁnitesimal variation of the oxygen concentration (δco e ), Faradaic and double layer currents, and over-potential (δηejωt). The total signal (voltage or current) is a summation of d.c. and a.c. parts of the signals and can be written as

V + δvejωt (2.19)

i + δiejωt (2.20)

jωt iF + δiF e (2.21)

jωt idl + δidl e (2.22)

η + δηejωt (2.23)

This measurement technique of superimposing a.c. signal over diﬀerent d.c. discharge currents is commonly referred to as a.c. voltammetry. The total impedance of the battery (using eqn. 2.6) is given by δv RΩδi δη δη δη Z = = − = RΩ = RΩ − δi δi − δiF + δidl − δiF + jωaLACdlδη

16 Z (ω)= R + Z (ω) Z (ω) (2.24) Ω F k dl where δη δη δk ZF (ω)= = (2.25) −δiF −δk δiF ∂V R = Ω (2.26) Ω − ∂i

CD = aLACdl (2.27) and 1 Zdl (ω)= (2.28) jωCD are the Faradaic component of the impedance, small-signal ohmic resistance, double layer capacitance, and the impedance due to the double layer at the electrode/electrolyte interface, respectively.

The negative sign in the deﬁnition of ZF and RΩ appears because any increase in the cell voltage results in a decrease of the cell current during the discharge of the cell. The Faradaic impedance δη δk δη can be computed using ZF = , where the ﬁrst ratio is obtained by linearizing eqn. 2.3, − δk δiF δk ∂k δk = ∂η δη and after substituting the result is

nβ η nβk a(ǫ)e− VT δk = − 0 δη (2.29) VT and the second ratio ∂k is obtained by substituting the small-signal perturbations into eqn. 2.7 ∂iF and solving the resulting expression

l jωt jωt iF + δiF = nAF (k + δke ) co (x)+ δco (x) e dx (2.30) Z 0 Expanding the above equation results in

l jωt jωt jωt jωt iF + δiF = nAF (kco (x)+ kδco (x) e + co (x) δke + δke δco (x) e ) dx Z0

Using eqn. 2.7 and neglecting δkδco (x) (since the value of their product is negligible with respect to the other terms), the above expression reduces to

l l

δiF = nAF k δco (x)+ nAF δk co (x) dx (2.31) Z0 Z0

17 The variation of the oxygen concentration can be computed by expanding eqn. 2.8 in power series and keeping only the ﬁrst-order terms

jωt 2 jωt ∂ co (x)+ δco e d co (x)+ δco e jωt jωt ǫ0 = Deﬀ 2 (k + δke ) co (x)+ δco (x) e (2.32) ∂t dx − where co (x) is the oxygen concentration at steady-state, given by eqn. 2.16. The diﬀerential equation for the small-signal oxygen concentration is obtained by linearizing eqn. 2.32

2 d δco (x) (∆k) δkco (x) 2 δco (x)= (2.33) dx − Deﬀ Deﬀ where ∆k = k + jωǫ0. Substituting the expression for co (x) (eqn. 2.16) into the above equation results in

2 d δco (x) (∆k) δkco∗ k k k 2 δco (x)= cosh x tanh l sinh x (2.34) dx − Deff Deff " sDeff ! − sDeff ! sDeff !#

The above equation is subjected to the following boundary conditions, which are obtained by lin-

dδco(x,t) earizing eqns. 2.11a and 2.11b: (δco (0,t) = 0) and = 0 . By taking into consideration dx x=l this boundary conditions the solution of the above non-homogeneous diﬀerential equation is

δkco∗ ∆k k ∆k ∆k δco (x)= cosh x cosh x tanh l sinh x jωǫ0 " sDeff ! − sDeff ! − sDeff ! sDeff ! k k + tanh l sinh x (2.35) sDeff ! sDeff !#

This equation gives the variation of the oxygen concentration in the electrolyte during the impedance measurement. The small-signal Faradaic current is obtained by substituting eqns. 2.16 and 2.35 into eqn. 2.31,

l l nAF kδkco∗ ∆k nAF kδkco∗ k δiF = cosh x dx cosh x dx jωǫ0 sDeff ! − jωǫ0 sDeff ! Z0 Z0 l nAF kδkc ∆k ∆k o∗ tanh l sinh x dx − jωǫ0 sDeff ! sDeff ! Z0 l nAF kδkc k k + o∗ tanh l sinh x dx (2.36) jωǫ0 sDeff ! sDeff ! Z0

18 After integration and simpliﬁcations, δiF reduces to

tanh ∆k l δi i k√k Deff k F = F + 1 (2.37) q δk k jωǫ0√∆k tanh k l − jωǫ0  Deff  q  The unit of δiF is Coulomb. Substituting eqn. 2.37, ∆k = k + jωǫ = k 1+ jωǫ0 , and δη = VT δk 0 k − δk nβk (which can be obtained from eqn 2.29) into eqn. 2.25 we obtain the followingh finali equation for the Faradaic impedance under d.c. discharge using

Z ZTafel (ω)= 0 (2.38) F k jωǫ k√k tanh + 0 l Deff 1 k + r jωǫ0 k  − jωǫ0√k+jωǫ0 tanh l  Deff q   where VT Z0 = (2.39) nβiF Tafel and ZF (ω) indicates that the Faradaic impedance is computed using the Tafel approximation. The value of the over-potential (η) depends on the d.c. discharge current and, hence, on the corresponding reaction rate k (η). The new reaction rate is obtained by solving nonlinear equation 2.18 Deff k iF = nAF kco∗ tanh l r k sDeff ! The total impedance (eqn. 1.10) is the parallel combination of the Faradaic impedance (eqn. 2.38) and the double layer impedance Z = R + Z Z (2.40) Ω F k dl where the ohmic resistance (RΩ) is the combined resistance of the ohmic contacts, electrolyte, and kinetic processes at the anode.

2.3.2 The case of low d.c. discharge currents

In the last section, the expression derived to compute the Faradaic impedance for LABs is valid only for large discharge currents (but not too large to cause the non-uniform deposition of

Li2O2 in the cathode during the impedance measurement). Figure 2.2 shows the range of values of the discharge current and reaction rate constant over which the Tafel equation can be assumed to valid, when the value of the over-potential (η) is less than 26 mV or close to the thermal voltage

19 10-4 0.3

) T 0.2 10-5 < 3V 2 0.1 10-6 (0 < - -2 < 0 0 10-7 -0.1

-8 Charge Tafel approximation

10 [V] (over-potential)

breaks down -0.2 2 Tafel approx-

Reaction rate constant [cm/s] constant rate Reaction -9 10 imation holds -0.3 (-2 > 3V ) Discharge T 10-10 10-6 10-5 10-4 10-3 Faradaic current [A]

Figure 2.2: The range of values of the d.c. discharge current and the reaction rate constant over which the Tafel equation is assumed to be valid.

(VT = 26 mV at T=300 K) the oxidation part of the Butler-Volmer equation cannot be neglected.

For η > 3VT , the reaction rates computed using the Tafel equation and Butler-Volmer equation are identical (the diﬀerence between them is about 0.3122%). When the values of the d.c. discharge current are very small or the reaction at the electrode/electrolyte interface is sluggish the Tafel approximation can no longer predict the reaction rate accurately. Under these conditions, the Faradaic impedance needs to be derived using the Butler-Volmer (BV) equation instead of the Tafel approximation. In this section, the Faradaic impedance is derived using the BV equation. Since the Tafel equation is an approximate form the more general BV equation, the ﬁnal expression derived in this section for the impedance spectra is valid at both low and high discharge currents. The reaction rate in the framework of BV kinetics is

−βn (1−β)n BV η η k = k a e VT e VT (2.41) 0 −

20 where the ﬁrst term on the right side is the forward reaction rate, the second term on the right side is the reverse reaction rate, and the superscript ’BV’ indicates that the reaction rate is computed using the Butler-Volmer equation. Since the equations for the steady-state current (eqn. 2.18) and a.c. and d.c. oxygen concentrations (eqn. 2.35 and d.c. eqn. 2.16, respectively) depend on the reaction rate but are independent on the functional form of the reaction rate, these equations are not derived again but are listed for completion

BV BV Deff k iF = nAF k co∗ BV tanh l rk s Deff    kBV kBV kBV co (x)= co∗ cosh x tanh l sinh x  s Deff  − s Deff  s Deff 

   δη     The expression that needs to be recomputed is δkBV . Linearizing eqn. 2.41 results in

(1−β)n −βn V η V η BV (β 1) ne T βne T δk = k0a − δη (2.42)  VT − VT 

This expression can be simpliﬁed slightly by collecting the common terms and using eqn. 2.41

BV (1−β)n δk n BV η = βk + k ae VT (2.43) − δη V 0 T The Faradaic impedance is computed using δη δkBV ZF (ω)= BV −δk δiF BV where δk /δiF is given by eqn. 2.37

BV k +jωǫ0 BV BV tanh D l BV δiF iF k √k eff k =  + 1 BV BV BV q BV δk k jωǫ0 k + jωǫ0 tanh k l − jωǫ0  Deff   p q  BV   BV and δη/δk is given by eqn. 2.42. It is important to note that the expression for δk /δiF depends on the value of the reaction rate constant but is independent of the theory that is used to calculate the value of the reaction rate constant. The final expression for the Faradaic impedance using the Butler-Volmer formalism is

BV Z0 f (β,η) ZF (ω)= (2.44) BV BV √ BV k +jωǫ0 k k tanh D l kBV r eﬀ ! 1 jωǫ +  0 BV kBV  − jωǫ0√k +jωǫ0 tanh l Deﬀ  r   

21 where VT Z0 = (2.45) niF and kBV f (β,η)= (1−β)n BV V η βk + k0ae T The Faradaic impedances computed using the BV and Tafel theories can be related using eqns. 2.44 and 2.38 βkBV ZBV (ω)= ZTafel (ω) (2.46) F F (1−β)n BV V η βk + k0ae T If the reaction at the cathode is assumed to be symmetrical (β = 0.5) [14, 44, 58, 79] the Butler- Volmer equation given by eqn. 2.41 can be rewritten as

n k = 2k a sinh η (2.47) − 0 2V T This simpliﬁed form of the BV equation can accurately predict the values of the reaction rate for very low as well as high discharge currents (provided there is uniform deposition in the cathode). δη The expression for δk can be obtained after linearizing eqn. 2.47; one obtains nkBV (β=0.5) n δkBV (β=0.5) = coth η δη (2.48) 2V 2V T T BV (β=0.5) Substituting δη/δk and δk/δiF (this ratio is also independent of the functional form of the reaction rate) into eqn. 2.37, the ﬁnal expression for the Faradaic impedance using the Butler- Volmer kinetics when β = 0.5 is

n Z0 tanh η BV (β=0.5) 2VT ZF (ω)= (2.49) kBV+jωǫ0 kBV√kBV tanh l k Deﬀ BV r 1 jωǫ + 0 kBV  − jωǫ0√kBV+jωǫ0 tanh l  Deﬀ r   where 2VT Z0 = − niF The Faradaic impedance computed in the framework of BV and Tafel theories can be related using eqns. 2.49 and 2.38 n ZBV (β=0.5) (ω)= ZTafel (ω) 2β tanh − η F F × 2V T

22 2.4 Simulation results

Simulation results for the electrochemical impedance spectra of a few representative LABs with organic electrolyte are presented in this section. The impedance spectra of these batteries are computed using the following steps:

1. Solve non-linear eqn.2.18 to obtain the value of the reaction rate k

2. Compute the Faradaic impedance for the desired range of frequencies using eqn. 2.38 and k from step 1

3. Compute the total impedance using Z = ZF + Zdl

Figure 2.3 presents the effect of the oxygen concentration on the impedance spectra for two different values of the diffusion coefficient. For large values of the initial oxygen concentration (co∗), the impedance of the double layer capacitance dominates, which is clearly visible on the Nyquist plot; the impedance due to oxygen diffusion is negligible. As the concentration of oxygen outside the battery decreases, the impedance due to oxygen diffusion increases up to the concentration of c = 3.26 10 6 mol/cm3. It is important to note that the impedances due to oxygen diffusion (i.e. o∗ × − the low frequency semicircle) and the one due to charge transfer (i.e. the low-medium frequency

c* -5 3 * -5 3 o = 3.26 × 10 mol/cm c (a) o = 3.26 × 10 mol/cm (b) c* -6 3 c* -6 3 120 o = 6.52 × 10 mol/cm 120 o = 6.52 × 10 mol/cm -6 3 c* c* -6 3 o = 3.26 × 10 mol/cm o = 3.26 × 10 mol/cm c* -6 3 c* -6 3 100 o = 1.63 × 10 mol/cm 100 o = 1.63 × 10 mol/cm c* -7 3 c* -7 3 o = 3.26 × 10 mol/cm o = 3.26 × 10 mol/cm 80 80 ] ] Ω Ω 60 60 -Z" [ -Z" -Z" [ -Z"

40 40

20 20

0 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160 Z' [Ω] Z' [Ω]

Figure 2.3: The eﬀect of the initial oxygen concentration on the impedance plot for two values of the system length: l = 100 nm (a) and l = 10 µm (b). The other simulation parameters are, C = 4 µF/cm2, D = 7 10 6 cm2/s, i = 200 µA/cm2, k = 1.3 10 8 cm/s and n = 3 d o × − F 0 × −

23 140 140 -6 -6 k0 = 1.3 × 10 cm/s (a) k0 = 1.3 × 10 cm/s (b) -8 -8 120 k0 = 1.3 × 10 cm/s 120 k0 = 1.3 × 10 cm/s -9 -9 k0 = 6.5 × 10 cm/s k0 = 6.5 × 10 cm/s -9 100 -9 100 k0 = 1.3 × 10 cm/s k0 = 1.3 × 10 cm/s ] -10 ] -10 k0 = 1.3 × 10 cm/s k0 = 1.3 × 10 cm/s Ω 80 Ω 80 -Z'' [ -Z'' 60 [ -Z'' 60

40 40

20 20

0 50 100 150 200 0 50 100 150 200 Z' [Ω] Z' [Ω]

Figure 2.4: The eﬀect of the reaction rate on the impedance plot for two values of the stoichiometric number of electrons: n = 2 (a) and n = 3 (b).

semicircle) are distinguishable from each other up to this oxygen concentration. For small values of the oxygen concentration (less than 3.26 10 6 ), the semicircles due to charge transfer resistance × − and oxygen diffusion start to overlap due to the double layer capacitance. The large value of Cd makes it difficult to distinguish the charge transfer and diffusion impedances. Figure 2.4 shows the effect of the reaction rate constant on the impedance spectra. When only

Li2O2 is formed (n = 2), the reaction rate has no eﬀect on the Nyquist plot (ﬁg. 2.4a), but, if other products are also formed (n = 2), the change in reaction rate can easily be observed on the Nyquist 6

120 120 n = 1 n = 1 n (a) n (b) = 2 100 = 2 100 n = 2.2 n = 2.2 n = 3 n = 3 ] ] 80 n = 3.2 80 n = 3.2 Ω Ω 60 60 -Z'' [ -Z'' -Z'' [ -Z'' 40 40

20 20

0 50 100 150 200 0 50 100 150 200 Z' [Ω] Z' [Ω]

Figure 2.5: The effect of the stoichiometric number of electrons on the impedance plot for two values of the oxidant diffusion coefficient: D = 7 10 7 cm2/s (a) and D = 1.4 10 5 cm2/s (b). o × − o × −

24 Table 2.1: List of simulation parameters for ﬁgures 2.4 and 2.5

Parameter Fig. 2.4 Fig. 2.5 Units 6 7 3 co∗ 3.26 10− 3.26 10− mol/cm × × 2 Cd 40 4 µF/cm k variable plotted 1.3 10 7 cm/s 0 × − l 1 10 4 10 10 4 cm × − × − 2 (a) n variable plotted 3 (b) 2 iF 200 500 µA/cm 7 6 7 10− (a) 2 Do 7 10− × 5 cm /s × 1.4 10− (b) × plot (fig. 2.4b). The other simulation parameters are listed in table 2.1 The number of electrons consumed in the Faradaic reaction are an important indicator for type of Faradaic reactions (reversible or irreversible) taking place in the cathode. The type of the reaction product has a large effect on the number of semicircles and total resistance of the system (fig. 2.5). The EIS spectra during d.c. discharge could be used to identify the stoichiometric number and hence the number and type of side reactions. The total resistance and the radius of the semicircles depend significantly on the oxygen diffusion coefficient of the electrolyte. The other

140 140 2 2 iF = 200 μA/cm (a) iF = 200 μA/cm (b) 120 2 120 2 iF = 500 μA/cm iF = 500 μA/cm 2 2 iF = 1 mA/cm iF = 1 mA/cm 100 2 100 2 iF = 2 mA/cm iF = 2 mA/cm ] ] 2 2 i = 5 mA/cm iF = 5 mA/cm

F Ω Ω 80 80 -Z'' [ -Z'' -Z'' [ -Z'' 60 60

40 40

20 20

0 50 100 150 200 0 50 100 150 200 Z' [Ω] Z' [Ω]

Figure 2.6: The effect of the d.c. discharge current on the impedance plot for two values of the oxygen diffusion coefficient: D = 3.5 10 6 cm2/s (a) and D = 7 10 5 cm2/s (b). o × − o × −

25 fi.26)tettlrssac srdcdadfrdc icag curren discharge d.c. for and reduced is resistance total values the large for 2.6b) However, (fig. observed. are diffusion) lay oxygen double the to to one corresponding other (one small semicircles For distinct two electrolyte. 2.6a) the (fig. through of diffuse resistance can total molecules the oxygen and curves easily Nyquist the of shape The coefficient. iue27 h ffc fteoye iuinceceto h impedanc the on coefficient diffusion oxygen capacitance: the layer of double effect the The 2.7: Figure iuainprmtr r itdi al 2.1 table in listed are parameters simulation -Z'' [ ] 100 120 140 iue26sosteeeto ieetdshrecret o w val two for currents discharge different of effect the shows 2.6 Figure Ω 20 40 60 80 01010200 150 100 50 0 D D D D D o o o o o = 7× 10 = 7× 10 = 7× 10 = 7× 10 = 7× 10 al .:Ls fsmlto aaeesfrfiue . n 2.7 and 2.6 figures for parameters simulation of List 2.2: Table -8 -7 -6 -5 -4 Parameter cm cm cm cm cm 2 2 2 2 2 Z' [ D C /s /s /s /s /s i k c n l F o ∗ 0 d o Ω ] C d aibeplotted variable 4 = 3 7 . 3 5 1 40 . × . 26 i.2.6 Fig. × µ 3 10 F × × 10 × / 2 − 10 cm 10 − 10 5 (a) 6 − − (b) 2 − 6 (a) 9 6 a and (a) 10 26 − aibeplotted variable 4 40

4 -Z'' [ ] Ω 100 120 140 3 1 C 20 40 60 80 . × . 26 i.2.7 Fig. × 3 d 01010200 150 100 50 0 10 × 40 = 200 10 × 3 − 10 − 10 D D D D D 6 6 o o o o o − = 7× 10 = 7× 10 = 7× 10 = 7× 10 = 7× 10 (a) µ (b) − 7 F 7 / cm radcag rnfradthe and transfer charge and er 2 mol mol -8 -7 -6 -5 -4 µ h atr eed nhow on depends battery the cm cm cm cm cm cm (b). auso xgndiffusivity oxygen of values A Units cm e fteoye diffusion oxygen the of ues ssalrta mA 1 than smaller ts cm / ftedffso coefficient diffusion the of Z' [ / / 2 2 2 2 2 2 cm / cm cm /s /s /s /s /s ltfrtovle of values two for plot e / s s Ω 2 3 3 ] (b) / cm 2 lcrlt naL-i ytmrdcstescn eiicethat semicircle second second the diffusion; reduces oxygen system by Li-air f a limited first, in not electrolyte observed: is are system effects the three v diffusivity, following small the with 2.7a) cathodes For (fig. capacitance response. impedance the on capacitance layer h iuinsmcrl aihswihsget htteoye di oxygen system. the the that for suggests factor which vanishes semicircle diffusion the iue29 h ffc ftedul ae aaiac nteipdnepl impedance the on capacitance layer electrons: double of the number of stoichiometric effect The 2.9: Figure v two for plot impedance the on electrons: length system of the number of effect The 2.8: Figure -Z'' [ ] -Z'' [ ] 100 120 140 Ω 100 120 140 Ω h gr . hw h ffc fteoye iuinfrtodifferent two for diffusion oxygen the of effect the shows 2.7 figure The 20 40 60 80 20 40 60 80 01010200 150 100 50 0 01010200 150 100 50 0 C C C C l l l C l l = 1 = 100nm = 10nm = 10 = 100 d d d d d = 400 = 80 = 20 = 400nF/cm = 4mF/cm μ μ m μ m μ μ m μ F/cm F/cm F/cm 2 2 2 n 2 2 Z' [ Z' [ a and (a) 2 = Ω Ω ] ] n n (b). 3 = a and (a) 2 = (a) (a) 27

n -Z'' [ ] Ω 100 120 140 -Z'' [ ] 100 120 140 (b). 3 = Ω 20 40 60 80 20 40 60 80 01010200 150 100 50 0 01010200 150 100 50 0 C C C C C l l l l l = 1 = 100nm = 10nm = 10 = 100 d d d d d = 4mF/cm = 400 = 80 = 20 = 400nF/cm μ μ m orsod oipdneoffered impedance to corresponds μ m μ μ m μ F/cm F/cm steoye iuiiyof diffusivity oxygen the as , F/cm uini olne limiting a longer no is ffusion 2 2 2 2 2 Z' [ rlrevle foxygen of values large or Z' [ le ftestoichiometric the of alues le ftedul layer double the of alues tfrtovle fthe of values two for ot Ω Ω ] ] auso h double the of values (b) (b) Table 2.3: List of simulation parameters for figures 2.8 and 2.9

Parameter Fig. 2.8 Fig. 2.9 Units 7 6 3 co∗ 3.26 10− 3.26 10− mol/cm × × 2 Cd 4 variable plotted µF/cm k 1.3 10 8 1.3 10 9 cm/s 0 × − × − l variable plotted 10 10 4 cm × − 2.2 (a) n 2.2 3 (b) 2 iF 200 µA/cm 7 6 7 10− (a) 2 Do 7 10− × 6 cm /s × 3.5 10− (b) × by oxygen diffusion starts to appear, indicating that system is starting to be limited by oxygen diffusion; third, as the oxygen diffusion coefficient is further reduced, the two semicircles, one due to charge transfer and the double layer and the other due to oxygen diffusion, combines to form a single semicircle and thus the dominating factor (charge transfer or oxygen diffusion) cannot be easily identified. For cathodes with large values of the double layer capacitance (fig. 2.7b) only two effects are observed: first, for large values of oxygen diffusivity, the system is not limited by oxygen diffusion; second, as the oxygen diffusion coefficient is further reduced, the two semicircles

1500

10 µA/cm2 1000 Tafel (β = 0.5) ] 20 µA/cm2 Ω

-Z'' [ -Z'' 500 50 µA/cm2 BV (β = 0.5)

100 µA/cm2 0 500 1000 1500 2000 2500 Z' [Ω]

Figure 2.10: Comparison between the impedance spectra computed using Tafel kinetics (eqn. 2.38) and Butler-Volmer kinetics (eqn. 2.44) for diﬀerent values of d.c. discharge current density (i).

28 Table 2.4: List of simulation parameters for ﬁgures 2.10 and 2.11.

Parameter ﬁg. 2.10 ﬁg. 2.11 Units a 105 cm2/cm3 6 3 co∗ 2.55 10− mol/cm × 2 Cd 10 µF/cm 6 2 Do 7 10− cm /s × 2 iF variable plotted 10 µA/cm k 5 10 9 variable plotted cm/s 0 × − l 0.02 cm n 2 ǫ 0.75

combines to form a single semicircle. The other simulation parameters are listed in table 2.2 (unless specified otherwise). Figure 2.8 shows the effect of the cathode length on the impedance spectra for two different values of n. When the discharge product is Li2O2 (n = 2, fig. 2.8a) the length of the cathode has no significant effect on the impedance response, however, for large cathode widths (l 10 µm) another ≥ semicircle due to oxygen diffusion starts to appear. But for a system with higher stoichiometric number (fig. 2.8b) there is a direct relationship between the length of the cathode and the total

1500 Tafel (β = 0.5)

k = 2×10-9 cm/s 1000 0 ] Ω k = 5×10-9 cm/s 0 -Z'' [ 500 k = 1×10-8 cm/s 0

BV (β = 0.5) 0 0 500 1000 1500 2000 2500 3000 Z' [Ω]

Figure 2.11: Comparison between the impedance spectra computed using the Tafel kinetics (eqn. 2.38) and Butler-Volmer kinetics (eqn. 2.44) for diﬀerent values of standard reaction rate constant.

29 Table 2.5: List of simulation parameters for the comparison between the impedance spectra computed using Tafel kinetics (eqn. 2.38), Butler-Volmer kinetics with symmetric charge transfer (β = 0.5) (eqn. 2.49), and Butler-Volmer kinetics without the symmetric charge transfer limitation (eqn. 2.44)

Parameter ﬁg. 2.12 ﬁg. 2.13 Units a 105 cm2/cm3 6 3 co∗ 2.55 10− mol/cm × 2 Cd 10 µF/cm 6 2 Do 7 10− cm /s × 2 iF variable plotted 10 µA/cm k 5 10 9 variable plotted cm/s 0 × − l 0.02 cm n 2 ǫ 0.75

resistance. The other simulation parameters are listed in table 2.3. Figure 2.9 shows the effect of the cathode length on the impedance spectra for two different values of the double layer capacitance. The double layer capacitance affects the number of peaks, while the total resistance remains constant. Small values of the cathode capacitance could be used

BV 800 β = 0.3 Tafel BV (β = 0.5) 600 β = 0.4 ] Ω 400 β = 0.5 -Z'' [

200 β = 0.7 0 0 500 1000 1500 Z' [Ω]

Figure 2.12: Comparison between the impedance spectra computed using Tafel kinetics (eqn. 2.38), Butler-Volmer kinetics with symmetric charge transfer (β = 0.5) (eqn. 2.49), and Butler-Volmer kinetics without the symmetric charge transfer limitation (eqn. 2.44) for diﬀerent values of the charge transfer coeﬃcient (β).

30 1500 k = 1×10-10 cm/s 0 k = 5×10-9 cm/s 1000 0 ] Ω k = 1×10-8 cm/s 0 -Z'' [ 500 BV (β = 0.55) BV (β = 0.5) k = 5×10-8 cm/s Tafel (β = 0.55) 0 0 0 500 1000 1500 2000 2500 3000 Z' [Ω]

Figure 2.13: Comparison between the impedance spectra computed using Tafel kinetics (eqn. 2.38), Butler-Volmer kinetics with symmetric charge transfer (eqn. 2.49), and Butler-Volmer kinetics without the symmetric charge transfer limitation (eqn. 2.44) for diﬀerent values of the standard reaction rate constant k0 when β = 0.55.

to distinguish a system with low oxygen diffusivity (fig. 2.9a) from a system with higher oxygen diffusion (fig. 2.9b). The simulation parameters are listed in table 2.3 (unless specified otherwise). Figure 2.10 presents the comparison between the impedance spectra computed using eqn. 2.38 (symbol) and eqn. 2.49 (continuous line) for different values of discharge current (fig. 2.10a) and for different values of the reaction rate constant k0 (fig. 2.10b). The parameters that are used for the simulation are listed in table 2.4. Figure 2.12 presents the comparison between the impedance spectra computed using eqn. 2.38 (delta symbol), eqn. 2.49 (open circle symbol), and eqn. 2.44 (continuous line) for different values of charge transfer coefficient (β). Figure. 2.13 presents a comparison between the impedance spectra computed using eqn. 2.38 (delta symbol), eqn. 2.49 (open circle symbol), and eqn. 2.44 (continuous line) for different values of the reaction rate constant k0. The parameters that are used for the simulation are listed in table 2.5.

31 CHAPTER 3

EIS OF LI-AIR BATTERIES UNDER D.C. CHARGE AND DISCHARGE

The model developed in the previous chapter for the EIS spectra can be applied only for the d.c. discharge of Li-air batteries because the Butler-Volmer (BV) equation that is used to describe the kinetics at the cathode was valid only under d.c. discharge. In this chapter a slightly more elaborate expression for the reaction kinetics, which is valid under both d.c. charge and d.c. discharge is presented. First, we present the reaction model, then we proceed with the derivation of the oxygen concentration and Faradaic current at steady-state and, then, we derive the expression for the complex impedance. As a particular case we will recover the results obtained in the previous chapter under high d.c. discharge.

3.1 Introduction

The Butler-Volmer relation can describe reaction kinetics at the electrode/electrolyte interface for both charging and discharging. While charging, the discharge products in Li-air batteries such as Li2O2 decomposes into lithium ions and oxygen; the lithium ions migrate through the electrolyte and the separator, and deposit on the lithium anode; the oxygen diﬀuses through the electrolyte and is returned to the atmosphere.

+ 2 Li +O2 +2e− ↽⇀ Li2O2 (3.1) −−−−

If the rate of the electrochemical reaction at the cathode can be descried by BV eq. (2.41), eqns. 2.49 and 2.44 can be used to describe the EIS during both charge and discharge. However, if the rate of the electrochemical reaction at the cathode deviates from this functional form, eqns. 2.49 and 2.44 do not correctly predict the EIS of LABs. In practical batteries, the electrochemical reactions are often irreversible and the forward and reverse reactions proceed at different rates. For this reason, in this chapter, we consider the case of Butler-Volmer kinetics with different forward and reverse reaction rates and non-equal electron transfer factors. Specifically, we assume that the specific

32 reaction rate in the cathode is given as

R = nF kr kf c (x) (3.2) c − o h i where the reaction rate constant for the forward reaction is given by

−βn f η f VT k = ak0 e . (3.3)

f where a is the speciﬁc area of the cathode, k0 (measured in cm/s) is the standard reaction rate constant for the forward reaction, β is the electron transfer coeﬃcient (0 <β< 1), n is the number of electrons transferred in electrochemical reaction, and η is the over-potential (it is the potential drop between the electrode surface and the bulk layer). The reaction rate constant for the reverse reaction is given by (1−β)n r r V η k = ak0e T (3.4)

r 2 where k0 is the standard reverse reaction rate (measured in mol/(cm s)).

Under equilibrium conditions, Rc = 0, η = 0 and co (x)= co∗, the reverse reaction rate is related to the forward reaction rate by (using eqn. 3.2)

r f k = k co∗ (3.5)

During charge kf > 0 and kr > 0. During discharge kf < 0 and kr < 0.

3.2 Steady-state analysis

The mass transport of oxygen inside the cathode is described using Fick’s second law of diffusion. It is important to note that in this analysis the effects of convection are neglected. For this reason the model presented in this section cannot be applied to compute the impedance response of Li-air flow batteries, where the electrolyte is continuously flowing.

3.2.1 Oxygen concentration at steady-state

Since the oxygen in the electrolyte carries no charge the migration term in the Nernst-Plank equation can be neglected and the mass-transport of oxygen in Li-air batteries under steady-state conditions can be described by 2 ∂ co(x) Rc 2 = 0 ∂x − nFDeﬀ

33 Using Rc from eqn. 3.2, the mass transport equation becomes

2 f r ∂ co (x) k k 2 + co (x) = 0 (3.6) ∂x Deﬀ − Deﬀ

The homogenous solution to the above diﬀerential equation 3.6 is

kf kf coc (x)= C1 cos x + C2 sin x sDeﬀ  sDeﬀ      where C1 and C2 are the integration constants. Assuming the particular solution to be a constant

cop (x)= C eqn. 3.6 gives ∂2C kf kr 2 + C = 0 ∂x Deﬀ − Deﬀ The particular solution becomes kr c (x)= C = op kf and the complete solution for eqn. 3.6 is

kf kf kr co (x)= C1 cos x + C2 sin x + f (3.7) sDeﬀ  sDeﬀ  k     The integration constants are computed using the following boundary conditions:

Boundary condition at x = 0

co(0) = co∗ (3.8a)

Boundary condition at x = l dc (x) o = 0 (3.8b) dx x=l

The expressions for the integration constants (C1 and C2) are obtained by using the boundary conditions (eqns. 3.8a and 3.8b) and eqn. 3.7,

kr C = c∗ (3.9) 1 o − kf kr kf C = c∗ tan l (3.10) 2 o − kf  D  s eﬀ  

34 The ﬁnal expression for the oxygen concentration distribution inside the cathode while charging is obtained by using eqns. 3.7, 3.9, and 3.9.

kf kf kf kr co(x)= cchg cos x + tan l sin x + f (3.11)  sDeff  sDeff  sDeff  k        where kr c = c∗ (3.12) chg o − kf 3.2.2 Faradaic current at steady-state

The Faradaic component (iF ) of the discharge current can be computed using l iF = A Rc dx (3.13) Z0 Substituting Rc from eqn 3.2 into the above equation, we obtain l f r iF = A nF k co (x) k dx (3.14) − 0 − Z h i Substituting co (x) from eqn. 3.11 and separating the integrals

l f f f f k k k iF = nF Ak cchg cos x + tan l sin x dx (3.15) − 0  sDeff  sDeff  sDeff  Z Splitting the integral into two terms and computing the first term gives  l f f l f f Deff k k k iF = nF Ak cchg  f sin x + tan l sin x dx − r k  sDeff  sDeff  0 sDeff  0 Z          Similarly, computing the second integral and applying the limits, yields 

f f f Deff k k k iF = nF Ak cchg sin l + tan l 1 cos l − k  sDeff ! sDeff   − sDeff  r        After simplifying the above equation and using eqn. 3.12, the final expression for the steady-state Faradaic current is r f f k Deff k i = nF Ak c∗ tan l (3.16) F − o − kf kf  D  r s eff   This equations expresses the Faradaic current as a function of the forward and backward reaction rate constants, effective diffusion coefficient of oxygen in the electrolyte, length of the cathode, and oxygen concentration in the organic electrolyte.

35 3.2.3 Steady-state Faradaic current during discharge

The model presented in eqn. 3.16 can describe the steady-state Faradaic current during both charge and discharge. However, during d.c. discharge, both kr and kr are negative, and eq. 3.16 can be rewritten into a slightly diﬀerent from more appropriate for the numerical implementation. Indeed, using the relation tanh (jx)= j tan (x), where where j = √ 1, eq. 3.16 becomes −

Deﬀ k iF = nF Akco∗ tanh l r k sDeﬀ ! In this way, we have have recovered the expression for the Faradaic current obtained by using Tafel relation (eqn. 2.18).

3.3 Small-signal analysis

Now we assume that an external sinusoidal signal of small amplitude with angular frequency ω is superimposed on d.c. value of the d.c. current. In the limit of small perturbations the oxygen concentration, over-potential, forward reaction rate constant, reverse reaction rate constant, and the Faradaic current can be written as

V + δvejωt,

i + δiejωt,

jωt iF + δiF e ,

jωt idl + δidl e , and

η + δηejωt respectively. (3.18) where j = √ 1 and δc (x), δη, and δi are the amplitudes of the a.c. signal. The oxygen − o F concentration in the cathode can be described using Fick’s second law of diﬀusion,

∂c (x) ∂2c (x) R ǫ o = D o c 0 ∂t eﬀ ∂x2 − nF

36 Using eqn. 3.18, the above equation can be rewritten as

jωt 2 jωt ∂ co (x)+ δco (x) e ∂ co (x)+ δco (x) e ǫ0 = Deff 2 ∂t ∂x + kf + δkf ejωt c (x)+ δc (x) ejωt kr + δkr ejωt (3.19) o o − Using eqn. 3.6 and neglecting the term δk1δco(x) (since their product is negligible), the above equation reduces to 2 f r ∂ δco (x) k jωǫ0 co (x) f δk 2 + − δco (x)+ δk = 0 ∂x Deff Deff − Deff Rearranging the above equation yields 2 f r ∂ δco (x) ∆k δk co (x) f 2 + δco (x)= δk (3.20) ∂x Deff Deff − Deff where∆kf = kf jωǫ . Substituting the expression for oxygen concentration c (x) from eqn. 3.11 − 0 o into eqn. 3.20

kf kf kf kr 2 f cchg cos x + tan l sin x + f r ∂ δco (x) ∆k Deff Deff Deff k f δk 2 + δco (x)+ δk = 0 ∂x Deff h q Dqeff q i − Deff r where c = c k . Rearranging the above equation leads to chg o∗ − kf kf kf kf f 2 f r cchg cos x + tan l sin x δk r f ∂ δco (x) ∆k δk Deff Deff Deff k δk 2 + δco (x)= f ∂x Deff Deff − h q Dqeff q i − k Deff (3.21) The homogeneous solution of the above equation is

∆kf ∆kf δcoc (x)= C1 cos x + C2 sin x (3.22) s Deff  s Deff      where C1 and C2 are the integration constants and δcoc (x) denotes the complementary solution for the small-signal oxygen concentration. Assuming the particular solution for the differential eqn. 3.21 to be kf kf δcop (x)= C3 cos x + C4 sin x + C5 (3.23) sDeff  sDeff      where C3, C4, and C5 are the integration constants . The complete solution for eqn. 3.21 is

∆kf ∆kf kf kf δco (x)= C1 cos x + C2 sin x + C3 cos x + C4 sin x + C5 s Deff  s Deff  sDeff  sDeff         (3.24)

37 The integration constants C3, C4, and C5 are obtained by replacing δco(x) by δcop (x) in eqn.3.21 where δcop (x) is given by (eqn. 3.23). The left hand side (L.H.S) of equation 3.21 is

2 kf kf ∂ C3 cos D x + C4 sin D x + C5 L.H.S = eff eff h q ∂x2 q i ∆kf kf kf + C3 cos x + C4 sin x + C5 Deff  sDeff  sDeff         After taking derivatives and rearranging the above equation, one obtains

f f f ∆k jωǫ0C3 k jωǫ0C4 k L.H.S = C5 cos x sin x (3.25) Deff − Deff sDeff  − Deff sDeff      The right hand side (R.H.S) of eq. 3.21 becomes

kf kf kf r c cos x + tan l sin x δkf r f δk chg Deff Deff Deff k δk R.H.S = f Deff − h q Dqeff q i − k Deff

Rearranging the above equation one obtains

f kf f r r f f f cchgδk tan l f k δk k δk cchgδk k Deff k R.H.S = f− cos x sin x k Deff − Deff sDeff  − Deffq sDeff      Since L.H.S. should be equal to R.H.S. for all the values of variable x, one can write

jωǫ C c δkf 0 3 = chg − Deff − Deff f c δkf tan k l jωǫ C chg Deff 0 4 = − Deff − Deffq ∆kf kf δkr krδkf C5 = f− Deff k Deff which gives the following values for the integration constants

f cchgδk C3 = (3.26) jωǫ0

f f cchgδk k C4 = tan l (3.27) jωǫ0 sDeﬀ  kf δkr krδkf  C = − (3.28) 5 ∆kf kf

38 Substituting the integration constants (C3, C4, and C5) back into the particular equation (eqn. 3.23)

f f f f f r r f cchgδk k k k k δk k δk δcop (x)= tan l sin x + cos x + −f f (3.29) jωǫ0  sDeff  sDeff  sDeff  ∆k k        Substituting the particular solution given in eqn. 3.29 into eqn. 3.24 the small-signal oxygen concentration becomes

f f f f ∆k ∆k cchgδk k δco(x)= C1 cos x + C2 sin x + cos x s Deff  s Deff  jωǫ0 sDeff        f r r f f f f k δk k δk cchgδk k k + −f f + tan l sin x (3.30) ∆k k jωǫ0 sDeff  sDeff      Integration constants C1 and C2 are obtained by applying the following boundaries conditions

Boundary condition at x = 0 δc (x) = 0 (3.31a) o |x=0 Boundary condition at x = l dδc (x) o = 0 (3.31b) dx x=l

which can be obtained by linearizing eqns. 3.8a and 3.8b respectively.

Integration constant C1 is obtained by applying boundary condition 3.31a to eqn.3.30

f f r r f cchgδk k δk k δk δco(0) = C1 + + −f f jωǫ0 ∆k k

The expression for integration constant C1 is

c δkf kf δkr krδkf C = chg + − (3.32) 1 − jωǫ ∆kf kf 0

Integration constant C2 is obtained by applying boundary condition 3.31b to eqn. 3.30

f f f r r f f f d ∆k ∆k k δk k δk cchgδk k C1 cos x + C2 sin x + −f f + cos x dx  s Deff  s Deff  ∆k k jωǫ0 sDeff          c δkf kf kf + chg tan l sin x = 0 (3.33) jωǫ  D   D  0 s eff s eff  x=l    

 39 After taking derivatives, the above equation becomes

f f f f f f k cchgδk k k ∆k ∆k tan l cos x C1 sin x sDeff jωǫ0 sDeff  sDeff  − s Deff s Deff        kf c δkf kf ∆kf ∆kf chg sin x + C cos x = 0 (3.34) − D jωǫ  D  D 2  D  s eff 0 s eff s eff s eff x=l    

After simpliﬁcations, the ﬁnal expression for the integration constant C2 is

∆kf C2 = C1 tan l (3.35) s Deﬀ    Substituting C1from eqn. 3.32, constant C2 can be computed as

c δkf kf δkr krδkf ∆kf C = chg + − tan l (3.36) 2 − jωǫ ∆kf kf  D  0 s eﬀ   Substituting C1 and C2 from eqns. 3.32 and 3.36 into equation 3.30, the ﬁnal expression for the small-signal oxygen concentration while becomes

f f f f f cchgδk k cchgδk k k δco(x)= cos x + tan l sin x jωǫ0 sDeff  jωǫ0 sDeff  sDeff        c δkf kf δkr krδkf ∆kf kf δkr krδkf chg + − cos x + − − jωǫ ∆kf kf  D  ∆kf kf 0 s eff   c δkf kf δkr krδkf ∆kf ∆kf chg + − tan l sin x (3.37) − jωǫ ∆kf kf  D   D  0 s eff s eff     The small-signal Faradaic current is calculated by applying perturbation theory to eqn. 3.14

l i + δi ejωt = nF A kf + δkf ejωt c (x)+ δc (x) ejωt dx F F − o o Z0 l kr + δkr ejωt dx (3.38) − Z0 Using eqn. 3.14 and neglecting δkf δc (x) (since their product is negligible), the small-signal × o Faradaic current becomes

l l l δi = nF A kf δc (x) dx + nF A δkf c (x) dx nF A δkr dx (3.39) − F o o − Z0 Z0 Z0

40 Computing the last integral the above equation can be represented as

δi = int + int nF Aδkrl (3.40) − F 1 2 − where l f int1 = nF Ak δco (x) dx (3.41) Z0 and l f int2 = nF Aδk co (x) dx (3.42) Z0 Next, integrals int1 and int2 will be evaluated separately and reintroduced into 3.40. Substituting

δco (x) (from eqn. 3.37) into eqn. 3.41

f l f f r r f l int1 cchgδk k k δk k δk f = cos x dx + −f f dx nF Ak jωǫ0 0 sDeff  ∆k k 0 Z Z   f f r r f l f f l f cchgδk k δk k δk ∆k ∆k ∆k + −f f cos x dx + tan l sin x dx − jωǫ0 ∆k k  0 s Deff  s Deff  0 s Deff   Z Z         c δkf kf l kf + chg tan l sin x dx (3.43) jωǫ0 sDeff  0 sDeff  Z     After taking the integrals and applying the limits the expression for int1 becomes

f f f r r f f cchgδk Deff k k δk k δk int1 = nF Ak f tan l + −f f l  jωǫ0 r k sDeff  ∆k k    c δkf kf δkr krδkf D ∆kf chg + − eff tan l (3.44) − jωǫ ∆kf kf ∆kf  D  0 r s eff   Integral int2 can be evaluated by substituting eqn. 3.11 into eqn. 3.42

l f r l int2 k k f = cchg cos x dx + f dx nF Aδk 0 sDeff  k 0 Z Z   kf l kf + cchg tan l sin x dx (3.45) sDeff  0 sDeff  Z     The final expression for int2 after integrating and applying boundary conditions is

f f r f Deﬀ k nF Aδk k int2 = nF Aδk cchg f tan l + f l (3.46) r k sDeﬀ  k  

41 Substituting int1 and int2 from eqns. 3.44 and 3.46 into eqn.3.40 leads to the following expression for δiF

f f f r r f f cchgδk Deff k k δk k δk r δiF = nF Ak f tan l + −f f l nF Aδk l −  jωǫ0 r k sDeff  ∆k k −    c δkf kf δkr krδkf D ∆kf chg + − eff tan l − jωǫ ∆kf kf ∆kf  D  0 r s eff   f r f nF Aδk k f Deff k + f l + nF Aδk cchg f tan l (3.47) k r k sDeff    The terms δkf and δkr can be computed by linearizing eqs. 3.3 and 3.4 βnkf δkf = − δη (3.48) VT (1 β)nkr δkr = − δη (3.49) VT Substituting eqns. 3.48 and 3.49 into eqn. 3.47 and rearranging

f δiF f Deff k = nF Ak f tan l − δη r k sDeff  ×   l nkr kf kf βnkf c βnc kf kf nkrl cot l chg chg cot l V ∆kf D  D  − jωǫ V − V − D  D  kf V T s eff s eff 0 T T s eff s eff T

    ∆kf r f f tan l 1 nk βnk c k Deff chg (3.50) f f f  −VT ∆k − jωǫ0 ∆k q k r tan D l eff  q f  f Deff k Since iF = nF Ak cchg f tan l the last equation becomes − k Deff q q δi i l nkr kf kf βnkf c βnc F = F cot l chg chg δη c V ∆kf D  D  − jωǫ V − V chg T s eff s eff 0 T T

   ∆kf r f f tan l f f r 1 nk βnk c k Deff k k nk l chg cot l (3.51) f f f f  −VT ∆k − jωǫ0 ∆k q k − sDeff sDeff  k VT r tan D l eff  q   δη f f  Rearranging the above equation and using ZF (ω)= and ∆k = k jωǫ0, the final expression − δiF − for the Faradaic impedance becomes Z Z (ω)= 0 (3.52) F f (β,ω,η)

42 where VT Z0 = (3.53) nβiF and

f k jωǫ0 tan − l kr kf kf Deff f (β,ω,η)=1+ r q k f − jωǫ kf jωǫ kf " co∗ kf (k jωǫ0) β 0 # s 0 tan l − − − Deff q kf jωǫ krl kf kf + 0 cot l (3.54) f f kr jωǫ0 − "βk (k jωǫ0) co∗ f # sDeff sDeff  − − k   The eqn. 3.52 describes the Faradaic impedance of Li-air batteries under d.c. charge current.

3.3.1 The Faradaic impedance during discharge

Eqns. 3.52-3.54 can compute the Faradaic impedance during both charge and discharge. Since kf and kr are negative during d.c. discharge eq. 3.16 can be rewritten into a slightly diﬀerent from which is sometimes more appropriate for the numerical implementation. Using relations cot (jx)= j coth (x) and tan (jx)= j tanh (x), eq. 3.54 becomes −

kf +jωǫ tanh 0 l | D| eff kr kf kf r ! f (β,ω,η)=1+ kr f f "β co∗ f (k jωǫ0) − jωǫ0 # sk jωǫ0 kf − k − − tanh l |Deff| r ! kf jωǫ krl kf kf + 0 | | coth | | l (3.55) f f kr jωǫ0 − "βk (k jωǫ0) co∗ f # sDeff sDeff  − − k   The Faradaic impedance during discharge can be calculated using eqns. 3.52, 3.55, 3.53, and 3.17. By putting kr = 0 and kf = k into eqn. 3.52 the expression for the Faradaic impedance under − Tafel discharge given in eqn. 2.38 (ZF (ω)) is easily recovered

Z0 ZF = k jωǫ k√k tanh + 0 l Deﬀ 1 k + r jωǫ0 k  − jωǫ0√k+jωǫ0 tanh l  Deﬀ q   Similarly, the Faradaic impedances developed for small-values of the d.c. discharge current (eqns. 2.44and 2.49) can also be recovered easily.

43 CHAPTER 4

EXPERIMENTAL VERIFICATION AND PARAMETER EXTRACTION

4.1 Introduction

Electrochemical impedance spectroscopy can be used to extract the values of many physical parameters of electrochemical systems. In this chapter a technique to extract information about the reaction rate, specific surface area, ohmic dissipation, and effective diffusion coefficient in the cathode is introduced. Low frequency impedance spectra in Li-air batteries can be separated into two cases depending on whether the Faradaic impedance is masked or not by the double layer capacitance: (1) the case when the effects of the Faradaic impedance are separated from the effects of the double layer capacitance so the Nyquist plot presents two different semicircles corresponding to the cathode diffusion-reaction layer, and (2) the case when the effects of the Faradaic impedance are hidden by the effects of the double layer capacitance and the Nyquist plot presents only one large semicircle corresponding to the cathode diffusion-reaction region. The first technique is more accurate as it uses more experimental data from the Nyquist plot, however, if the specific area of the cathode is larger than 104 cm2/cm3 it might be impossible to identify the two semicircles, in which case one needs to use the second technique. In the following analysis only the low-frequency semicircle corresponding to the cathode is considered and all the other semicircles attributed to the anode or anode-separator interface are eliminated from the Nyquist plot.

4.2 Parameter extraction at low cathode speciﬁc areas

When the specific area of the cathode is relatively low the impedance spectra present two semicircles: one due to oxygen diffusion and the other to charge transfer, as shown in Fig. 4.1. To find equations for the effective oxygen diffusion coefficient and reaction rate it is instrumental to rewrite the Faradaic impedance obtained in chapter 2 into an alternate form

2 VT ωǫ0λ (iF ) l ZF (iF ,ω)= F , (4.1) nβiF × " Deﬀ λ (iF )#

44 where λ = Deff . The value of lambda depends on the d.c. value of the discharge current and kf \ can be computedq by solving the following nonlinear equation nF Ac D l i = o∗ eff tanh (4.2) F λ λ which is obtained from eqn. 2.18. Function F is complex and depends on the d.c. value of the discharge current and cathode length. It is defined as jΩ F (Ω,L)= (4.3) tanh(√1+jΩL) + jΩ 1 √1+jΩ tanh(L) − Function F (Ω,L) has the following mathematical properties: jΩ(1+ jΩ) jΩ jΩ lim F (Ω,L)= = 2 = (4.4) l 2 (jΩ) 1 1 1 √1+ jΩ 1+(jΩ) − + jΩ 1+ →∞ − 1+jΩ √1+jΩ − √1+jΩ jΩ(1+ jΩ) lim F (Ω,L) = lim = 1 (4.5) l 0 l 0 1+ jΩ 1+(jΩ)2 → → − lim F (Ω,L) = 1 (4.6) l →∞ 1 lim F (Ω,L)= [1, 2] (4.7) l 0 1 l ∈ → 2 + sinh(2l) Im[F (Ω,L)] 0 (4.8) ≤ The total impedance Z expressed in eqn. 2.24 can be rewritten into a normalized form by using eqn. 4.1 RF (ω/ω0, l/λ) Z = RΩ + (4.9) 1+ jωRCDF (ω/ω0, l/λ) where RΩ is the sum of parasitic resistances such as resistance of the electrolyte, electrical contacts, separator, anode-separator interface, solid electrolyte interface layer, etc., and V R = T (4.10) nβi

Deﬀ ω0 = 2 , (4.11) ǫ0λ Under steady-state conditions, the Faradaic steady-state current is equal to the total current at steady-state (iF = i). Using the mathematical properties of function F , the expressions for the impedance at low and high frequencies are : 2 sinh (2L) lim Z = RΩ + R (4.12) Ω 0 sinh (2L) + 2L →

45 -Z '' F R2 R1 ω

Z ' F RΩ=Z∞ RΩ+R2 Z0

Figure 4.1: Possible low frequency impedance spectra of Li-air batteries with low speciﬁc area of the cathode. The dashed continuation line at high frequencies show that the Nyquist spectra might contain other semicircles, which are usually due to the anode and anode-separator interface.

lim Z = RΩ (4.13) Ω →∞ Denoting the impedance at low frequency (ω 0) by Z and at very high frequencies (ω ) → 0 → ∞ by Z , equation 4.9 becomes ∞ F (ω/ω , l/λ) Z = Z + 0 (4.14) ∞ 2 sinh(2L) + jωC F (ω/ω , l/λ) (Z0 Z∞) sinh(2L)+2L D 0 −

The last equation needs to be fitted to the experimental impedance spectra to compute Z0, Z , ∞ CD, ω0, and λ. Once these parameters are determined the effective oxygen diffusion coefficient can be computed using the value of λ and solving the following equation

iλ l D = coth (4.15) eﬀ 2AF c λ o∗ If the value of the over-potential η is estimated using the d.c. measurements then the reaction rate f 1 (k is expressed in s− ) can be computed using:

nβ f Deﬀ η k a = e VT . (4.16) 0 λ2

It is important to note that most of the ﬁtting parameters in eqn. 4.14 can be determined easily from the experimental data. Indeed, Z0 and Z are the low and high frequencies impedances ∞ and can be read directly from the data, while parameter λ can be computed using the ratio of

46 the diameter of the low frequency semicircle to the diameter of the high frequency semicircle (see notations in Fig. 4.1), i.e.:

R 1 sinh (2L) 2L sinh (2l/λ) 2l/λ 1 = 1= − = − (4.17) R 1 L sinh (2L) + 2L sinh (2l/λ) + 2l/λ 2 2 + sinh(2L) − which can be derived from eqn. 4.7. The parameter determination technique can be summarized as follows: one first computes ratio l/λ by solving nonlinear eqn. 4.17, then the effective oxygen diffusion coefficient from 4.15 and the reaction rate from 4.16. The remaining two parameters, CD and ω0, can be identified using nonlinear least-square estimations or alternative numerical techniques.

4.3 Parameter extraction at high cathode speciﬁc areas

When the speciﬁc area of the cathode is relatively high the Faradaic component of the impedance is masked by the high capacitance of the double layer and the impedance spectrum of the cathode presents only one semicircle as shown in Fig. 4.2. In this case, eqns. 4.14-4.16 still hold, however, radii R1 and R2 cannot be identiﬁed using eqn. 4.17. In this case it is more convenient to determine resistance R12 from the experimental spectra and use the following equation

V 2 sinh (2L) V 2 sinh (2l/λ) R = T = T (4.18) 12 nβi sinh (2L) + 2L nβi sinh (2l/λ) + (2l/λ)

In this case, the ratio l/λ is computed by solving eqn. 4.18 and, then, the effective oxygen diffusion coefficient and the reaction rate are calculated using eqns. 4.15 and 4.15 respectively. The value of charge transfer coefficient (n) that is needed to compute l/λ from eqn. 4.18 can be computed either using discharge curves or by fitting the experimental and theoretical values of frequency ω0.

4.4 Experimental veriﬁcation

The validity of the analytical model for Li-air batteries under high d.c. discharge is tested by comparing the theoretical predictions (obtained by eqns. 2.18 and 2.38) to experimental data by Adams et al. [76], who measured the impedance spectra for rechargeable Li-air batteries with cathodes made using diﬀerent fabrication techniques: separate cast and dual cast cathodes with

47 -Z '' F R12 ω

Z ' F RΩ=Z∞ Z0

Figure 4.2: Possible low frequency impedance spectra of Li-air batteries with high speciﬁc area of the cathode. The dashed lines show that the Nyquist spectra might contain other semicircles at high frequencies, which are usually due to the anode and anode-separator interface.

soaked and embedded electrolytes, room temperature ionic liquid embedded cathodes, and PTFE- 2 calendered cathodes. The capacitance of the diffusion layer Cdl = 7 µF/cm and the specific area of the cathode a = 5 104 cm2/cm3 were measured experimentally. The large value of the specific × area of the cathode suggests that the Faradaic impedance is hidden by the double layer capacitance and this effect is indeed observed in the experimental results (fig. 4.3) that show only a single semicircle at the low frequency with the diameter equal to R12. The low and high frequency intercepts on fig. 4.3 provide the values of Z0 = 19 Ω and Z = 570 Ω from the experimental data. ∞ Solving eqn. 4.18 the value of L is computed as

L = l/λ = 1.96

For a cathode with l = 0.01 mm (such as the one used in Ref. [76]), eqn. 4.15 gives an effective value of the diffusion coefficient of D = 5.6 10 6 cm2/s. For a standard cathode with porosity eff × − ǫ = 75% and Bruggeman coefficient of 1.5, the bulk diffusion coefficient is D = 8.6 10 6 cm2/s, 0 o × − which is in good agreement with predictions by Read et al. [54] who obtained D = 7 10 6 cm2/s. o × − The theoretical predictions are compared to experimental data in Fig. 4.3. The frequency was varied between 1 mHz and 1 MHz in agreement with the experimental data. The low frequency points were not used in the parameter determination because, as mentioned by the authors, those

400 Analytical results Experimental results 300 ]

Ω Li (anode)-separator interface (cathode) [

200 R

Z'' 12 -

100

0 0 100 200 300 400 500 600 Z ' [ Ω]

Figure 4.3: Comparison between the theoretical and experimental impedance spectra. R12 is the diameter of the second semicircle along the real axis and can be used to determine the effective value of the oxygen diffusion coefficient and the reaction rate in the cathode using equations 4.15, 4.16, and 4.18.

points were subject to high experimental errors due to the long measurement time. Notice that the experimental spectra does not start with a slope of 45◦ at high frequencies like in the case of traditional Gerischer type impedance spectra (see appendix for more information) but at a much higher slope as predicted by the analytical model for high d.c. discharge current. The high frequency semicircle of the cell simulated in Fig. 4.3 is due to the anode-separator interface and is modeled with a constant phase element represented using 1/ (jω)α Q, where parameters α = 0.8 and Q = 8.2 10 3 are measured experimentally [76]. × − The analytical model predicts the experimental values of resistance R12 (which is denoted as

R2 in Ref. [76]) remarkably well. Indeed, at relatively large discharge currents, eqn. 4.2 predicts

2VT 2 large values for L, so R12 = nβi . Hence, for a d.c. discharge current of 0.1 mA/cm and n = 2, the predicted value of R is the 1 kΩ, which agrees very well with the value of 1 0.02 kΩ presented 12 ± 2 in Fig. 14 from Ref. [76]. At a discharge current of 0.6mAcm− , by keeping the same value of the charge transfer coeﬃcient n = 2, the predicted value of R12 comes out to be 166 Ω, which agrees relatively well with the value of (145 25 Ω) data from the same reference. The slight ± overestimate in the value of R12 is due to the fact that in reality n increases slightly with the value

49 Z F R Ω

Ohmic losses C (electrolyte,deposit D layer, etc.)

Figure 4.4: Small-signal equivalent circuit of Li-air batteries. ZF denotes the Faradaic impedance, CD is the capacitance of the double layer, RΩ is the combined resistance of the electrolyte, Li-ions, and electrons in the cathode matrix. of the discharge current. In addition, it is shown experimentally that at relatively large discharge currents the values of this resistance is independent of the cathode fabrication technique and oxygen concentration, in agreement with the analytical model (eqn. 4.18, which is independent of oxygen concentration, speciﬁc area, and tortuosity). At low discharge currents, the value of L become sensitive to the oxygen concentration and can vary from L = when the oxygen concentration is large to L = 0 when the oxygen concentration ∞ is very small. Hence, by decreasing the oxygen concentration one can double the value of R12 from R to 2R (as implied by 4.18 and 4.7). This eﬀect is clearly observed in the experimental data presented in Ref. [76], where R12 increases from 860 Ω to 1570 Ω when the oxygen concentration is decreased from 100% to 5%.

4.5 Equivalent circuit model (ECM)

In this section, a technique to extract electrochemical parameters of a Li-air battery using a small-signal equivalent circuit is developed. Using the ECM technique, the total impedance (Z) of the circuit can be represented as a resistor RΩ in series with a parallel combination of the double-layer capacitance (CD) with impedance the Faradaic impedance (ZF ) (ﬁg. 4.4 represents the equivalent circuit model) 1 Z = RΩ + ZF (4.19) jωCD k

where RΩ represents the combined resistance of the electrolyte, anode, and anode-separator interface, which might play a signiﬁcant role in Li-air batteries as suggested by Adams et al. [76] In

50 -Z '' F semicircle

0.47 R ω

Ωmax

Z ' R 1.52R 2R F

Figure 4.5: Nyquist plot of the Faradaic impedance at large discharge currents and cathode widths (i.e. l λ). ≫

such cases RΩ might be modeled with resistors in series with one or more parallel resistor-capacitor (or resistor-constant phase element) pairs connected in series. The circuit corresponding to model

4.19 is represented in Fig. 4.4. In this circuit, the Faradaic impedance ZF is given in terms of the geometric and material properties of the battery by using an alternate equation 4.1. This circuit can be compared to the standard Randles equivalent circuit [80]. Depending on the value of λ it is worthwhile investigating two special cases: one for large currents and cathode widths (l λ) and the other one for small currents and cathode widths ≫ (l λ). The condition l λ implies that ≪ ≫ il nFc∗D (4.20) A ≫ o eﬀ which can be obtained directly from 4.2. Using typical values for a Li-air battery with organic electrolyte, n = 2 [76], c = 3.26 10 6 mol/cm3 [50], D = 7 10 6 centim/s, and using the o∗ × − o × − brugg Bruggeman relation Deﬀ = ǫ Do, the condition 4.20 becomes

il 6 A 4.4 10− (4.21) A ≫ × cm This condition is often satisﬁed during the normal operation of Li-air batteries particularly when the discharge current and cathode widths are large enough. If both the cathode width and the discharge current are large enough so that condition 4.20 is satisﬁed, L 1 and the total Faradaic ≫ impedance can be approximated to

VT jΩ ZF (4.22) ≃ nβi jΩ 1+ 1 − √1+jΩ

51 ZF R R C RΩ

Figure 4.6: Approximate small-signal equivalent circuit of Li-air batteries operating at large discharge currents and with large cathode width (l λ). The values of R , C , R, and C can be ≫ Ω D expressed in terms of physical parameters using eqns. 2.26, 2.27, 4.10, and 4.28, respectively. This circuit should be used with care in practical applications and, instead, one should use the more general circuit from Fig. 4.4.

The last equation was derived using approximations 4.4-4.8. The Nyquist diagram of the Faradaic impedance is represented in Fig. 4.5 with continuous line. Notice that the real part of the impedance

VT 2VT extends from R = nβi at high frequencies to nβi at low frequencies and the absolute value of

1.521 VT the imaginary part has a maximum at Ωmax = 0.647, for which Re [ZF (Ωmax)] = nβi× and Im[Z (Ω )] = 0.471 V . At this Ω can be written as F max × T 2 ωmaxǫ0λ =Ωmax = 0.647 (4.23) Deﬀ The impedance spectra represented in Fig. 4.5 can be approximated with a semicircle with radius equal to R and centered at 1.5 R, where R is given by eqn. 4.10 (see the dashed line in Fig. 4.5). × It is known that such a semicircle can be modeled by a resistor R in series with another resistor of the same value R in parallel with a capacitor C (see the network encircled with dash line in Fig. 4.6). The impedance of this network is 1 1 Z = R 1+ = R 1+ (4.24) F 0 1+ jωRC 1+ jω/ω max where 1 ω = (4.25) max RC

52 is the frequency at which the absolute value of the imaginary component of ZF 0 is maximum. Eqns. 4.10, 4.23, and 4.24 can be solved to compute C as a function of the parameters of the battery

ǫ λ2 C = 0 (4.26) RDeﬀΩmax

The diﬀusion length under the condition of large cathode widths (l/λ ) can be written as → ∞ nF AD c λ = eﬀ o∗ (4.27) i

Using the eqns. 4.26 and 4.27 the capacitance C can be written as

n3F 2A2βǫ D (c )2 C = 0 eﬀ o∗ (4.28) VT Ωmaxi

Hence, the small-signal equivalent circuit of Li-air batteries can be approximated with the one shown in Fig. 4.6 with R and C given by equations 4.10 and 4.28, where Ωmax = 0.647. The circuit represented in Fig. 4.6 and eqn. 4.28 should be used with care in practical applications because it requires large discharge currents under which the porosity of the cathode can change significantly in time over the duration of the impedance measurements. One can show that Z (Ω) Z(Ω) < 4.3% for all values of Ω. Hence, in the case of Li-air k F 0 − k batteries with wide cathode width, the error in using the small-signal equivalent circuit shown in Fig. 4.6 instead of the complex circuit elements shown in Fig. 4.4 is 4.3%. Equations 4.10 and 4.4 are instrumental because they relate the small-signal circuit elements to the geometrical and electrochemical properties of the battery. By properly fitting the experimental data to the small- signal equivalent circuit shown in Fig. 4.6 one can extract information about the effective oxygen diffusion coefficient and the reaction rate constant.

53 CHAPTER 5

EFFECT OF FINITE OXYGEN DISSOLUTION ON THE IMPEDANCE SPECTRA

5.1 Introduction

In the previous chapters, the concentration of oxygen at x = 0 is considered to be a constant and equal to co∗, however, in reality in an open system such as Li-air, the external partial pressure of oxygen, the solubility of oxygen in the electrolyte, and the value ﬁnite value of the discharge current should play an important role in this battery. The amount of oxygen dissolved into an electrolyte solution from the atmosphere, at low concentrations, can be described using Henry’s law,

po = kH co (5.1) where po is partial pressure of oxygen in the atmosphere, co is the dissolved oxygen concentration in the electrolyte, and kH is the Henry’s law constant and it depends on the solute and solvent in the electrolyte and the ambient temperature. Another common method of quantifying solubility of gas in a liquid is called Bunsen solubility constant. Bunsen solubility coeﬃcient α is described as the volume of the gas (at STP conditions) absorbed by a liter of the electrolyte at a temperature T . The relation between Bunsen solubility coeﬃcient, partial pressure of oxygen, and dissolved oxygen concentration is [81], αpo co = (5.2) RT0 where α is the Bunsen constant (cm3 gas/cm3 liquid), R is the gas constant (8314.4621 kPa cm3/(mol K)), po is the partial pressure of the ambient oxygen (kPa), T0 is the STP temperature (273.15 K). The Bunsen solubility constant can be related to Henry’s law constant using the following relation

RT c α = 0 = o (5.3) kH po

54 h liquid the ersne sn ice etrn h oosctoeat cathode porous using the represented (entering are circles fluxes using which represented in process, dissolution the of htol ml rcino h a particles, gas the of fraction small a only that ocnrto ftegsdsovdi h ovn at solvent the in dissolved gas the of concentration h u eed hte h atr sdshrigo charging. or discharging electrochemical is the battery of the rate whether the depends flux on the depends strength whose , oxygen, of k rsne n[2,tedsouino xgni oee saceia r chemical a as as expressed modeled is effect is oxygen oxygen kinetic of of the dissolution to the important [82], is in it presented and fails law wh Henry’s instance discharge, for conditions, non-equilibrium under however, law, where b stert foye xtn h battery, the exiting oxygen of rate the is h owr aecntn a edtrie yuigHrzKusne Hertz-Knudsen using by determined be can constant rate forward The ne qiiru odtos h islto foye nteelec the in oxygen of dissolution the conditions, equilibrium Under iue51 islto ieisi ihu-i atr ihoxygen with battery Lithium-air a in kinetics Dissolution 5.1: Figure N o steflxo a neigtesystem, the entering gas of flux the is . oe o islto kinetics dissolution for Model 5.2

Anode N k f o x= l = = Separator f k f f (2 p itn h lcrlt/a nefc ( interface electrolyte/gas the hitting , p

55 Porous o o πM k f − steprilpesr ftegas, the of pressure partial the is x=0 Cathode stert foye neigit h battery, the into entering oxygen of rate the is O k b k 2 x c RT b o 0 (5.4) (0) .Fgr . hw ipie view simplified a shows 5.1 Figure 0. = k (5.5) ) x f ) uigdshreteei flux a is there discharge During 0). = O 2 ntebteyi ne hreor charge under is battery the en rosadoye oeue are molecules oxygen and arrows rlt sdsrbdb Henry’s by described is trolyte .B olwn h approach the following By s. ato n h u density flux the and eaction ecin h ieto of direction The reaction. isligat dissolving uto n assuming and quation x c )enters 0) = o x 0 sthe is (0) 0 = where MO2 is the molecular weight of oxygen. At equilibrium (No = 0), the concentration of oxygen in the electrolyte is computed using Henry’s law. The value of the Henry’s law constant can be obtained by imposing equilibrium conditions. Setting No = 0, eqn. 5.4 gives

kf co(0) 1 = = kH− (5.6) kb po The concentration at the electrode/gas interface (x = 0) is computed using Fickian diﬀusion

N = D c (0) (5.7) o − eff∇ o where Deff is the diffusion coefficient of oxygen in the electrolyte. Assuming 1-D ∂c (0) N = D o (5.8) o − eff ∂x

The expression for dissolved oxygen concentration at x = 0 is obtained by using equations 5.4 and 5.8 ∂c (0) k p k c (0) = D o f o − b o − eﬀ ∂x kf and kb are related using eqn. 5.6 ∂c (0) k (p k c (0)) = D o f o − H o − eﬀ ∂x

After rearranging the above equation, the ﬁnal expression for oxygen concentration dissolved at the electrode/gas boundary is

∂c (x) k k k p o = f H c (0) f o (5.9) ∂x D o − D x=0 eﬀ eﬀ

5.2.1 Steady-state analysis

Under steady-state conditions the modiﬁed Fick’s second law of diﬀusion reduces to

2 d co(x) kco(x) 2 = 0 (5.10) dx − Deﬀ where k is the reaction rate that can be described using the Butler-Volmer equation. In this chapter we will assume the simpliﬁed case of β = 0.5, but the same analysis can be performed using more accurate equations for the reaction rate such as the one employed in the previous chapter. Hence, we will assume n k = 2k a sinh η (5.11) − 0 2V T

56 The solution of eqn. 5.10 is

x x c (x)= C cosh + C sinh (5.12) o 1 λ 2 λ Deﬀ where λ = k , λ describes the diﬀusion length for oxygen in the electrolyte and C1 and C2 are integrationq constants. These integration constants are obtained by applying the following boundary conditions:

Boundary condition at x = 0

∂c (x) k k k p o = f H c (0) f o (5.13a) ∂x D o − D x=0 eﬀ eﬀ

Boundary condition at x = l dc (x) o = 0 (5.13b) dx x=l

C2 is obtained by applying boundary condition (5.13b) to eqn. 5.12 dc (x) C x C x o = 1 sinh + 2 cosh = 0 dx λ λ λ λ x=l and, after rearranging l C = C tanh (5.14) 2 − 1 λ Next, substitute the expression for C2 into eqn 5.12

x l x c (x)= C cosh tanh sinh (5.15) o 1 λ − λ λ The integration constant C1 is obtained by applying boundary condition (eqn. 5.13a) to eqn. 5.15 and get ∂c (x) C l k k k p o = 1 tanh = f H c (0) f o . ∂x − λ λ D o − D x=0 eﬀ eﬀ

Rewriting the above equation and using c (0) from eqn. 5.15, the above equation reduces to o

λ kf po kf kH C1 = l C1 tanh Deﬀ − Deﬀ × λ

After rearranging, the ﬁnal expression for integration constant C1 is

λkf po C1 = Co = l (5.16) "tanh λ Deﬀ + λkf kH #

57 The ﬁnal expression to compute the spatial variation of the oxygen concentration in the cathode of Li-air batteries by assuming ﬁnite oxygen dissolution kinetics is

x l x c (x)= C cosh tanh sinh (5.17) o o λ − λ λ where Co is a constant that is given by eqn. 5.16 and depends upon diﬀusion length of the oxygen

(λ), diffusion coefficient of oxygen in the electrolyte (Deff), partial pressure of oxygen in the region just outside the cathode (po), rate of oxygen dissolving in the electrolyte (kf ), and Henry’s law constant (kH ). The Faradaic current is computed using

l iF = A Rc dx (5.18) Z0 where Rc is given by

Rc = nFkco (x) (5.19)

Using eqn. 5.17 and Rc, the relation for Faradaic current becomes

l x l x iF = nF AkCo cosh tanh sinh dx 0 λ − λ λ Z 2 Deff After integration, applying boundary conditions, and using λ = k the final expression for the Faradaic current becomes nF AC D l i = o eff tanh (5.20) F λ λ 5.3 Impedance spectra under d.c. discharge

The small-signal Faradaic current is computed by applying the perturbation theory to eqn. 5.18. The small-signal representation of Faradaic current, over-potential and oxygen concentration jωt jωt jωt jωt jωt is δiF e , δηe , and δco(x)e respectively. Substituting iF + δiF e , k + δke , and co (x)+ jωt δco (x) e into l iF = nF Ak co (x) dx, (5.21) Z0 leads to l jωt jωt jωt iF + δiF e = nF A k + δke co (x)+ δco (x) e dx (5.22) Z0

58 l l jωt jωt iF + δiF e = nF A k co (x) dx + k δco (x) e dx Z0 Z0 l l jωt jωt jωt +δke δco (x) e dx + δke co (x) dx (5.23) Z0 Z0 Using equation 5.18 and neglecting the third term (since the product δc δk is negligible) in eqn. o × 5.23 results in the following expression

l l δiF = nF A k δco (x) dx + δk co (x) dx (5.24) Z0 Z0 The last equation gives the expression of small-signal Faradaic current, where k is given by eqn.

5.11, co (x) is given by eqn. 5.17, δk is obtained by linearizing eqn. 5.11, andδco (x) is obtained by solving the small-signal form of the Fick’s second law of diﬀusion

2 d δco (x) (k + jωǫ0) δkco (x) 2 δco (x)= (5.25) dx − Deﬀ Deﬀ

The homogeneous solution of the previous diﬀerential equation is

x x δc = C cosh + C sinh (5.26) o,c 1 δλ 2 δλ where C and C are the integration constants and δλ = Deﬀ . The particular solution of the 1 2 (k+jωǫ0) second-order non-homogeneous diﬀerential equation is q

x x δc = C cosh + C sinh (5.27) o,p 3 λ 4 λ where C3 and C4 are the integration constants. The complete solution is

x x x x δc (x)= C cosh + C sinh + C cosh + C sinh (5.28) o 1 δλ 2 δλ 3 λ 4 λ The expressions of integration constants C3 and C4 are obtained by substituting eqn. 5.27 into eqn. 5.25 and then comparing the left and right sides of the resulting equation. Substituting 5.27 into the L.H.S of eqn. 5.25, the L.H.S. of this equation becomes

2 d x x (k + jωǫ0) x x L.H.S = 2 C3 cosh + C4 sinh C3 cosh + C4 sinh dx λ λ − Deﬀ λ λ h i h i and after simpliﬁcation

jωǫ0 x x L.H.S = C3 cosh + C4 sinh (5.29) − Deﬀ λ λ h i

59 Now, substituting 5.27 into the R.H.S of eqn. 5.25 gives

δkC x l x R.H.S = o cosh tanh sinh (5.30) Deﬀ λ − λ λ Equating the L.H.S. and the R.H.S. gives

x x x l x jωǫ C cosh + C sinh = δkC cosh tanh sinh − 0 3 λ 4 λ o λ − λ λ h i The expressions for integration constants C3 and C4 are obtained by comparing

δkCo C3 = (5.31) − jωǫ0 δkC l C = o tanh (5.32) 4 jωǫ λ 0

Substituting the expressions of C3 and C4 into eqn. 5.28, the complete solution for eqn. 5.25 becomes

x x δkCo x δkCo l x δco (x)= C1 cosh + C2 sinh cosh + tanh sinh (5.33) δλ δλ − jωǫ0 λ jωǫ0 λ λ The values of integration constants C1 and C2 are obtained by imposing the following boundary conditions

Boundary condition at x = 0

∂δc (x) k k o = f H δc (0) (5.34a) ∂x D o x=0 eﬀ

Boundary condition at x = l

dδc (0) o =0 (5.34b) dx x=l

Integration constant C2 is obtained using the boundary condition 5.34b

l C = C tanh (5.35) 2 − 1 δλ and C1 is obtained by applying boundary condition 5.34a. Substituting eqn. 5.33 into eqn. 5.34b, taking the derivative, and applying the boundary conditions results in

C δkC l k k 2 + o tanh = f H δc (0) δλ jωǫ λ λ D o 0 eﬀ

60 Substituting C2 from eqn. 5.35 leads to

C l δkC l k kH − 1 tanh + o tanh = f δc (0) δλ δλ jωǫ λ λ D o 0 eﬀ

Next, by evaluating δco(x) at x = 0 in eqn. 5.33 and using the last equation gives

C l δkC l k k δkC 1 tanh = o tanh f H C o δλ δλ jωǫ λ λ − D 1 − jωǫ 0 eﬀ 0

Rearranging the above equation yields the value of integration constant C1

1 tanh l + kf kH δkCo λ λ Deff C1 = (5.36) jωǫ0 1 tanh l + kf kH δλ δλ Deff By replacing constant C1 in eqn. 5.35 results in the final expression for integration constant C2

1 tanh l + kf kH δkCo λ λ Deff l C2 = tanh (5.37) − jωǫ0 1 tanh l + kf kH δλ δλ δλ Deff The complete solution of the differential eqn. 5.25 after replacing the integration constants C1

(eqn. 5.36) and C2 (eqn. 5.37) into eqn. 5.33 is

1 tanh l + kf kH δkCo λ λ Deff x l x δco (x)= cosh + tanh sinh jωǫ0  1 tanh l + kf kH δλ λ λ δλ δλ Deff  k k 1 tanh l + f H x λ λ Deff l x cosh tanh sinh − λ − 1 tanh l + kf kH δλ δλ  δλ δλ Deff  The final expression for the small-signal oxygen concentration δco (x) is

1 tanh l + kf kH δkCo λ λ Deﬀ x l x δco (x)= cosh tanh sinh jωǫ0 1 tanh l + kf kH δλ − δλ δλ δλ δλ Deﬀ δkCo x l x + cosh + tanh sinh (5.38) jωǫ0 − λ λ λ Finally, the small-signal Faradaic current δiF can be computed

l l δiF = nF A k δco(x) dx + δk co(x) dx Z0 Z0 = int1 + int2 (5.39)

61 where the integral was divided into two parts: int1 and int2. Integral int2 is computed by substituting co(x) from eqn. 5.17 into eqn. 5.39

l x l l x int2 = nF AδkCo cosh dx nF AδkCo tanh sinh dx 0 λ − 0 λ λ Z Z Solving the integral results in l int = nF AλδkC tanh (5.40) 2 o λ Next, int1 is computed by substituting δco (x) with eqn. 5.38

k kH 1 tanh l + f l l nF kAδk Co λ λ Deff x l x int1 = cosh dx tanh sinh dx kf kH jωǫ0 1 tanh l + 0 δλ − 0 δλ δλ δλ δλ Deff Z Z nF kAδk C l l x l x + o tanh sinh dx cosh dx jωǫ0 0 λ λ − 0 λ Z Z after integrating and applying boundary conditions the final expression for int1 is

1 tanh l + kf kH δλ λ λ Deﬀ l λ l int1 = nF kAδk Co tanh tanh (5.41) jωǫ0 1 tanh l + kf kH δλ − jωǫ0 λ  δλ δλ Deﬀ   Substituting integrals int1 (eqn. 5.41) and int2 (eqn. 5.40) back into eqn. 5.39 and rearranging the expression for δiF gives

1 tanh l + kf kH δλ λ λ Deff l λ l λ l δiF = nFkδkACo tanh tanh + tanh jωǫ0 1 tanh l + kf kH δλ − jωǫ0 λ k λ  δλ δλ Deff   The final form of the small-signal Faradaic current δiF is

k k 1 tanh l + f H l iF δk kδλ λ λ Deﬀ tanh δλ k δiF = + 1 (5.42) k jωǫ λ 1 l kf kH l jωǫ  0 tanh + tanh λ − 0 δλ δλ Deﬀ   The Faradaic impedance is given by,

δn δη δk ZF (ω)= = (5.43) −δiF −δk δiF where δk is given by eqn. 5.42 and δη is obtained by linearizing eqn. 5.11 and is given by δiF δk

δk 2V n = T tanh η (5.44) δη nk 2V T

62 The Faradaic impedance is given by

tanh n η 2VT 2VT ZF (ω)= k k − niF 1 tanh( l )+ f H l kδλ λ λ Deff tanh( δλ ) k k k l + 1 jωǫ0λ 1 l f H tanh jωǫ0 tanh( )+ ( λ ) − δλ δλ Deff tanh n η 2VT 2VT ZF (ω)= k k l f H l − niF 2 tanh( )+λ kδλ λ Deff tanh( δλ ) k 2 k k l + 1 jωǫ0λ l f H tanh jωǫ0 tanh( )+δλ ( λ ) − δλ Deff Since δλ = Deff and λ = Deff the final expression for Faradaic current for Li-air batteries (k+jωǫ0) k under d.c. dischargeq using dissolutionq kinetics can be rewritten as

Z tanh n η 0 2VT ZF (ω)= (5.45) k jωǫ tanh + 0 l k k√k Deff 1 + r fdiss (ω,η) − jωǫ0 jωǫ0√k+jωǫ0 tanh k l Deff q where Z0 is 2VT Z0 = − (5.46) niF and √D k tanh k l + k k eff Deff f H fdiss (ω,η)= (5.47) q k+jωǫ0 D (k + jωǫ0) tanh l + k k eff Deff f H p q 5.4 The case of high dissolution kinetics

In chapters 2-4 it was assumed that the rate at which oxygen is dissolving in the electrolyte is very high (i.e. the consumption of oxygen in the Faradaic reaction is much slower than the amount of O2 dissolving in the electrolyte). In such a case, kf is very high, the dissolution of oxygen is no longer the limiting factor. The Faradaic impedance in this case is obtained by setting k or f → ∞ 1/k 0 in eqn. 5.47. f → To take the limit k it is instrumental to rewrite eqn. 5.47 in terms of 1/k f → ∞ f 1 √D k tanh k l + k kf eff Deff H fdiss (ω,η)= (5.48) 1 q k+jωǫ0 D (k + jωǫ0) tanh l + k kf eff Deff H p q

63 1 Under the condition of high dissolution kinetics ( 0) fdiss (ω,η) = 1 and the expression for kf → the Faradaic impedance (eqn. 5.45) reduces to

Z tanh n η 0 2VT ZF = (5.49) k jωǫ tanh + 0 l √ Deﬀ 1 k + k k r  − jωǫ0 jωǫ0√k+jωǫ0 tanh k l  Deﬀ q   which is identical to eqn. 2.49 obtained in chapter 2.

64 CHAPTER 6

FINITE ELEMENT MODELING OF EIS

6.1 Introduction

The advantage of expressing the EIS spectra into a compact analytical form is that the final results are computationally simple and, in the same time, provide a significant amount of phenomenological information about the system. Compact models can describe a system and predict its behavior with high accuracy using only few number of parameters. Unfortunately, analytical solutions are usually restricted to regular geometries and simple boundary conditions, such as the ones that were used in the previous chapters. In order to better understand the behavior of the system one needs to solve the transport equations using more comprehensive models and computationally expensive techniques such as finite difference, finite element, or finite volume modeling. In previous chapters, many factors such the discharge product resistivity, electrolyte concentration, electron conductivity in the current collector backbone, diffusion of lithium, consumption of lithium were neglected. If these effects were included in the analysis, it was much harder or even impossible to find a closed form solution for the EIS spectra. In this chapter a novel approach to compute the impedance spectra using finite element simulations is introduced. This chapter is structured as follows. In the next section the finite element model for the computation of impedance spectra is summarized. Then, the numerical technique that was developed to compute the impedance spectra in Li-air batteries under d.c. discharge is presented and the numerical implementation of the algorithm is discussed. Then, detailed numerical analysis of the impedance spectra under different discharge conditions and state of charge is presented.

6.2 Finite element model

The ﬁnite element model used to compute the impedance spectra is based on a transport model that was originally developed by Andrei et al. [50]. The model has been subsequently improved and calibrated by a number of other research groups in the energy storage community parto [83] \

65 and is based on the theory of concentrated solutions originally proposed by Newman and Thomas- Alyea [84]. The mass and charge transport in the electrolyte is described by the following set of partial diﬀerential equation (PDEs):

electrolyte drift-diffusion equations • ∂ (φ φ ) . (κ φ + κ ln c ) R = aC − Li , (6.1) ∇ eff∇ Li D∇ Li − c D ∂t ∂ (ǫc ) 1 t+ I t+ Li = . (D c ) − R Li ·∇ , (6.2) ∂t ∇ Li,eff∇ Li − F c − F oxygen diffusion • ∂ (ǫc ) R o = . (D c ) c , (6.3) ∂t ∇ o,eff∇ o − nF electron conductivity • ∂ (φ φ ) . (σ φ)+ R = aC − Li (6.4) ∇ eff∇ c D ∂t where cLi is the concentration of the electrolyte (which is equal to the concentration of Li ions), φ is the potential of electrons, φLi is the electrostatic potential of the electrolyte, ILi is the electrolyte current density equal to κ φ κ ln c , C is the capacitance of the double layer per unit − eff∇ Li − D∇ Li D area, a is the specific area of the electrode, ǫ is the porosity (equal to the volume fraction of the electrolyte), t+ is the transference number of the positive ion, and F is the Faraday constant. The reaction rate at the cathode, Rc, is given by the Butler-Volmer equation (see eqn. 2.41)

− βn η (β 1)n η R = nFc k a e VT e VT (6.5) c o 0 − where k0 is the reaction rate constant, n is the number of electrons involved in the cathode reaction, and VT is the thermal voltage. The over-potential is given by

η = φ φ + E0 V , (6.6) Li − − dis where Vdis is the voltage drop across the discharge product. The pores in the cathode are assumed to be cylindrical in shape with single radius (see ﬁg. 6.1), the voltage drop across the discharge product is [85] R ρ ǫ V = c dis r¯ln 0 , (6.7) dis 2 ǫ and the speciﬁc area becomes 2ǫ a = , (6.8) r¯

66 Figure 6.1: Cross-section of a cathode with cylindrical pores with a single radius

where ρdis is the resistivity of the discharge product andr ¯ is the radius of the pores. The eﬀects of the pore radius distribution and diﬀerent pore structures on the impedance spectra are analyzed later in this chapter. The discharge products in the cathode are: lithium peroxide (2Li+ +O +2e– Li O ), 2 −−→ 2 2 lithium oxide (4Li+ +O +4e– 2 Li O), and lithium superoxide (Li+ +O +e– LiO ) 2 −−→ 2 2 −−→ 2 [50, 61, 86, 87]. Since all these discharge produces are insoluble in the organic electrolyte, they deposit on the surface of the cathode, and the porosity changes according to the following equation

∂ǫ Mdis 2ǫ = Rc , (6.9) ∂t − nF ρm,dis r¯ where Mdis is the molar mass and ρm,dis is mass density of discharge product.

67 The eﬀective oxygen and electrolyte diﬀusion constant and the electrolyte conductivity constants are given by Bruggeman equations

brugg Do,eff = ǫ Do, (6.10) d ln co D = ǫbruggLi D 1 , (6.11) Li,eff Li − d ln c Li bruggκ κeff = ǫ κ, (6.12) while the diffusional conductivity of the electrolyte is

2RTκ (t+ 1) ∂f κ = eff − 1+ , (6.13) D F ∂ ln c Li The electrolyte drift and diffusion equations [i.e. equations 6.1 and 6.2] are derived from the transport equations of the positive and negative ions of the concentrated solution. A more detailed derivation for equations and can be found in references [84] and [88]. Equations 6.1- 6.4, and eqn. 6.9 are subject to boundary and initial conditions. The modified diffusion equation (2.9) is subject to the following initial and boundary conditions:

Initial condition

co(x, 0) = co∗

where co∗ is the bulk oxygen concentration

Boundary condition at x = 0

co(0,t)= co∗

Boundary condition at x = l dc (x,t) o = 0 dx x=l

and are solved using the Newton technique.

6.3 Computation of impedance spectra

In this section, the technique to compute the impedance spectra in electrochemical systems using ﬁnite element modeling is presented. The impedance spectra are computed by applying the small-signal analysis theory to the transport equations and solving the ﬁnal linear system of equations numerically.

68 The impedance of the battery can be expressed in complex form as

δv Z = δi

In order to compute δv any small variation in the discharge current δiejωt translates into local vari- jωt jωt jωt ations of the over-potentialδηe , oxygen concentration δco e , lithium concentration δcLi e , jωt jωt electron potential δφe , electrolyte potential δφLi e , as well as of the currents, fluxes, reaction rates at the anode and cathode, and other state variables, where η, co, cLi, φ, and φLi, are complex functions of position that need to be determined. This fact can be expressed in a compact mathematical form by introducing the vector of state variables, y. In finite element simulations, the state variables are defined separately at each node of the mesh, and can be usually written as in vector form as

T y =[co,1,...,co,N ,cLi,1,...,cLi,N ,φLi,1,...,φLi,N ,φ1,...,φN ] (6.15) where co,i, cLi,i, φLi,i, and φi denote the values of the oxygen concentration, lithium concentration, lithium potential and electron potential at each node i = 1,...,N, where N is the total number of nodes, and superscript T denotes the transpose of the vector. Using small-signal perturbations, the state vector can be written as jωt y = Y0 + δy e (6.16) where Y0 and δy are column vectors that denote the d.c. and a.c. values of the state variables at each mesh point. The d.c. values of the state vector can be computed numerically by solving the transport equations for the “unperturbed”battery. Indeed, if i is d.c. value of the discharge current, the transport equations 6.1-6.4 can be written in discretized form as:

F (Y0,i) = 0 (6.17) where F is a vector whose components are the discretized diﬀerential equation at each node. If the value of the discharge current i is given, one can solve transport equations 6.17 numerically to obtain the value of state variable Y0. The a.c. values of the state vector can be computed by solving

jωt jωt F Y0 + δy e ,i + δie = 0 (6.18) 69 Using small-signal perturbations and eqn. 6.17, the above equation can be rewritten as

jωt jωt FY (Y 0,i) δy e + F i (Y 0,i) δie = 0 (6.19) where FY (Y 0,i) is the Jacobian of the transport system and

dF i (y,i) F i (Y 0,i)= (6.20) di y = Y 0

i = i

are the derivatives of the discretized transport equations with respect to the discharge current. If the value of the a.c. current perturbation δi is given, Eqn. 6.19 can be written as a linear system of equations for the a.c. values of the state variables, δy

FY (Y ,i) δy = F i (Y ,i ) δi (6.21) 0 − 0 0

The potential at the air side of the cathode is equal to the value of the electrostatic potential at the air side of the cathode, v = φ . Hence, δv is a component of the state vector δy:δv = δφ . |x=l |x=l Once the system equations are solved numerically, the impedance spectra are computed using the impedance equation (Z = δv/δi). The model described was implemented in our in-house ﬁnite element simulator developed at Florida State University, RandFlux [89]. The transport equations were discretized on a 1-D non- uniform grid in the space coordinate and the backward Euler method was applied to discretize the time coordinate. The Jacobian matrix, FY , is computed numerically and the transport equations are solved self-consistently using the Newton technique. The simulation time required to compute the discharge curve for a given discharge current is less than one minute for a grid with 100 mesh points on a 3 GHz single-processor workstation. The computational overhead to compute the impedance at one frequency is mostly given by the time required to solve the linear sparse system 6.21, which is of the order 10-100 milliseconds. The simulation time to evaluate the impedance spectra at a few hundred frequency points is less than 10 seconds. The parameters used in simulations are presented in Table 1. The width of the air electrode is 200 µm, the separator is 50 µm, and the metal anode is separated from the membrane by a 50 nm layer of organic electrolyte (also known as anode protective layer, APL).

70 Table 6.1: List of parameters used in the ﬁnite element simulations

Parameter Value 3 Capacitance of the double layer (CD) 10 µF/cm APL thickness (LA) 50 nm Separator thickness (Ls) 50 µm Cathode thickness (Lc) 200 µm Bruggeman constants (brugg, bruggLi, and bruggκ) 1.5 Temperature (T) 300 K Initial electrolyte concentration (c (t = 0)) 1 10 3 mol/cm3 Li × − Open cell voltage (E0) 2.956 V Initial porosity of the cathode (ǫ) 0.75 Porosity of the separator (ǫ) 0.75 Molecular weight of Li2O2 45.88 g/mol

Mass density of Li2O2 (ρm,Li2O2 ) 2.14 g/mol

6.4 Simulation results

To verify the numerical implementation, the numerical results of the ﬁnite element model are compared with analytical results obtained using the model presented in chapter 2.3.2. According to the analytical model, the impedance of the battery can be written in compact form as

ZF (ω) Z (ω)= RΩ + (6.22) 1+ jωalACdZF (ω) where l is the width of the cathode, a is the cross-sectional area of the cathode, and RΩ is the combined resistance of the ohmic contacts, electrolyte, and kinetic processes at the anode. The

Faradaic impedance, ZF (ω), depends on the d.c. value of the discharge current iF according to eqn. 2.38 VT ZF (ω)= k jωǫ k√k tanh + 0 l Deff nβi 1 k + r F jωǫ0 k  − jωǫ0√k+jωǫ0 tanh l  Deff q Fig. 6.2 presents a comparison between the finite element simulation (symbols) and the analytical model (continuous line) for different values of the d.c. discharge current density ranging from 0.1 mA/cm2 to 1 mA/cm2. A very good agreement between the analytical results and simulations is observed. The slight deviations at high discharge currents are due to the fact that the analytical model does not take into account the high current effects such as voltage losses in the electrolyte,

71 150 Analytical model 2 Finite element simulation 0.1mA/cm

100 2 ] 1mA/cm 2

Ω 0.2mA/cm

-Z'' [ 50 0.3mA/cm2 0.5mA/cm2

0 0 50 100 150 200 250 300 Z' [Ω]

Figure 6.2: Comparison between the ﬁnite element simulation (symbols) and the analytical model (continuous) for diﬀerent values of the d.c. discharge current density.

separator, and the kinetic phenomena at the anode and non-linear deposition of the deposit products in the cathode (the analytical model assumes that the porosity in the cathode uniform and that the porosity remains constant during the time of impedance measurement).

Dependence of EIS spectra on the state of discharge. To analyze the contribution of the discharge product on the impedance spectra of a Li-air battery, the EIS spectra are simulated at different state-of-discharge for two values of the discharge current: 0.1 mA/cm2 (fig. 6.3) and 1 mA/cm2 (fig. 6.4). Fig. 6.3 presents the distribution of the discharge product (light green region), porosity distribution (red curve), and the concentration of oxygen (blue curve) in the cathode at (a) 20%, (b) 50%, (c) 90%, and (d) 100% state-of-discharge. The discharge product deposits uniformly inside the cathode at low discharge currents until the pores on the air side of the cathode are completely blocked and oxygen can no longer enter the battery. The deposit product is considered to be resistive in nature and adds to the total charge transfer resistance of the battery. Fig. 6.3e represents the impedance response computed at different states of discharge. The contribution of the resistance of the deposit layer to the total impedance is negligible when the state-of-discharge is less than 30%, but the contribution of the deposit layer is significantly higher and dominate the EIS spectra when the impedance response is measured near the end of the discharge cycle. It is

72 (a) (b) Li2O2 Li2O2

Li O Li2O2 2 2

Li2O2 Li2O2

(e) State of discharge 300 90% 80% 70% 50% ] 0% Ω 200 -Z'' [ -Z''

100

0 0 100 200 300 400 500 Z' [Ω]

Figure 6.3: Distribution of the discharge product (a)-(d) for diﬀerent states of discharge: 20%, 50%, 90%, and 100% (for a discharge current density of 0.1 mA/cm2) and impedance spectra (e) simulated for diﬀerent values of the state-of-discharge.

important to mention that the impedance contribution due to electrolyte decomposition, corrosion of the electrode, and separator deformation are not considered in this analysis. In ﬁg. 6.4 the distribution of the discharge product (a-d) and the impedance response (e) for diﬀerent states of discharge are presented for a discharge current of 1 mA/cm2. Under heavy discharge (i.e. large values of the discharge current) a large amount of the discharge products deposits near the air-side of the cathode as compared to the separator-side of the cathode. This

73 (a) (b)

Li2O2 Li2O2

Li2O2

Li2O2 Li2O2

(e) State of discharge 90% 100 80% 70% 50%

] 0% Ω

-Z'' [ -Z'' 50

0 0 50 100 150 Z' [Ω]

Figure 6.4: Distribution of the discharge product (a)-(d) for diﬀerent states of discharge: 20%, 50%, 90%, and 100% (for a discharge current density of 1 mA/cm2) and impedance spectra (e) simulated for diﬀerent values of the state-of-discharge.

non-uniform deposition leads to under-utilization of the air cathode and is one of the causes for the reduction of the speciﬁc capacity of the battery. The Nyquist plot (ﬁg. 6.4e) shows that the total impedance of the cathode increases by more than 300% when the EIS is computed at 90% state-ofdischarge as compared on a fresh battery. It is important to note that the time taken to perform an actual EIS at 90% state-of-discharge may exceed the total discharge time of a Li-air battery for some values of the discharge current, cathode geometry, cathode materials, and concentration of

3.0

0.1 mA/cm 2 2.9 10% 50% 70% 80% 90% 2.8 10% 50%

70% 2.7 80% 90% Discharge voltage [V] 2.6 1 mA/cm 2

2.5 0 500 1000 1500 2000 2500 Specific capacity [mAh/g]

Figure 6.5: Cell voltage as a function of speciﬁc capacity for the two discharge currents used in the simulations presented in Fig. 6.3 and Fig. 6.4. The symbols represent the state of discharge at which the impedance spectra are simulated.

the ambient oxygen; these effects are not considered in our analysis. Figure 6.5 represents discharge curves for two discharge currents: 0.1 mA/cm2 and 1 mA/cm2 used in the simulations presented in Fig. 6.3 and Fig. 6.4. The symbols represent the state-ofdischarge at which the impedance spectra are simulated. Figure 6.6(a-d) shows the distribution of the discharge product (light green region), porosity distribution (red curve), and the concentration of oxygen (blue curve) in the cathode for four different values of resistivity of the discharge product: (a) ρ =1 1010 Ωcm, (b) ρ =1 1011 Ωcm, dis × dis × (c) ρ =1 1012 Ω cm, and (d) ρ =1 1013 Ωcm. The resistivity of the deposit layer strongly dis × dis × influences the deposition profile of the discharge product in the cathode: for large values of resistivity the discharge products are deposited uniformly throughout the cathode and , in this way, the battery is slightlybetter utilized as compared to the case when discharge products have a low resistivity. In addition to the added electric resistance, the deposition of the discharge product also causes physical changes in the cathode such as reduced specific area, reduced porosity, and increased tortuosity. Figure 6.6e shows that the impedance response simulated at 50% state-of-

75 (a) (b)

Li2O2 Li2O2

Li O 2 2 Li2O2

(e) 125 1013 Ωcm 1012 cm 100 Ω 1011 Ωcm 1010 Ωcm 109 Ωcm

] 75 Ω

-Z'' [ -Z'' 50

1 mHz 0 0 50 100 150 Z' [Ω]

Figure 6.6: (a)-(d) show the distribution proﬁle of the discharge product and (e) shows the impedance spectra for diﬀerent values of Li2O2 resistivity at 50% state-of-discharge. The simulations are carried out for d.c. discharge current density of 1 mA/cm2.

discharge for a discharge current of 1 mA/cm2 for four diﬀerent values of resistivity of the discharge product. The resistivity of lithium peroxide strongly inﬂuences the overall impedance of the battery, the total impedance increases 1600% when the resistivity of lithium peroxide is changed from ρ =1 1010 Ω cm to ρ =1 1013 Ωcm. dis × dis × The resistivity of the discharge products plays an important role in the “sudden death”of Li- air batteries (end of the discharge cycle). Figure 6.7 shows voltage discharge curves simulated

2.8

2.6

2.4

SOD = 50%

2.2 rho = 10 13 Ωcm 12 Discharge voltage [V] rho = 10 Ωcm rho = 10 11 Ωcm 10 2.0 rho = 10 Ωcm rho = 10 9 Ωcm 0 200 400 600 800 1000 Specific capacity [mAh/g]

Figure 6.7: Cell voltage as a function of speciﬁc capacity for diﬀerent values of resistivity of the discharge product. The dotted line and the black circle symbols denote the 50% state of discharge.

for different values of the resistivity of the discharge product. The large variation in the specific capacities for various values of the discharge product resistivity can be related to the difference in discharge mechanism at low and high values of the resistivity. For small values of the resistivity, the sudden death in lithium air batteries may occur because the pores near the air-side are filled with the product and blocks the oxygen from entering the battery. For large values of resistivity, the sudden death in lithium air batteries may occur because the potential drop due to the resistive layer can no longer sustain the electrochemical reaction. The concentration of dissolved oxygen strongly influence the specific capacity of practical Li-air batteries. The concentration of oxygen in the electrolyte can be increased by either using electrolytes with high oxygen solubility or by increasing the partial pressure of oxygen that is supplied to the battery. Figure 6.8a shows the effect of the external oxygen concentration on the specific capacity of a Li-air battery. For small values of the partial pressure of oxygen the redox reaction is restricted to the region near the cathode/air interface and the major portion of the cathode plays no part in this reaction, thus reducing the specific capacity of the battery significantly. However, as the concentration of oxygen in the electrolyte is increased the oxygen is able to travel throughout

(a) 2.8 200% (b) 500 100% 200% 50% 100% 21% 400 50% 10% 21% 2.7 5% 10%

] 300 5% Ω

-Z''[ 200 2.6 Discharge voltage [V] 100 1 mHz

2.5 0 0 500 1000 1500 2000 0 100 200 300 400 500 600 700 Specific capacity [mAh/g] Z' [ Ω]

Figure 6.8: Voltage discharge curves (a) and simulated EIS spectra (b) for diﬀerent values of oxygen concentration in the atmosphere (relative to 1 atmosphere).

the cathode and take part in the redox reaction, thus increasing the total specific capacity of the battery Figure 6.8b represents the impedance spectra simulated at 50% state-of-discharge for different values of oxygen concentration in the atmosphere. The two main contributors of the low frequency resistance are: the low Faradaic reaction rate and the high oxygen diffusion resistance. For small values of the oxygen concentrations, the system is starved of oxygen, which causes a large increase in the diffusion resistance as shown by the high total impedance at low frequencies. For large values of the oxygen concentrations, the diffusion impedance reduces drastically and thus reduces the total impedance of the battery at low frequencies, which suggests that the system is no longer limited by diffusion.

78 CHAPTER 7

CONCLUSIONS

The Faradaic impedance of Li-air batteries under d.c. discharge was investigated and closed- form equations for the total electrochemical impedance were developed. The closed-form solutions were derived in terms of the effective value of the diffusion coefficient of oxygen, specific cathode area, porosity, discharge current, reaction rate, oxygen concentration, and cathode width. It was shown that the relatively low effective diffusion coefficient of the oxygen in the cathode results in two slightly asymmetrical semicircles on the Nyquist diagram: one at low frequencies, where the oxygen diffusion dominates the operation of the cell and one at medium frequencies due to the combined effects of the double-layer capacitance and Faradaic processes. Depending on the values of the effective diffusion coefficient, oxygen concentration, porosity, and cathode width the two semicircles can appear separately on the Nyquist plot or can merge into one semicircle. A physics based model is developed to compute the electrochemical impedance during charging and discharging. The closed-form equations depends explicitly on kinetic reaction rate of lithium peroxide formation and on the reaction rate for the dissolution of the discharge product. The analytical model is valid for small and large values of the current density. However, this model cannot describe the impedance response in batteries with large or non-uniform deposition of discharge product such as Li2O2 and LiO2. A technique to extract physical parameters of Li-air batteries such as the effective diffusion coefficient of oxygen in the cathode and the reaction rate coefficient is proposed using the analytical result for the spectral impedance. This technique was tested on the experimental data published by Adams et al. [76] for rechargeable organic Li-air batteries with different cathode structures. A very good agreement between our theoretical predictions and the published experimental results was obtained. The theory can predict quantitatively well the value of resistances on the Nyquist diagram for a large range of the oxygen concentrations and discharge currents. A small-signal equivalent circuit for Li-air batteries is also developed and the elements of the circuit are expressed in terms physical parameters of the battery such as the oxygen diffusion coefficient, oxygen concentration,

79 discharge current, and other material and kinetic parameters, which make the model instrumental for extracting information about the material structure, reaction processes, and diffusion in the cathode. A high discharge current approximation is also presented for the small-signal equivalent circuit model. The approximate circuit contains only elementary components such as resistors and capacitors and can be implemented numerically easily in circuit simulators and used to fit the experimental data. A technique to compute the impedance spectra in electrochemical systems using finite element modeling is also developed and used to investigate the effects of the discharge products on the impedance spectra of Li-air batteries with organic electrolyte under d.c. discharge. It is shown that the total impedance of the Li-air batteries increases by more than 200% when the impedance spectra are computed at 90% state-of-discharge as compared to on a fresh battery. The change in the total impedance increases when the value of the d.c. discharge current is increasing. The value of the resistivity of the deposit layer not only determines the sudden death mechanism, either due to pore clogging or due to very large voltage drop across the lithium peroxide layer, but also affects the total impedance of these batteries. The overall impedance of the battery can change by more than one order of magnitude, depending on whether the resistivity of the discharge product is small or large. The concentration of oxygen in the atmosphere strongly influences the total impedance of the battery, therefore, using electrolytes with high oxygen solubility and high partial pressure of oxygen can reduce the total resistance of the battery.

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88 BIOGRAPHICAL SKETCH

Mohit Mehta received his Bachelors of Engineering in Electronics and Telecommunications from Pune University, India in the summer of 2010. He joined Florida State University in Fall 2011 for his Doctorate of Philosophy in Electrical Engineering. He is currently working on Modeling and simulation of electrochemical impedance spectra in Li-air batteries under the guidance of Dr. Petru Andrei.