Florida State University Libraries 2015 Modeling, Simulation, and Experimental Verification of Impedance Spectra in Li-Air Batteries Mohit Rakesh Mehta Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ENGINEERING MODELING, SIMULATION, AND EXPERIMENTAL VERIFICATION OF IMPEDANCE SPECTRA IN LI-AIR BATTERIES By 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 Copyright c 2015 Mohit Rakesh Mehta. All Rights Reserved. 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 com- puted 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 stoichio- metric 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 val- ues 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 stoichio- metric 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 elec- trolyte, 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 dis- charge 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.