Modeling Li + Ion Battery Electrode Properties

Transcription

1 Modeling Li + Ion Battery Electrode Properties June 20, / 39

2 Students Annalinda Arroyo: Rensselaer Polytechnic Institute Thomas Bellsky: Michigan State University Anh Bui: SUNY at Buffalo Haoyan Chi: SUNY at Buffalo Ioana Cipcigan: University of Maryland, Baltimore County Michael Franklin: Claremont Graduate University Luyen Nguyen: University of Delaware Javed Siddiqeu: George Mason University Sumanth Swaminathan: Northwestern University Olga Trichtchenko: McGill University Chen Zhang: SUNY at Buffalo 2 / 39

3 Contributing Faculty and TIAX Team Contributing Faculty Daniel M. Anderson: George Mason University Sean Bohun: Ontario Institute of Technology Chris Breward: University of Oxford, UK Joseph D. Fehribach: Worcester Polytechnic Institute Luis Melara: University of Colorado, Boulder Colin P. Please: Southampton, UK Giles Richardson: Nottingham, UK Bogdan Vernescu: Worcester Polytechnic Institute TIAX Jacqueline Ashmore David Clatterbuck 3 / 39

4 Outline / 39

5 The Big Picture 5 / 39

6 The Little Picture Figure: Cathode schematic. 6 / 39

7 The Really Little Picture Figure: Cathode particles with different geometric properties. 7 / 39

8 Project Goals TIAX would benefit from algorithms, methods, models, scaling relations, or frameworks to analyze the effect of different particle characteristics on electrode properties. 8 / 39

9 Conservation Laws v.2 Conservation of concentrations: c 1 t + q 1 = 0 c 2 t + q 2 = 0 c 1 = c 2 where c 1 = [Li + ] and c 2 = [PF ], and the flux is given by: 6 q 1 = D 1 ( c 1 + F RT c 1 Φ e ) q 2 = D 2 ( c 2 F RT c 2 Φ e ) 9 / 39

10 Butler-Volmer Kinetics v.2 At the electrolyte/solid interface, reaction rate R: ( F k 1 c s exp 2RT (φ s φ e u(c s )) R = q 1 ˆn Ω = q s ˆn Ω = ) k 2 c exp ( F 2RT (φ s φ e u(c s )) This is the reaction governing the departure of Li + ions from the solid into the electrolyte. ) 10 / 39

11 Dimensionless Equations v.2 Dimensionless conservation: c t + α λ [ c ] λc φ e = 0 c t + D 2α λ [ c ] + λc φ e = 0 where α/λ dimensionless diffusivity of Li + λ = Fφ 0 RT >> 1 Dimensionless Butler-Volmer reaction: R = k 1c s λ ( ) λ exp 2 (φ s φ e u(c s)) k 2c λ exp ( λ 2 (φ s φ e u(c s)) ) 11 / 39

12 Electrode Equations v.2 c s : Concentration of Li in solid electrode particles. Diffusion equation for c s coupled to Laplace s equation 2 ψ = 0 for potential in particles. Various conditions on interface relating ψ, c s in electrode particles, φ, c in electrolyte to reaction rate R. 12 / 39

13 Model Problem v.2 13 / 39

14 Nondimensional Equations v.2 Electrolyte χ O(1), λ O(100), Σ O(1) α λ = O(1), β = O(1) Electrode c t = α ( ) 1 + D2 2 c λ D 2 (c φ) = 1 ( ) 1 D2 2 c λ 1 + D 2 c s t = β 2 c s 2 ψ = 0 14 / 39

15 Nondimensional Equations v.2 Boundary Conditions c φ n Ω = R 2, ψ n Ω = R Σ, c n Ω = λr 2 c s n Ω = R χ R = k 1c s λ ( ) λ exp 2 (φ s φ e u(c s)) k ( 2c λ exp λ ) 2 (φ s φ e u(c s)) 15 / 39

16 Expanding in powers of 1/λ v.2 Since 1/λ << 1, expand dependent variables as: c(z, t) = c 0 (z, t) + 1 λ c 1(z, t) + c s (z, t) = c s0 (z, t) + 1 λ c s1(z, t) + φ(z, t) = φ 0 (t) + 1 λ φ 1(z, t) + etc. Φ(z, t) = Φ 0 (t) + 1 λ Φ 1(z, t) + 16 / 39

17 v.2 Qualitative Results At O(1) we find Φ 0 determined by Butler-Volman conditions is constant. By proceeding to O(1/λ) can find evolution of Φ 1, the correction to the potential drop across electrode. Show timescale analysis would yield evolution of Φ 0 (t/λ) over long times as c s0 (t/λ). 17 / 39

18 v.2 Unsystematically averaged macroscale model Some wild assumptions transport of ions occurs in spaces between spheres (which form a porous medium). spheres provide source of ions. Rate given by solving the microscale problem and is proportional to surface area per sphere and number of spheres per control-layer volume 18 / 39

19 v.2 Unsystematically averaged macroscale model Some wild assumptions transport of ions occurs in spaces between spheres (which form a porous medium). spheres provide source of ions. Rate given by solving the microscale problem and is proportional to surface area per sphere and number of spheres per control-layer volume 19 / 39

20 v.2 The Model Notation: C is the concentration of lithium ions (also counterions) in the electrolyte at a certain height Φ is the electric field at a certain height C s is the concentration of lithium atoms at a certain height θ is the liquid volume fraction = 1 4/3πa 3 N/V 20 / 39

21 The Model v.2 Notation: C is the concentration of lithium ions (also counterions) in the electrolyte at a certain height Φ is the electric field at a certain height C s is the concentration of lithium atoms at a certain height θ is the liquid volume fraction = 1 4/3πa 3 N/V C t = ( λ 1 θ C ) + 2πa2 N z z V R (1) ( λ 2 θ C z z λ 3θC Φ ) = 4πa2 N z V R (2) dc s dt = 4πa2 N V R (3) 21 / 39

22 Initial conditions v.2 Boundary conditions C s = C H(m 1 z), C = C 0, (4) C z = 0, Φ = 0, Φ = J at z = 0 (5) z C z = 0, at z = L (6) 22 / 39

23 v.2 Numerical Solution of the Model 23 / 39

24 v.2 Flux stuff Simple model says that reaction at surface has Butler-Volmer form particles small enough that diffusion ensures that c s is constant in each sphere, i.e. C s is constant at each level (NB - not what TIAX wants!). 24 / 39

25 Flux stuff Simple model says that v.2 reaction at surface has Butler-Volmer form particles small enough that diffusion ensures that c s is constant in each sphere, i.e. C s is constant at each level (NB - not what TIAX wants!). R = C k s 1 N eφ U(Cs/N) k 2 Ce (Φ U(C s/n)) cathode (7) R = 0 in between (8) R = k 3 C s N eφ V(C s/n) + k 4 Ce (Φ V(C s/n)) anode (9) 25 / 39

26 Flux stuff Simple model says that v.2 reaction at surface has Butler-Volmer form particles small enough that diffusion ensures that c s is constant in each sphere, i.e. C s is constant at each level (NB - not what TIAX wants!). R = C k s 1 N eφ U(Cs/N) k 2 Ce (Φ U(C s/n)) cathode (7) R = 0 in between (8) R = k 3 C s N eφ V(C s/n) + k 4 Ce (Φ V(C s/n)) anode (9) Better plan: solve microscale problem and relate what s going on in each sphere to the reaction on the surface Can get c s as an infinite sum of exponentials and will (probably) get Abel-esque equation for c s on the surface. 26 / 39

27 v.2 What can we do with model? Vary surface area while keeping volume fraction constant. Vary volume fraction while keeping surface area constant. Consider populations of spheres - the surface area will become average surface area and formula for θ will change. 27 / 39

28 v.2 Nondimensional Steady-State Potential Equations PDE : { u + = 0 (ɛ u s ) = g x Ω e x Ω s BC : n u + = ɛ n u s = i 0 sinh(u s u + ) x Γ where u s : Li atom potential in the solid particles u + : Li ion potential in the electrolyte ɛ := κ s /κ + 28 / 39

29 v.2 Bounds on Effective Cathode Conductivity u : Homogenized Li potential throughout the cathode (κ u) = θ p g x Cathode 29 / 39

30 v.2 Bounds on Effective Cathode Conductivity u : Homogenized Li potential throughout the cathode (κ u) = θ p g x Cathode ( ) 1 κ 1 m 0 θ p c 30 / 39

31 v.2 Bounds on Effective Cathode Conductivity u : Homogenized Li potential throughout the cathode (κ u) = θ p g x Cathode ( ) 1 κ 1 m 0 θ p c κ 1 ɛ + θ e i 0 + θ p λ ɛ i 0 λ θ eλɛ θ pɛi / 39

32 Kinetic Monte Carlo Method Model of circular lithium metal oxide particles: Initial battery setup allows for variable particle size and packing. 32 / 39

33 Kinetic Monte Carlo Method Based on KMC models of Schulze (2002, 2006) and Voter. Atoms in the solid hop with rates determined by the number of neighbors. Atoms hop out of the solid into electrolyte with specified rate. Lithium ions diffuse away in the electrolyte instantaneously. 33 / 39

34 34 / 39

35 Volume Fraction Results: Donev, Science 2004 Ellipsoids with an aspect ratio close to M&M s candies can randomly pack more densely, up to φ = According to their experiments, the aspect ratio α 1.3 gives the best density φ with no significant orientational ordering. Higher density related to the larger number of particle contacts required to mechanically stabilize the packing. 35 / 39

36 Volume Fraction 36 / 39

37 Made progress on several areas TIAX requested, including: 1 Developed thorough microscopic and macroscopic models. 2 Scaling relations. 3 Various numerical approaches. 37 / 39

38 Thank you Questions? 38 / 39

39 West, K., Jacobsen, T. Modeling of Porous Insertion Electrodes with Liquid Electrolyte, J. Electrochem. Soc. 129, (1982). Fuller, T., Doyle, M., Newman, J. Simulation and Optimization of the Dual Lithium Ion Insertion Cell, J. Electrochem. Soc. 141, 1 10 (1994). Doyle, M., Fuller, T., Newman, J. Modeling of Galvanostatic Charge and Discharge of the Lithium/Polymer/Insertion Cell, J. Electrochem. Soc. 140, (1993). Ionix ionixpower.com/lithium-ion-battery.htm, url:, (2008). 39 / 39