Mathematical Modeling All Solid State Batteries

Transcription

1 Katharina Becker-Steinberger, Stefan Funken, Manuel Landsdorfer, Karsten Urban Institute of Numerical Mathematics Konstanz, Mathematical Modeling All Solid State Batteries

2 Page 1/31 Mathematical Modeling All Solid State Batteries Introduction Derivation of the 1-D model equations Model assumptions Transport equations Boundary conditions Classification Numerics Steady state solution Transient solution Towards Optimization Weak formulation Performance citeria Incorperating mechanical effects Cahn-Hilliard Approach Electrochemical interfacial kinetics

3 Page 2/31 Mathematical Modeling All Solid State Batteries Introduction Introduction Project LISA: Lithium insertation compounts for solar devices Part of the BMBF network: fundamental research on renewable energies Compound Project: Uni Kiel, TU Darmstadt, ZSW Ulm, Uni Ulm First example for considered material-configuration: Discharge of a Lithium Ion Cell flux of electrons Li Li 5+x BaLa 2 Ta 2 O x T io 2 Anode Electrolyte Cathode flux of Lithium Ions Solid state diffusion deflector deflector intercalted Lithium Crystal Mathematical modeling and optimization of all solid state thin film lithium ion batteries

4 Page 3/31 Mathematical Modeling All Solid State Batteries Derivation of the 1-D model equations Part I: Derivation and Numerics of a model for all solid-state lithium ion batteries

5 Page 4/31 Mathematical Modeling All Solid State Batteries Model assumptions Derivation of the 1-D model equations Model assumptions z y Schematic 3-D Cell Deflector Anode Isotropic in y, z dimensions Perfect planar interfaces x Electrolyte Cathode Anode Electrolyte Cathode Ω 1 Ω 2 Model Geometry

6 Page 4/31 Mathematical Modeling All Solid State Batteries Model assumptions Derivation of the 1-D model equations Model assumptions flux of electrons Isotropic in y, z dimensions Perfect planar interfaces Reactions only at the interfaces Anode Electrolyte Cathode Li Li + + e Li + + e Li deflector deflector

7 Page 4/31 Mathematical Modeling All Solid State Batteries Model assumptions Derivation of the 1-D model equations Model assumptions Isotropic in y, z dimensions Perfect planar interfaces flux of electrons Reactions only at the interfaces Anode Electrolyte Cathode No repulsive ion-ion interaction D Li + = const. Li Li + + e Li + + e Li deflector deflector

8 Page 4/31 Mathematical Modeling All Solid State Batteries Model assumptions Derivation of the 1-D model equations Model assumptions Isotropic in y, z dimensions Perfect planar interfaces Reactions only at the interfaces No repulsive ion-ion interaction D Li + = const. Potentiostatic discharge

9 Page 4/31 Mathematical Modeling All Solid State Batteries Model assumptions Derivation of the 1-D model equations Model assumptions Potential Isotropic in y, z dimensions Perfect planar interfaces Reactions only at the interfaces No repulsive ion-ion interaction D Li + = const. Potentiostatic discharge No potential variation as function of status of charge (SOC) or intercalated lithium

10 Page 4/31 Mathematical Modeling All Solid State Batteries Model assumptions Derivation of the 1-D model equations Model assumptions Discharge of a Lithium Ion Cell Isotropic in y, z dimensions flux of electrons Perfect planar interfaces Reactions only at the interfaces Anode Electrolyte flux of Lithium Ions Cathode No repulsive ion-ion interaction D Li + = const. Potentiostatic discharge deflector deflector No potential variation as function of status of charge (SOC) or intercalated lithium No mechanical effects of lithium intercalation Crystal

11 Page 5/31 Mathematical Modeling All Solid State Batteries Transport equations Derivation of the 1-D model equations Overview Lithium Foil Solid Electrolyte Intercalation Cathode Ω 1 Ω 2 Discharge

12 Page 5/31 Mathematical Modeling All Solid State Batteries Transport equations Derivation of the 1-D model equations Overview Lithium Foil Solid Electrolyte Intercalation Cathode Ω 1 Ω 2 Discharge Ion diffusion Electric potential Variables: C Li +(X, t), Φ(X, t)

13 Page 5/31 Mathematical Modeling All Solid State Batteries Transport equations Derivation of the 1-D model equations Overview Lithium Foil Solid Electrolyte Intercalation Cathode Ω 1 Ω 2 Discharge Ion diffusion Electric potential Variables: C Li +(X, t), Φ(X, t) Intercalted particle diffusion Variable: C Li (X, t)

14 Page 5/31 Mathematical Modeling All Solid State Batteries Transport equations Derivation of the 1-D model equations Overview Lithium Foil Solid Electrolyte Intercalation Cathode Ω 1 Ω 2 Discharge Ion diffusion Electric potential Variables: C Li +(X, t), Φ(X, t) Intercalted particle diffusion Variable: C Li (X, t) Chemical reaction (deintercalation) Stern Layer potential drop Chemical reaction (intercalation) Stern Layer potential drop

15 Page 6/31 Mathematical Modeling All Solid State Batteries Transport equations Derivation of the 1-D model equations Transport equations Lithium Foil Solid Electrolyte L 1 L 2 Discharge Intercalation Electrode Continuity equation: C i t = J i X, i = Li +, Li C i - concentration of species i J i - flux of species i in...

16 Page 6/31 Mathematical Modeling All Solid State Batteries Transport equations Derivation of the 1-D model equations Transport equations Lithium Foil Solid electrolyte Solid Electrolyte L 1 L 2 Discharge Intercalation Electrode Continuity equation: C i t = J i X, C Nernst Planck flux: J Li + = D Li + Li + X ) Poisson equation: X Intercalation electrode ( ε El Φ X i = Li +, Li C i - concentration of species i J i - flux of species i in... = ρ El B Li +C Li + Φ X Fickian flux: J Li = D Li C Li X

17 Page 6/31 Mathematical Modeling All Solid State Batteries Transport equations Derivation of the 1-D model equations Electrostatic Field E E =0 E Solid electrolyte C Nernst Planck flux: J Li + = D Li + Li + X Poisson equation: X ( εel Φ X ) B Li +C Li + Φ X = ρ El = F (C Li + C } Anions ) {{} =fixed

18 Page 7/31 Mathematical Modeling All Solid State Batteries Transport equations Derivation of the 1-D model equations Non-dimensional transport equations Non-dimensional variables without subindices. Solid electrolyte Nernst Planck equation: τ c = (A 1 c + A 1 c Φ) in Q 1 := Ω 1 (0, T ) Poisson equation: Φ = 1 2ε 2 (c c Anion ) in Q 1 := Ω 1 (0, T ) eff Intercalation electrode Diffusion equation: τ ρ = (A 2 ρ) in Q 2 := Ω 2 (0, T )

19 Page 8/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Derivation of Robin Boundary conditions for Φ solid bulk material E E Φ(X 1 ) = Φ 0 Φ S E =0 E Φ 0 Φ(X 1 ) Φ S Stern layer X 1 =0

20 Page 8/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Derivation of Robin Boundary conditions for Φ solid bulk material E E Φ(X 1 ) = Φ 0 Φ S E =0 E Φ 0 Φ(X 1 ) Stern layer X 1 =0 Φ S Treat the Stern Layer as (plate) capacitor Ĉ S = Q = 1 Aε 0 ε r E Φ S Φ S Φ S = 1 C S ε 0 ε r Φ X 1 X1=X R

21 Page 8/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Derivation of Robin Boundary conditions for Φ solid bulk material E E Φ(X 1 ) = Φ 0 Φ S E =0 E Φ 0 Φ(X 1 ) Stern layer X 1 =0 Φ S Robin Boundary condition for Φ: (Bazant, Chu, Bayly, 2005) Treat the Stern Layer as (plate) capacitor Ĉ S = Φ n + C S Φ ε 0 ε X1=X = C S Φ 0 r R ε 0 ε r Q = 1 Aε 0 ε r E Φ S Φ S Φ S = 1 C S ε 0 ε r Φ X 1 X1=X R

22 Page 9/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Mathematical description of chemical reactions Basics: Assume a fist order redox reaction A B C A t = k f C A }{{} A B + k b C B }{{} B A C A, C B - concentrations of A, B, k f, k b - reaction rate coefficients.

23 Page 9/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Mathematical description of chemical reactions Basics: Assume a fist order redox reaction A B C A t C A, C B - concentrations of A, B, k f, k b - reaction rate coefficients. Transition state theory: k i = k BT h = k f C A }{{} A B exp with G 0 i - Gibbs free energy of activation, T - Temperature, k B - Bolzmann constant, h - Planck constant. ( + k b C B }{{} B A ) G 0 i RT

24 Page 10/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Mathematical description of chemical reactions Electrochemistry: Chemical reactions are accelerated due to electric potential differences between reacting phases. Energy G 0 f k f = k BT h k b = k BT h ( ) G 0 f exp RT ( ) G 0 b exp RT Φ α Reaction coordinate

25 Page 10/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Mathematical description of chemical reactions Electrochemistry: Chemical reactions are accelerated due to electric potential differences between reacting phases. Energy G f G 0 f G b k f = k BT h k b = k BT h ( ) (G 0 f α Φ) exp RT ( ) G 0 b exp RT Φ α Reaction coordinate

26 Page 10/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Mathematical description of chemical reactions Electrochemistry: Chemical reactions are accelerated due to electric potential differences between reacting phases. Energy G f G 0 f G b ( ) k f = ˆk +α Φ f exp RT ( ) k b = k BT (G 0 b +(1 α) Φ) exp h RT Φ α Reaction coordinate

27 Page 10/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Mathematical description of chemical reactions Electrochemistry: Chemical reactions are accelerated due to electric potential differences between reacting phases. Energy G f G 0 f G b ( ) k f = ˆk +α Φ f exp RT ( ) k b = ˆk (1 α) Φ b exp RT Φ α Reaction coordinate

28 Page 10/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Mathematical description of chemical reactions Electrochemistry: Chemical reactions are accelerated due to electric potential differences between reacting phases. Frumkin-Butler-Volmer equation C A t = ˆk f C A exp (α Φ) + ˆk b C B exp ( (1 α) Φ)

29 Page 10/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Mathematical description of chemical reactions Electrochemistry: Chemical reactions are accelerated due to electric potential differences between reacting phases. Frumkin-Butler-Volmer equation C A t = ˆk f C A exp (α Φ S ) + ˆk b C B exp ( (1 α) Φ S ) Φ S (X, t) = Φ 0 Φ(X, t) - potential drop in the Stern layer C i (X, t) - concentration of species i = A, B

30 Page 10/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Mathematical description of chemical reactions Electrochemistry: Chemical reactions are accelerated due to electric potential differences between reacting phases. Frumkin-Butler-Volmer equation C A t = ˆk f C A exp (α Φ S ) + ˆk b C B exp ( (1 α) Φ S ) Φ S (X, t) = Φ 0 Φ(X, t) - potential drop in the Stern layer C i (X, t) - concentration of species i = A, B With η = Φ Φ eq the reaction rate is given by the classical Butler-Volmer equation (Doyle, Fuller, Newman, 1993)

31 Page 11/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Relate (surface) concentration to concentration flux Definition: Surface concentration (in a continuum mechanical sense) C := C (X ) [ ] mol X ΣA d V m 2 Surface reaction: C A t = k f C A + k b C B

32 Page 11/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Relate (surface) concentration to concentration flux Definition: Surface concentration (in a continuum mechanical sense) C := C (X ) [ ] mol X ΣA d V m 2 Surface reaction: Γ A n C A t = k f C A + k b C B Interpretation: Surface reaction as outward flux of C A on Σ A := Γ A (0, T ). nj A =

33 Page 11/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Relate (surface) concentration to concentration flux Definition: Surface concentration (in a continuum mechanical sense) C := C (X ) [ ] mol X ΣA d V m 2 Γ A n Γ A n Surface reaction: C A t = k f C A + k b C B surface reaction Interpretation: Surface reaction as outward flux of C A on Σ A := Γ A (0, T ). nj A = C A t

34 Page 11/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Relate (surface) concentration to concentration flux Definition: Surface concentration (in a continuum mechanical sense) C := C (X ) [ ] mol X ΣA d V m 2 Γ A n Γ A n Surface reaction: C A t = k f C A + k b C B surface reaction Interpretation: Surface reaction as outward flux of C A on Σ A := Γ A (0, T ). nj A = C A t = k f C A k b C B = k f d V C A ΣA k b d V C B ΣA

35 Page 11/31 Mathematical Modeling All Solid State Batteries Boundary conditions Derivation of the 1-D model equations Relate (surface) concentration to concentration flux Definition: Surface concentration (in a continuum mechanical sense) C := C (X ) [ ] mol X ΣA d V m 2 Γ A n Γ A n Surface reaction: C A t = k f C A + k b C B surface reaction Interpretation: Surface reaction as outward flux of C A on Σ A := Γ A (0, T ). Neumann Boundary Condition for C A : nj A = ˆk f d V [C A exp (α(φ 0 Φ))] ΣA ˆk b d V [C B exp ( (1 α)(φ 0 Φ))] ΣA

36 Page 12/31 Mathematical Modeling All Solid State Batteries Classification Classification - In contrast to Bazant or other Poisson-Nernst-Planck (PNP) systems for semiconductors Model for a complete time dependent battery discharge PNP system coupled with intercalation electrode Frumkin-Butler-Volmer equation as coupling boundary condition for intercalation

37 Page 12/31 Mathematical Modeling All Solid State Batteries Classification Classification - In contrast to Bazant or other Poisson-Nernst-Planck (PNP) systems for semiconductors Model for a complete time dependent battery discharge PNP system coupled with intercalation electrode Frumkin-Butler-Volmer equation as coupling boundary condition for intercalation... Newman Solid electrolyte with fixed anion structure Double layer potential drop:... diffuse: calculated with Poisson equation... Stern: calculated with Robin boundary conditions Frumkin-Butler-Volmer equation: reaction accelerated due to Stern layer potential drop Advantage/Disadvantage: based on non measurable paramerters

38 Page 13/31 Mathematical Modeling All Solid State Batteries Classification Known mathematical Results for the related models......for nonstationary PNP systems: with homogeneous Neumann boundary conditions Existence of a unique weak solution (Gajewsky, Gröger, 1986), Asymptotic convergence to steady state (Biler, Hebisch, Nadzieja, 1994) with exponential rate (Arnold, Markowich, Toscani, 2000), Convergent finite element based discretization (Prohl, Schmunck, 2009)

39 Page 13/31 Mathematical Modeling All Solid State Batteries Classification Known mathematical Results for the related models......for nonstationary PNP systems: with homogeneous Neumann boundary conditions Existence of a unique weak solution (Gajewsky, Gröger, 1986), Asymptotic convergence to steady state (Biler, Hebisch, Nadzieja, 1994) with exponential rate (Arnold, Markowich, Toscani, 2000), Convergent finite element based discretization (Prohl, Schmunck, 2009) with homogeneous Neumann boundary conditions and inhomogeneous Dirichlet boundary conditions for species, Robin boundary condition for electrostatic potential (Gröger, 1984) Existence of a unique weak solution Existence of a unique weak solution to the stationary problem

40 Page 13/31 Mathematical Modeling All Solid State Batteries Classification Known mathematical Results for the related models......for nonstationary PNP systems: with homogeneous Neumann boundary conditions Existence of a unique weak solution (Gajewsky, Gröger, 1986), Asymptotic convergence to steady state (Biler, Hebisch, Nadzieja, 1994) with exponential rate (Arnold, Markowich, Toscani, 2000), Convergent finite element based discretization (Prohl, Schmunck, 2009) with homogeneous Neumann boundary conditions and inhomogeneous Dirichlet boundary conditions for species, Robin boundary condition for electrostatic potential (Gröger, 1984) Existence of a unique weak solution Existence of a unique weak solution to the stationary problem...for the macroscopic Newman system: (Wu, Xu, Zou, 2006) Local existence of a unique weak solution for the multidimensional case Global existence of a unique weak solution in the 1-D case

41 Page 14/31 Mathematical Modeling All Solid State Batteries Numerics Numerical simulation - Overview Numerical Simulation Initial condition of a charged cell Discharge Charge Time dependent Stationary

42 Page 15/31 Mathematical Modeling All Solid State Batteries Steady state solution Numerics Initial Condition of a charged cell Stationary system in the electrolyte ( τ c = 0): 0 = 2 c(, τ) + ( c(, τ) ) ϕ(, τ), x 1 x 1 x 1 0 = ε 2 2 x 1 ϕ(, τ) (c(, τ) c A). Numerical Simulation Initial condition of a charged cell Discharge Time Stationary dependent Charge

43 Page 15/31 Mathematical Modeling All Solid State Batteries Steady state solution Numerics Initial Condition of a charged cell Stationary system in the electrolyte ( τ c = 0): 0 = 2 c(, τ) + ( c(, τ) ) ϕ(, τ), x 1 x 1 x 1 0 = ε 2 2 x 1 ϕ(, τ) (c(, τ) c A). Numerical Simulation Initial condition of a charged cell Discharge Time Stationary dependent Charge Boundary conditions on x 1 {0, 1}: c(x 1, τ) + c(x 1, τ) ϕ(x 1, τ) = 0, x 1 x 1 ϕ(x 1, τ) + 1 x 1 γε ϕ(x 1, τ) = ϕ 0, with λ S := ε 0 ε r /C S, γ := λ S /λ D, λ D := ε b RT 2F 2 C bulk Li +.

44 Page 15/31 Mathematical Modeling All Solid State Batteries Steady state solution Numerics Initial Condition of a charged cell Numerical Simulation Stationary system in the electrolyte ( τ c = 0): 0 = 2 c(, τ) + ( c(, τ) ) ϕ(, τ), x 1 x 1 x 1 0 = ε 2 2 x 1 ϕ(, τ) (c(, τ) c A). Initial condition of a charged cell Discharge Time Stationary dependent Charge Boundary conditions on x 1 {0, 1}: c(x 1, τ) + c(x 1, τ) ϕ(x 1, τ) = 0, x 1 x 1 ϕ(x 1, τ) + 1 x 1 γε ϕ(x 1, τ) = ϕ 0, with λ S := ε 0 ε r /C S, γ := λ S /λ D, λ D := ε b RT 2F 2 C bulk Li +. Additional weak constraint 1 0 c(x 1 ) dx 1 = c A.

45 Page 16/31 Mathematical Modeling All Solid State Batteries Steady state solution Numerics Stationary solution with reaction - Variation of λ D Numerical Simulation Initial condition of a charged cell Discharge Charge Time dependent Stationary 3 Potential and Concentration Distribution across the Solid Elektrolyte Potential Drop ΦS 100 Paramter Setting: ε1 = 1.00e+02 L1 = 1.00e-05 D1 = 1.00e-13 λd = 5.00e-10 λd =(εε1rt /2F 2 c M ax Li ) 1/ 2 Φ 2 c + Li Bulk Li + concentration: c B Li u l k =3.0e+04 [mol/m 3 ] Max. Li + concentration: c M Li ax =3.4e+06 [mol/m 3 ] 1 Remark: Frumkin Butler Volmer Eq. R exp(±α Φ S ) ΦS

46 Page 16/31 Mathematical Modeling All Solid State Batteries Steady state solution Numerics Stationary solution with reaction - Variation of λ D Numerical Simulation Initial condition of a charged cell Discharge Charge Time dependent Stationary Potential and Concentration Distribution across the Solid Elektrolyte 100 Potential Drop ΦS 80 Paramter Setting: ε1 = 1.00e+02 L1 = 1.00e-05 D1 = 1.00e-13 λd = 1.00e-10 λd =(εε1rt /2F 2 c M ax Li ) 1/ 2 Φ c + Li Bulk Li + concentration: c B Li u l k =3.0e+04 [mol/m 3 ] Max. Li + concentration: c M Li ax =2.9e+06 [mol/m 3 ] Remark: Frumkin Butler Volmer Eq. R exp(±α Φ S ) 20 ΦS x 1 0

47 Page 16/31 Mathematical Modeling All Solid State Batteries Steady state solution Numerics Stationary solution with reaction - Variation of λ D Numerical Simulation Initial condition of a charged cell Discharge Charge Time dependent Stationary Potential and Concentration Distribution across the Solid Elektrolyte Potential Drop ΦS Paramter Setting: Φ 2 50 c + Li ε1 = 1.00e+02 L1 = 1.00e-05 D1 = 1.00e-13 λd = 6.23e-11 λd =(εε1rt /2F 2 c M ax Li ) 1/ 2 Bulk Li + concentration: c B Li u l k =3.0e+04 [mol/m 3 ] Max. Li + concentration: c M Li ax =2.6e+06 [mol/m 3 ] Remark: Frumkin Butler Volmer Eq. R exp(±α Φ S ) ΦS

48 Page 17/31 Mathematical Modeling All Solid State Batteries Transient solution Numerics Model equations for discharge d-dimensional equation system τ c = A 1( c + c ϕ) in Q 1, 0 = ϕ + f (c) in Q 1, τ ρ = A 2 ρ in Q 2,

49 Page 17/31 Mathematical Modeling All Solid State Batteries Transient solution Numerics Model equations for discharge d-dimensional equation system τ c = A 1( c + c ϕ) in Q 1, 0 = ϕ + f (c) in Q 1, τ ρ = A 2 ρ in Q 2, Boundary conditions n A 1 ( c + c ϕ) = R(c, ρ, ϕ) on Σ 1, Initial values ϕ n + αϕ = g on Σ 1, n A 2 ρ = R(c, ρ, ϕ) on Σ 2, c(x, 0) = c 0 (x) in Ω 1, ρ(x, 0) = ρ 0 (x) in Ω 2,

50 Page 17/31 Mathematical Modeling All Solid State Batteries Transient solution Numerics Model equations for discharge d-dimensional equation system τ c = A 1( c + c ϕ) in Q 1, 0 = ϕ + f (c) in Q 1, τ ρ = A 2 ρ in Q 2, with f (c) := 1/(2ε 2 )(c c A ), Boundary conditions n A 1 ( c + c ϕ) = R(c, ρ, ϕ) on Σ 1, Initial values ϕ n + αϕ = g on Σ 1, n A 2 ρ = R(c, ρ, ϕ) on Σ 2, c(x, 0) = c 0 (x) in Ω 1, ρ(x, 0) = ρ 0 (x) in Ω 2,

51 Page 17/31 Mathematical Modeling All Solid State Batteries Transient solution Numerics Model equations for discharge d-dimensional equation system τ c = A 1( c + c ϕ) in Q 1, 0 = ϕ + f (c) in Q 1, τ ρ = A 2 ρ in Q 2, Boundary conditions n A 1 ( c + c ϕ) = R(c, ρ, ϕ) on Σ 1, Initial values ϕ n + αϕ = g on Σ 1, n A 2 ρ = R(c, ρ, ϕ) on Σ 2, c(x, 0) = c 0 (x) in Ω 1, with f (c) := c A ), 1/(2ε { 2 )(c ρ(x, 0) = ρ 0 (x) in Ω 2, R(c, ρ, ϕ) := k c,1 ce αc 1 (ϕ0 ϕ) k a,1 c M e αa 1 (ϕ0 ϕ), on Σ 11, k c,2 ce αc 2 ϕ k a,2 ρ N ρe αa 2 ϕ, on Σ 12,

52 Page 17/31 Mathematical Modeling All Solid State Batteries Transient solution Numerics Model equations for discharge d-dimensional equation system τ c = A 1( c + c ϕ) in Q 1, 0 = ϕ + f (c) in Q 1, τ ρ = A 2 ρ in Q 2, Boundary conditions n A 1 ( c + c ϕ) = R(c, ρ, ϕ) on Σ 1, Initial values ϕ n + αϕ = g on Σ 1, n A 2 ρ = R(c, ρ, ϕ) on Σ 2, c(x, 0) = c 0 (x) in Ω 1, with f (c) := c A ), 1/(2ε { 2 )(c ρ(x, 0) = ρ 0 (x) in Ω 2, R(c, ρ, ϕ) := k c,1 ce αc 1 (ϕ0 ϕ) k a,1 c M e αa 1 (ϕ0 ϕ), on Σ 11, k c,2 ce αc 2 ϕ k a,2 ρ N ρe αa 2 ϕ, on Σ 12,

53 Page 17/31 Mathematical Modeling All Solid State Batteries Transient solution Numerics Model equations for discharge d-dimensional equation system τ c = A 1( c + c ϕ) in Q 1, 0 = ϕ + f (c) in Q 1, τ ρ = A 2 ρ in Q 2, Boundary conditions n A 1 ( c + c ϕ) = R(c, ρ, ϕ) on Σ 1, Initial values ϕ n + αϕ = g on Σ 1, n A 2 ρ = R(c, ρ, ϕ) on Σ 2, with f (c) := 1/(2ε 2 )(c c A ), R(c, ρ, ϕ) := g := { ϕ 0, on Σ 11, 0, on Σ 12, α := 1/(γε). c(x, 0) = c 0 (x) in Ω 1, ρ(x, 0) = ρ 0 (x) in Ω 2, { k c,1 ce αc 1 (ϕ0 ϕ) k a,1 c M e αa 1 (ϕ0 ϕ), on Σ 11, k c,2 ce αc 2 ϕ k a,2 ρ N ρe αa 2 ϕ, on Σ 12,

54 Page 17/31 Mathematical Modeling All Solid State Batteries Transient solution Numerics Model equations for discharge d-dimensional equation system τ c = A 1( c + c ϕ) in Q 1, 0 = ϕ + f (c) in Q 1, τ ρ = A 2 ρ in Q 2, Boundary conditions n A 1 ( c + c ϕ) = R(c, ρ, ϕ) on Σ 1, Initial values ϕ n + αϕ = g on Σ 1, n A 2 ρ = R(c, ρ, ϕ) on Σ 2, with f (c) := 1/(2ε 2 )(c c A ), R(c, ρ, ϕ) := { ϕ 0, on Σ 11, c(x, 0) = c 0 (x) in Ω 1, ρ(x, 0) = ρ 0 (x) in Ω 2, { k c,1 ce αc 1 (ϕ0 ϕ) k a,1 c M e αa 1 (ϕ0 ϕ), on Σ 11, k c,2 ce αc 2 ϕ k a,2 ρ N ρe αa 2 ϕ, on Σ 12, g := Differences to 1-D case 0, on Σ 12, α := 1/(γε). Additional isolation boundary conditions on all other boundaries

55 Page 17/31 Mathematical Modeling All Solid State Batteries Transient solution Numerics Model equations for discharge d-dimensional equation system τ c = A 1( c + c ϕ) in Q 1, 0 = ϕ + f (c) in Q 1, τ ρ = A 2 ρ in Q 2, Boundary conditions n A 1 ( c + c ϕ) = R(c, ρ, ϕ) on Σ 1, Initial values ϕ n + αϕ = g on Σ 1, n A 2 ρ = R(c, ρ, ϕ) on Σ 2, with f (c) := 1/(2ε 2 )(c c A ), R(c, ρ, ϕ) := { ϕ 0, on Σ 11, g := 0, on Σ 12, α := 1/(γε). c(x, 0) = c 0 (x) in Ω 1, ρ(x, 0) = ρ 0 (x) in Ω 2, { k c,1 ce αc 1 (ϕ0 ϕ) k a,1 c M e αa 1 (ϕ0 ϕ), on Σ 11, k c,2 ce αc 2 ϕ k a,2 ρ N ρe αa 2 ϕ, on Σ 12, Nonlinearities

56 Page 18/31 Mathematical Modeling All Solid State Batteries Transient solution Numerics Time dependent discharge - D 1 = Numerical Simulation Initial condition of a charged cell Discharge Charge Time dependent Stationary

57 Page 19/31 Mathematical Modeling All Solid State Batteries Transient solution Numerics Time dependent discharge - D 1 = Numerical Simulation Initial condition of a charged cell Discharge Charge Time dependent Stationary

58 Page 20/31 Mathematical Modeling All Solid State Batteries Weak formulation Towards Optimization With suitable test functions c, ϕ, ρ denote A 1 (w, ϕ), c := A 1 ( w + w ϕ) c dx Ω 1 G 1 (w, ϕ, u), c := R(w, ϕ, u) c dx Γ 1 System of equations c (τ) + A 1 (c(τ), ϕ(τ)) = G 1 (c(τ), ϕ(τ), ρ(τ)),

59 Page 20/31 Mathematical Modeling All Solid State Batteries Weak formulation Towards Optimization With suitable test functions c, ϕ, ρ denote A 1 (w, ϕ), c := A 1 ( w + w ϕ) c dx Ω 1 G 1 (w, ϕ, u), c := R(w, ϕ, u) c dx Γ 1 Bϕ, ϕ := ϕ ϕ dx + (αϕ g) ϕ dx Ω 1 Γ 1 System of equations c (τ) + A 1 (c(τ), ϕ(τ)) = G 1 (c(τ), ϕ(τ), ρ(τ)), Bϕ(τ) = f (c(τ)),

60 Page 20/31 Mathematical Modeling All Solid State Batteries Weak formulation Towards Optimization With suitable test functions c, ϕ, ρ denote A 1 (w, ϕ), c := A 1 ( w + w ϕ) c dx Ω 1 G 1 (w, ϕ, u), c := R(w, ϕ, u) c dx Γ 1 Bϕ, ϕ := ϕ ϕ dx + (αϕ g) ϕ dx Ω 1 Γ 1 A 2 (u, ϕ), ρ := A 2 u ρ dx Ω 2 G 2 (w, ϕ, u), ρ := R(w, ϕ, u) ρ dx Γ 2 System of equations c (τ) + A 1 (c(τ), ϕ(τ)) = G 1 (c(τ), ϕ(τ), ρ(τ)), Bϕ(τ) = f (c(τ)), ρ (τ) + A 2 (ρ(τ)) = G 2 (c(τ), ϕ(τ), ρ(τ)),

61 Page 20/31 Mathematical Modeling All Solid State Batteries Weak formulation Towards Optimization With suitable test functions c, ϕ, ρ denote A 1 (w, ϕ), c := A 1 ( w + w ϕ) c dx Ω 1 G 1 (w, ϕ, u), c := R(w, ϕ, u) c dx Γ 1 Bϕ, ϕ := ϕ ϕ dx + (αϕ g) ϕ dx Ω 1 Γ 1 A 2 (u, ϕ), ρ := A 2 u ρ dx Ω 2 G 2 (w, ϕ, u), ρ := R(w, ϕ, u) ρ dx Γ 2 System of equations c (τ) + A 1 (c(τ), ϕ(τ)) = G 1 (c(τ), ϕ(τ), ρ(τ)), Bϕ(τ) = f (c(τ)), ρ (τ) + A 2 (ρ(τ)) = G 2 (c(τ), ϕ(τ), ρ(τ)), c(x, 0) = c 0 (x) ρ(x, 0) = ρ 0 (x).

62 Page 21/31 Mathematical Modeling All Solid State Batteries Performance citeria Towards Optimization Specific energy E = 1 M A T Average specific power P = 1 M A T 0 T 0 iv dt iv dt Specific Energy [W h/kg ] 10 2 Ragone Plot for Anode particle radii from 4.5µm to 9.5µm RA = 4.5µ m RA = 5µ m RA = 5.5µ m RA = 6.5µ m RA = 7.5µ m RA = 8.5µ m RA = 9.5µ m Specific Power [W /kg ] T - discharge time, M A - mass per unit area, V - cell potential,

63 Page 21/31 Mathematical Modeling All Solid State Batteries Performance citeria Towards Optimization Specific energy E = 1 M A T Average specific power P = 1 M A T 0 T 0 iv dt iv dt Specific Energy [W h/kg ] 10 2 Ragone Plot for Anode particle radii from 4.5µm to 9.5µm RA = 4.5µ m RA = 5µ m RA = 5.5µ m RA = 6.5µ m RA = 7.5µ m RA = 8.5µ m RA = 9.5µ m Specific Power [W /kg ] T - discharge time, M A - mass per unit area, V - cell potential, Total (discharge) current density: i = i(t; V ) (for galvanostatic discharge to a given applied voltage):

64 Page 21/31 Mathematical Modeling All Solid State Batteries Performance citeria Towards Optimization Specific energy E(i; V ) = 1 M A T Average specific power P(i; V ) = 1 M A T 0 T 0 iv dt iv dt Specific Energy [W h/kg ] 10 2 Ragone Plot for Anode particle radii from 4.5µm to 9.5µm RA = 4.5µ m RA = 5µ m RA = 5.5µ m RA = 6.5µ m RA = 7.5µ m RA = 8.5µ m RA = 9.5µ m Specific Power [W /kg ] T - discharge time, M A - mass per unit area, V - cell potential, Total (discharge) current density: i = i(t; V ) (for galvanostatic discharge to a given applied voltage): sum of conduction and displacement current densities i(t) = i c (X, t) ε b t X Φ(X, t) i c = F [D Li + X c(x, t) + B Li + X Φ(X, t)], A electrode area, F Faraday constant.

65 Page 21/31 Mathematical Modeling All Solid State Batteries Performance citeria Towards Optimization Specific energy E = 1 M A T Average specific power P = 1 M A T 0 T 0 iv dt iv dt Specific Energy [W h/kg ] 10 2 Ragone Plot for Anode particle radii from 4.5µm to 9.5µm RA = 4.5µ m RA = 5µ m RA = 5.5µ m RA = 6.5µ m RA = 7.5µ m RA = 8.5µ m RA = 9.5µ m Specific Power [W /kg ] T - discharge time, M A - mass per unit area, V - cell potential, Total (discharge) current density: i = i(t; V ) (for galvanostatic discharge to a given applied voltage): sum of conduction and displacement current densities i(t) = i c (X, t) ε b t X Φ(X, t) i c = F [D Li + X c(x, t) + B Li + X Φ(X, t)], A electrode area, F Faraday constant.

66 Page 22/31 Mathematical Modeling All Solid State Batteries Performance citeria Towards Optimization Denote c(τ) := c(τ; µ), ϕ(τ) := ϕ(τ; µ), ρ(τ) := ρ(τ; µ) max E(c, ϕ, ρ; µ), max P(c, ϕ, ρ; µ) subject to c (τ) + A 1 (c(τ), ϕ(τ)) = G 1 (c(τ), ϕ(τ), ρ(τ)), on Q 1, Bϕ(τ) = f (c(τ)), on Q 1, ρ (τ) + A 2 (ρ(τ)) = G 2 (c(τ), ϕ(τ), ρ(τ)), on Q 2, c(x, 0) = c 0 (x), on Ω 1, ρ(x, 0) = ρ 0 (x), on Ω 2.

67 Page 22/31 Mathematical Modeling All Solid State Batteries Performance citeria Towards Optimization Denote c(τ) := c(τ; µ), ϕ(τ) := ϕ(τ; µ), ρ(τ) := ρ(τ; µ) max E(c, ϕ, ρ; µ), max P(c, ϕ, ρ; µ) subject to Question: Reasonable µ? Material parameters c (τ) + A 1 (c(τ), ϕ(τ); µ) = G 1 (c(τ), ϕ(τ), ρ(τ)), on Q 1, Diffusion coefficients - D Li +, D Li Bϕ(τ) = f (c(τ)), on Q 1, ρ (τ) + A 2 (ρ(τ); µ) = G 2 (c(τ), ϕ(τ), ρ(τ)), on Q 2, c(x, 0) = c 0 (x), on Ω 1, ρ(x, 0) = ρ 0 (x), on Ω 2.

68 Page 22/31 Mathematical Modeling All Solid State Batteries Performance citeria Towards Optimization Denote c(τ) := c(τ; µ), ϕ(τ) := ϕ(τ; µ), ρ(τ) := ρ(τ; µ) max E(c, ϕ, ρ; µ), max P(c, ϕ, ρ; µ) subject to Question: Reasonable µ? Material parameters c (τ) + A 1 (c(τ), ϕ(τ)) = G 1 (c(τ), ϕ(τ), ρ(τ)), on Q 1, Diffusion coefficients - D Li +, D Li Permittivity of the electrolyte - ε Bϕ(τ) = f (c(τ); µ), on Q 1, ρ (τ) + A 2 (ρ(τ)) = G 2 (c(τ), ϕ(τ), ρ(τ)), on Q 2, c(x, 0) = c 0 (x), on Ω 1, ρ(x, 0) = ρ 0 (x), on Ω 2.

69 Page 22/31 Mathematical Modeling All Solid State Batteries Performance citeria Towards Optimization Denote c(τ) := c(τ; µ), ϕ(τ) := ϕ(τ; µ), ρ(τ) := ρ(τ; µ) max E(c, ϕ, ρ; µ), max P(c, ϕ, ρ; µ) subject to Question: Reasonable µ? Material parameters c (τ) + A 1 (c(τ), ϕ(τ)) = G 1 (c(τ), ϕ(τ), ρ(τ)), on Q 1, Diffusion coefficients - D Li +, D Li Permittivity of the electrolyte - ε Stiffness of the intercalation electrode Bϕ(τ) = f (c(τ)), on Q 1, ρ (τ) + A 2 (ρ(τ)) = G 2 (c(τ), ϕ(τ), ρ(τ)), on Q 2, c(x, 0) = c 0 (x), on Ω 1, ρ(x, 0) = ρ 0 (x), on Ω 2.

70 Page 22/31 Mathematical Modeling All Solid State Batteries Performance citeria Towards Optimization Denote c(τ) := c(τ; µ), ϕ(τ) := ϕ(τ; µ), ρ(τ) := ρ(τ; µ) max E(c, ϕ, ρ; µ), max P(c, ϕ, ρ; µ) subject to Question: Reasonable µ? Material parameters c (τ) + A 1 (c(τ), ϕ(τ); µ) = G 1 (c(τ), ϕ(τ), ρ(τ)), on Q 1, Diffusion coefficients - D Li +, D Li Permittivity of the electrolyte - ε Stiffness of the intercalation electrode Geometric parameters Bϕ(τ) = f (c(τ)), on Q 1, ρ (τ) + A 2 (ρ(τ); µ) = G 2 (c(τ), ϕ(τ), ρ(τ)), on Q 2, c(x, 0) = c 0 (x), on Ω 1, ρ(x, 0) = ρ 0 (x), on Ω 2. Component sizes: for 1-D length of electrolyte and electrode - L 1, L 2

71 Page 22/31 Mathematical Modeling All Solid State Batteries Performance citeria Towards Optimization Denote c(τ) := c(τ; µ), ϕ(τ) := ϕ(τ; µ), ρ(τ) := ρ(τ; µ) max E(c, ϕ, ρ; µ), max P(c, ϕ, ρ; µ) subject to Question: Reasonable µ? Material parameters c (τ) + A 1 (c(τ), ϕ(τ)) = G 1 (c(τ), ϕ(τ), ρ(τ); µ), on Q 1, Diffusion coefficients - D Li +, D Li Permittivity of the electrolyte - ε Stiffness of the intercalation electrode Geometric parameters Bϕ(τ) = f (c(τ)), on Q 1, ρ (τ) + A 2 (ρ(τ)) = G 2 (c(τ), ϕ(τ), ρ(τ); µ), on Q 2, c(x, 0) = c 0 (x), on Ω 1, ρ(x, 0) = ρ 0 (x), on Ω 2. Component sizes: for 1-D length of electrolyte and electrode - L 1, L 2 Contact surface

72 Page 22/31 Mathematical Modeling All Solid State Batteries Performance citeria Towards Optimization Denote c(τ) := c(τ; µ), ϕ(τ) := ϕ(τ; µ), ρ(τ) := ρ(τ; µ) max E(c, ϕ, ρ; µ), max P(c, ϕ, ρ; µ) subject to Question: Reasonable µ? Material parameters c (τ) + A 1 (c(τ), ϕ(τ)) = G 1 (c(τ), ϕ(τ), ρ(τ)), on Q 1, Diffusion coefficients - D Li +, D Li Permittivity of the electrolyte - ε Stiffness of the intercalation electrode Geometric parameters Bϕ(τ) = f (c(τ)), on Q 1, ρ (τ) + A 2 (ρ(τ)) = G 2 (c(τ), ϕ(τ), ρ(τ)), on Q 2, c(x, 0) = c 0 (x), on Ω 1, ρ(x, 0) = ρ 0 (x), on Ω 2. Component sizes: for 1-D length of electrolyte and electrode - L 1, L 2 Contact surface Composition of the electrode (not incorperated jet)

73 Page 23/31 Mathematical Modeling All Solid State Batteries Incorperating mechanical effects Part II: Incorperating mechanical effects

74 Page 24/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Total free energy Homogeneous free energy Frèchet derivation Composition dependent energy Strain energy density Strain Chemical potential Stress Generell diffusion Γ = B µ Composition dependent reaction rate Reformulation Fick s diffusion with concentration dependent coefficient Γ = D(c) c

75 Page 25/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Total free energy G[C Li, e] = Ω f 0 (C Li ) C Li K C Li + W(C Li, e) dx }{{} =:F (C Li (X ), C Li (X )), def. g:= C Li Total free energy Homogeneous free energy Frèchet derivation Composition dependent energy Strain energy density Strain Chemical Potential Stress Generell diffusion Γ = B µ Composition dependent reaction rate Reformulation Fick s diffusion with concentration dependent coefficient Γ = D(c) c

76 Page 25/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Total free energy G[C Li, e] = Ω f 0 (C Li ) C Li K C Li + W(C Li, e) dx }{{} =:F (C Li (X ), C Li (X )), def. g:= C Li Total free energy Homogeneous free energy Free energy (per molecule) of a homogeneous system Frèchet derivation Composition dependent energy Strain energy density Strain Chemical Potential Stress Generell diffusion Γ = B µ Composition dependent reaction rate Reformulation Fick s diffusion with concentration dependent coefficient Γ = D(c) c

77 Page 25/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Total free energy G[C Li, e] = Ω f 0 (C Li ) C Li K C Li + W(C Li, e) dx }{{} =:F (C Li (X ), C Li (X )), def. g:= C Li Total free energy Homogeneous free energy Free energy (per molecule) of a homogeneous system Composition dependent energy Composition dependent energy Frèchet derivation Strain energy density Strain Chemical Potential Stress Generell diffusion Γ = B µ Composition dependent reaction rate Reformulation Fick s diffusion with concentration dependent coefficient Γ = D(c) c

78 Page 25/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Total free energy G[C Li, e] = Ω f 0 (C Li ) C Li K C Li + W(C Li, e) dx }{{} =:F (C Li (X ), C Li (X )), def. g:= C Li Total free energy Homogeneous free energy Free energy (per molecule) of a homogeneous system Composition dependent energy Strain energy density (in a material point) Frèchet derivation Chemical Potential Generell diffusion Γ = B µ Composition dependent energy Strain energy density Strain Stress Composition dependent reaction rate Reformulation Fick s diffusion with concentration dependent coefficient Γ = D(c) c

79 Page 25/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Total free energy G[C Li, e] = Ω f 0 (C Li ) C Li K C Li + W(C Li, e) dx }{{} =:F (C Li (X ), C Li (X )), def. g:= C Li Total free energy Homogeneous free energy Free energy (per molecule) of a homogeneous system Composition dependent energy Strain energy density (in a material point) Frèchet derivation Generell diffusion Γ = B µ Chemical Potential Composition dependent energy Strain energy density Strain Stress Composition dependent reaction rate Reformulation Fick s diffusion with concentration dependent coefficient Γ = D(c) c Chemical potential µ Li := δg[c Li] δc Li (X )

80 Page 26/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Cahn-Hilliard equation Instead of Fick s Diffusion... t C Li = (D Li C Li )

81 Page 26/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Cahn-Hilliard equation Instead of Fick s Diffusion... t C Li = (D Li C Li )..diffusion in the chemical potential gradient t C Li = (B Li µ Li ) ( = B Li δg[c ) Li] δc Li (X )

82 Page 26/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Cahn-Hilliard equation Instead of Fick s Diffusion... t C Li = (D Li C Li )..diffusion in the chemical potential gradient t C Li = (B Li µ Li ) ( = B Li δg[c ) Li] δc Li (X ) δg[c Li ] δc Li (X ) = F d F C Li dx g = f (C Li(X )) K C Li (X ) + W(C Li(X ), e(x )) C Li C Li

83 Page 26/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Cahn-Hilliard equation Instead of Fick s Diffusion... t C Li = (D Li C Li )..diffusion in the chemical potential gradient t C Li = (B Li µ Li ) ( = B Li δg[c ) Li] δc Li (X ) δg[c Li ] δc Li (X ) = F d F C Li dx g = f (C Li(X )) K C Li (X ) + W(C Li(X ), e(x )) C Li C Li Total free energy Homogeneous free energy Composition dependent energy Frèchet derivation Strain energy density Strain Chemical Potential Stress Generell diffusion Composition dependent Γ = B µ reaction rate Reformulation Fick s diffusion with concentration dependent coefficient Γ = D(c) c ( f (C (X )) = t C = B K C (X ) + C Problem: fourth oder PDE ) W(C (X ), e(x )) C

84 Page 27/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Strain energy density W.l.o.g.... Linearised mechanical strain (relative to the referenz configuration) e(u(x )) = 1 2 ( u(x ) + u(x ) T )

85 Page 27/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Strain energy density W.l.o.g.... Linearised mechanical strain (relative to the referenz configuration) e(u(x )) = 1 2 ( u(x ) + u(x ) T ) Stress-free strain (dependent of local cristal structure and local composition) e 0 (C Li ) = M C Li

86 Page 27/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Strain energy density W.l.o.g.... Linearised mechanical strain (relative to the referenz configuration) e(u(x )) = 1 2 ( u(x ) + u(x ) T ) Stress-free strain (dependent of local cristal structure and local composition) Frèchet derivation Generell diffusion Γ = B µ Total free energy Chemical Potential Homogeneous free energy Composition dependent energy Strain energy density Strain Stress Composition dependent reaction rate e 0 (C Li ) = M C Li... with Hooks law of linear elasticity follows Reformulation Fick s diffusion with concentration dependent coefficient Γ = D(c) c W(C Li, e) = 1 2 (e e0 (C Li )) : E(C Li ) ( e e 0 (C Li ) ) }{{}}{{} =: e(c Li ) σ( e(c Li ))

87 Page 28/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Equation System in the intercalation electrode Generalized diffusion equation t C Li = B Li µ Li µ = δg[c Li, u] δc Li = f (C Li) C Li K C Li + W(C Li, u) C Li

88 Page 28/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Equation System in the intercalation electrode Generalized diffusion equation t C Li = B Li µ Li µ = δg[c Li, u] δc Li = f (C Li) C Li Quasi-static mechanic equilibrium (define F := u) σ = 0 σ = δg[c Li, u] δu σ = σ T = W(C Li, u) F K C Li + W(C Li, u) C Li Hidden assumption: Mechanical equilibrium is attaint on a much faster time scale than diffusion takes place.

89 Page 28/31 Mathematical Modeling All Solid State Batteries Cahn-Hilliard Approach Incorperating mechanical effects Equation System in the intercalation electrode Ficks diffusion t C Li = D Li C Li Quasi-static mechanic equilibrium (define F := u) σ = 0 σ = δg[c Li, u] δu σ = σ T = W(C Li, u) F

90 Page 29/31 Mathematical Modeling All Solid State Batteries Electrochemical interfacial kinetics Incorperating mechanical effects Electrochemical interfacial kinetics Energy G f G 0 f G b Frumkin-Butler-Volmer equation (electrochemical driving force) Φ α Reaction coordinate So far, reaction rate depending on potential drop in the stern layer [ ] ] R elec = k f,i C A e α( Φ S ) k b,i [C B e (1 α) Φ S

91 Page 29/31 Mathematical Modeling All Solid State Batteries Electrochemical interfacial kinetics Incorperating mechanical effects Electrochemical interfacial kinetics Energy G f G 0 f G b Frumkin-Butler-Volmer equation (electrochemical driving force) Φ α Reaction coordinate So far, reaction rate depending on potential drop in the stern layer [ ] ] R elec = k f,i C A e α( Φ S ) k b,i [C B e (1 α) Φ S

92 Page 29/31 Mathematical Modeling All Solid State Batteries Electrochemical interfacial kinetics Incorperating mechanical effects Electrochemical interfacial kinetics Total free energy Homogeneous free energy Composition dependent energy Frèchet derivation Strain energy density Strain Chemical potential Stress Frumkin-Butler-Volmer equation (electrochemical driving force) Generell diffusion Γ = B µ Reformulation Fick s diffusion with concentration dependent coefficient Γ = D(c) c Composition dependent reaction rate So far, reaction rate depending on potential drop in the stern layer [ ] ] R elec = k f,i C A e α( Φ S ) k b,i [C B e (1 α) Φ S... incorperate thermodynamic driving force Reaction rate also depending on the composition [ ] ] R chem = k f,i C A e β(µ A µ B ) e α( Φ S ) k b,i [C B e β(µ A µ B ) e (1 α) Φ S

93 Page 30/31 Mathematical Modeling All Solid State Batteries Electrochemical interfacial kinetics Incorperating mechanical effects Boundary conditions... Γ C Γ D... for the diffusion equation Ω 1 Ω 2 Γ R discharge Γ I n (D Li + C Li +) = R chem (C Li +, C Li, Φ, µ) n (D Li + C Li +) = 0 on Γ R on Γ I

94 Page 30/31 Mathematical Modeling All Solid State Batteries Electrochemical interfacial kinetics Incorperating mechanical effects Boundary conditions... Γ C Γ D... for the diffusion equation Ω 1 Ω 2 Γ R discharge Γ I n (D Li + C Li +) = R chem (C Li +, C Li, Φ, µ) n (D Li + C Li +) = 0 on Γ R on Γ I... for the mechanical equation σn = 0 (surface forces in equilibrium) on Γ F

95 Page 30/31 Mathematical Modeling All Solid State Batteries Electrochemical interfacial kinetics Incorperating mechanical effects Boundary conditions... Γ C Γ D... for the diffusion equation Ω 1 Ω 2 Γ R discharge Γ I n (D Li + C Li +) = R chem (C Li +, C Li, Φ, µ) n (D Li + C Li +) = 0 on Γ R on Γ I... for the mechanical equation σn = 0 (surface forces in equilibrium) on Γ F... or treat electrolyte as rigid body u = 0 (fixed body) on Γ D n u 0 (contact condition) on Γ C

96 Page 30/31 Mathematical Modeling All Solid State Batteries Electrochemical interfacial kinetics Incorperating mechanical effects Boundary conditions... Γ C Γ D... for the diffusion equation Ω 1 Ω 2 Γ R discharge Γ I n (D Li + C Li +) = R chem (C Li +, C Li, Φ, µ) n (D Li + C Li +) = 0 on Γ R on Γ I... for the mechanical equation σn = 0 (surface forces in equilibrium) on Γ F... or treat electrolyte as rigid body u = 0 (fixed body) on Γ D n u 0 (contact condition) on Γ C Question: Is there an attainable quasi-static mechanical equilibrium under these boundary conditions? = Change to dynamic elasticity, moving boundaries

97 Page 31/31 Mathematical Modeling All Solid State Batteries Further Questions? Questions and Discussion