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Manuel LandstorferInstitute of Numerical Mathematics15.09.2009

Mathematical Modeling All Solid StateBatteries

Toskana


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Page 1/23 Mathematical Modeling All Solid State Batteries

Introduction

Transport Equtions

Boundary Condtions for

(Flux) boundary conditions for c

Summary: Equation System and Conclusion


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Page 2/23 Mathematical Modeling All Solid State Batteries Introduction

Introduction

Project LISA: Lithium Insertationsverbindungen fur Solare Anwendungen

Part of the BMBF network: fundamental research on renewable energies Compound Project: Uni Kiel, TU Darmstadt, ZSW Ulm, Uni Ulm

Uni Ulm: Mathematical Modeling of all solid state thin film lithium ionbatteries


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Introduction

Project LISA: Lithium Insertationsverbindungen fur Solare Anwendungen

Part of the BMBF network: fundamental research on renewable energies Compound Project: Uni Kiel, TU Darmstadt, ZSW Ulm, Uni Ulm

Uni Ulm: Mathematical Modeling of all solid state thin film lithium ionbatteries

Tasks

Developing a rigorous Mathematical Model

Account for various effects on various time/length scales

Numerics and Simulation Validation with Experimental Data


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What is a Lithium Ion Battery?

Li Li++ e

Li++ e

Li

AE

ux of electrons

Anode CathodeElectrolyte

flux of Lithium Ions

deflector

deflector

Discharge of a Lithium Ion Cell

Crystal

LixTi1xO2LixCo1xO2

C


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What is a Lithium Ion Battery?

Li Li++ e

Li++ e

Li

AE

ux of electrons

Anode CathodeElectrolyte

flux of Lithium Ions

deflector

deflector

Discharge of a Lithium Ion Cell

Crystal

LixTi1xO2LixCo1xO2

C

Solid state diffusion


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Page 4/23 Mathematical Modeling All Solid State Batteries Transport Equtions

Outline

Introduction

Transport Equtions





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Transport Equations

L1 L2

Solid

Electrolyte

Lithium

Foil

Discharge

Intercalation

Electrode


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Transport Equations

L1 L2

Solid

Electrolyte

Lithium

Foil

Discharge

Intercalation

Electrode

Solid Electrolyte

Nernst Planck Flux: Li+ = DLi+cLi+ cLi+

Continuity Equation:

tcLi+ = Li+

Poisson Equation: Electrolyte = Electrolyte


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Transport Equations

Solid Electrolyte



tcLi+ = Li+


/


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Transport Equations

E

Solid Electrolyte



tcLi+ = Li+


P 5/23


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Transport Equations

EE

E = 0

Solid Electrolyte



tcLi+ = Li+



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Page 5/23 M th ti l M d li All S lid St t B tt i T t E ti


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Transport Equations

L1 L2

Solid

Electrolyte

Lithium

Foil

Discharge

Intercalation

Electrode

Solid Electrolyte



tcLi+ = Li+

Poisson Equation: Electrolyte = F(cLi+ cAnions =const

)

Nernst Einstein Relation: =DLi+F

RT



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Transport Equations

L1 L2

Solid

Electrolyte

Lithium

Foil

Discharge

Intercalation

Electrode

Solid Electrolyte

Nernst Planck Flux: Li+ = DLi+cLi+ DLi+F

RTcLi+


tcLi+ = Li+

Poisson Equation: Electrolyte = F(cLi+ cAnions)

Intercalation Electrode

Fickian Flux: s = DscLi+


tcs = s



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Non-dimensional Transport Equations

Non-dimensional Variables

x1 = X1

L1x1 [0, 1] x2 = X

2

L2x2 [0, 1]

c(x1, ) =cLi+

cMaxLi+

(x2, ) =cs

cMaxs

(x1

) =F

RT = t



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g / Mathematical Modeling All Solid State Batteries Transport Equtions


Non-dimensional Variables

x1 =X1

L1x1 [0, 1] x2 =

X2

L2x2 [0, 1]

c(x1, ) =cLi+

cMaxLi+

(x2, ) =cs

cMaxs

(x1) =F

RT

= t

Solid Electrolyte

Nernst Planck Equation: L1

c = (D1c + D1c) in 1

Poisson Equation: 2eff =1

2 (c cAnion) in 1


Diffusion Equation: L2

= (D2) in 2



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/ g p q


x1

1

x2

2

Solid

ElectrolyteIntercalation

Electrode

Solid Electrolyte


c = (D1c + D1c) in 1


2(c cAnion) in 1



= (D2) in 2

Page 7/23 Mathematical Modeling All Solid State Batteries Boundary Condtions for


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Outline

Introduction

Transport Equtions






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Derivation of Robin boundary conditions for

EE

E = 0



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EE

E = 0



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solid!bulk!material

E



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0

S

x1 = 0

(x1)

solid!bulk!material

E

Stern Layer

(x1) = 0 S



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0

S

x1 = 0

(x1)

solid!bulk!material

E

Stern Layer

(x1) = 0 S



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0

S

x1 = 0

(x1)

solid!bulk!material

E

Stern Layer

(x1) = 0 S

treat the Stern layer as (plate)capacitor

CS =Q

S=

1

SA0r E

S = A

CS0r

d

dx

= 1

CS0r

d

dxx=x1



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0

S

x1 = 0

(x1)

solid!bulk!material

E

Stern Layer

(x1) = 0 S

treat the Stern layer as (plate)capacitor

CS =Q

S=

1

SA0r E

S = A

CS0r

d

dx

= 1

CS0r

d

dxx=x1

Robin boundary condition for : n

+ CS0r

x=x1 = CS0r 0

Page 9/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c


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Outline

Introduction

Transport Equtions






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Mathematical description of chemical reactions

Basics:

A redox reaction A B, as a reaction of 1. order is described via

d[A]

dt= kf[A] + kb [B],

where [A] and [B] are the concentrations ofA,B, respectively and k1,2 arereaction rate coefficients. (standard notation in chemistry).



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Basics:


d[A]

dt= kf[A]

AB+ kb [B]

BAwhere [A] and [B] are the concentrations ofA,B, respectively and k1,2 arereaction rate coefficients. (standard notation in chemistry).



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Basics:


dcA

dt= kfcA

AB+ kbcB

BA

where [A] and [B] are the concentrations ofA,B, respectively and k1,2 arereaction rate coefficients. (standard notation in chemistry).



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Basics:

A redox reaction A B, as a reaction of 1. order is described viawhere [A] and [B] are the concentrations ofA,B, respectively and k1,2 arereaction rate coefficients. (standard notation in chemistry).

Transiton State Theory:

ki =kBT

hexp

GiRT

with G is the Gibbs free energy of activation.



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Electrochemistry:

Chemical reactions are accelerated due to electric potential differences betweenreacting phases.



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Electrochemistry:


Reaction coordinate

Energy

G0f

kf =kBT

hexp

G0fRT

kb =kBT

h

expG0b

RT



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Electrochemistry:


Reaction coordinate

Energy

G0f

kf =kBT

hexp

G0fRT

kb =kBT

h

expG0b

RT


f


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Electrochemistry:


Gb

Reaction coordinate

Energy

G

f

G0f

kf =kBT

hexp

GfRT

kb =kBT

h

expGb

RT


M h i l d i i f h i l i


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Electrochemistry:


Gb

Reaction coordinate

Energy

G

f

G0f

kf =kBT

hexp

(G0f )

RT

kb =kBT

h

expGb

RT


M th ti l d i ti f h i l ti


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Electrochemistry:


Gb

Reaction coordinate

Energy

G

f

G0f

kf = kfexp

+

RT

kb =kBT

hexp

(G0b +)

RT


M th ti l d i ti f h i l ti


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Electrochemistry:


Gb

Reaction coordinate

Energy

G

f

G0f

kf = kfexp

+

RT

kb = kbexp RT




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Electrochemistry:


dcA

dt

= kf cA exp () + kb cB exp ()




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Electrochemistry:


dcA

dt

= kf cA exp () + kb cB exp ()

Main Question:

How to relate chemical reactions to flux boundary conditions?


Relate (surface) concentration to concentration flux


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Everything yet said about chemical reactions was for particles per Volume

molm3

.




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molm3

.

Definition: Surface concentration (in a continuum mechanical sense)

c = c(x)x

dV

mol

m2




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molm3

.


c = c(x)x

dV

mol

m2

Surface reaction:dcA

dt= kf cA + kb cB




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molm3

.


c = c(x)x

dV

mol

m2


dt= kf cA + kb cB

A

n




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molm3

.


c = c(x)x

dV

mol

m2


dt= kf cA + kb cB

surface reaction

A

nA

n




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molm3

.


c = c(x)x

dV

mol

m2

Surface reaction:dcAdt

= kf cA + kb cB

surface reaction

A

nA

n




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molm3

.


c = c(x)x

dV

mol

m2


dt= kf cA + kb cB

Interpretation: Surface reaction as outward flux ofcA.

n A = dcA

dt

x




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( )


molm3

.


c = c(x)x

dV

mol

m2


dt= kf cA + kb cB


n A = dcA

dt

x

= kf cA kb cB




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( )


molm3

.


c = c(x)x

dV

mol

m2


dt= kf cA + kb cB


n A = dcA

dt

x

= kf cA kb cB = kfdVcAx

kbdVcBx




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( )


molm3

.


c = c(x)x

dV

mol

m2

Surface reaction: dcAdt

= kf cA + kb cB


n A = kfdV cA(x) e

((0(x)))

x kbdV

cB e

((0(x)))

x

Neumann boundary condition for cA

Page 13/23 Mathematical Modeling All Solid State Batteries Summary: Equation System and Conclusion

Outline


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Introduction

Transport Equtions





Equation System


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Transport and Potential Equations


c = (D1c + D1c) in 1


2(c cAnion) in 1


= (D2) in 2


Equation System


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Transport and Potential Equations


c = (D1c + D1c) in 1


2(c cAnion) in 1


= (D2) in 2

n (D1c + D1c) = R(c, cLi,0 , 1) for x1 = 0

n (D1c + D1c) = R(c, |x2=0, 0, 2) for x1 = 1

n +CS

0r =

CS

0r 0 for x1 = 0

n + CS0r

= 0 for x1 = 1

n D2 = R(, c|x1=1, 0, 2)x2=0

n D2 = 0x2=1

R(cA, cB ,, i) = kf,i

cA e

(())

kb,i

cB e

(())


Numerical implementation


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Comsol Multiphysics: Some useful facts




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FEM for space discretization

Various methods for time discretization

Possibility to solve coupled PDE systems (Multiphysics)




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Include own code (C,C++)

Write Matlab scripts using Comsol functions

Easy to use




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Include own code (C,C++)

Write Matlab scripts using Comsol functions

Easy to use

Example: Time dependent 2D diffusionequation


Solution: Full coupled PDE system


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Battery thickness: 100nm

Distance electrode/electrolyte: 0.1nm


Solution: Full coupled PDE system cLi


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Solution: full coupled PDE system cS


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Still (a lot of) problems...


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Conclusion


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Next topics: Short term

Finish Comsol implementation

Produce examples for the Vienna ECS Meeting

Literature work on some constants

Next topics: Mid term

Switch from constant D2 to D2() Material expansion due to intercalation

Anisotropic effects

Next topics: Long term

Heterogenous Multi Scale Method on coupling boundaries Ab inito computation of model parameters

Yet unknown ideas, suggestions ...


Further Questions?


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Questions and Discussion

i

t(r, t) =

2

2m2(r, t) + V(r)(r, t)

Ab initio - The Schr odinger Equation

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