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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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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
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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Page 3/23 Mathematical Modeling All Solid State Batteries Introduction
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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Page 3/23 Mathematical Modeling All Solid State Batteries Introduction
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
Boundary Condtions for
(Flux) boundary conditions for c
Summary: Equation System and Conclusion
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Page 5/23 Mathematical Modeling All Solid State Batteries Transport Equtions
Transport Equations
L1 L2
Solid
Electrolyte
Lithium
Foil
Discharge
Intercalation
Electrode
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Page 5/23 Mathematical Modeling All Solid State Batteries Transport Equtions
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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Page 5/23 Mathematical Modeling All Solid State Batteries Transport Equtions
Transport Equations
Solid Electrolyte
Nernst Planck Flux: Li+ = DLi+cLi+ cLi+
Continuity Equation:
tcLi+ = Li+
Poisson Equation: Electrolyte = Electrolyte
/
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Page 5/23 Mathematical Modeling All Solid State Batteries Transport Equtions
Transport Equations
E
Solid Electrolyte
Nernst Planck Flux: Li+ = DLi+cLi+ cLi+
Continuity Equation:
tcLi+ = Li+
Poisson Equation: Electrolyte = Electrolyte
P 5/23
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Page 5/23 Mathematical Modeling All Solid State Batteries Transport Equtions
Transport Equations
EE
E = 0
Solid Electrolyte
Nernst Planck Flux: Li+ = DLi+cLi+ cLi+
Continuity Equation:
tcLi+ = Li+
Poisson Equation: Electrolyte = Electrolyte
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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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Page 5/23 Mathematical Modeling All Solid State Batteries Transport Equtions
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 = F(cLi+ cAnions =const
)
Nernst Einstein Relation: =DLi+F
RT
Page 5/23 Mathematical Modeling All Solid State Batteries Transport Equtions
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Page 5/23 Mathematical Modeling All Solid State Batteries Transport Equtions
Transport Equations
L1 L2
Solid
Electrolyte
Lithium
Foil
Discharge
Intercalation
Electrode
Solid Electrolyte
Nernst Planck Flux: Li+ = DLi+cLi+ DLi+F
RTcLi+
Continuity Equation:
tcLi+ = Li+
Poisson Equation: Electrolyte = F(cLi+ cAnions)
Intercalation Electrode
Fickian Flux: s = DscLi+
Continuity Equation:
tcs = s
Page 6/23 Mathematical Modeling All Solid State Batteries Transport Equtions
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Page 6/23 Mathematical Modeling All Solid State Batteries Transport Equtions
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
Page 6/23 Mathematical Modeling All Solid State Batteries Transport Equtions
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g / Mathematical Modeling All Solid State Batteries Transport Equtions
Non-dimensional Transport Equations
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
Intercalation Electrode
Diffusion Equation: L2
= (D2) in 2
Page 6/23 Mathematical Modeling All Solid State Batteries Transport Equtions
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/ g p q
Non-dimensional Transport Equations
x1
1
x2
2
Solid
ElectrolyteIntercalation
Electrode
Solid Electrolyte
Nernst Planck Equation: L1
c = (D1c + D1c) in 1
Poisson Equation: 2eff =1
2(c cAnion) in 1
Intercalation Electrode
Diffusion Equation: L2
= (D2) in 2
Page 7/23 Mathematical Modeling All Solid State Batteries Boundary Condtions for
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Outline
Introduction
Transport Equtions
Boundary Condtions for
(Flux) boundary conditions for c
Summary: Equation System and Conclusion
Page 8/23 Mathematical Modeling All Solid State Batteries Boundary Condtions for
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Derivation of Robin boundary conditions for
EE
E = 0
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Derivation of Robin boundary conditions for
EE
E = 0
Page 8/23 Mathematical Modeling All Solid State Batteries Boundary Condtions for
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Derivation of Robin boundary conditions for
solid!bulk!material
E
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Derivation of Robin boundary conditions for
0
S
x1 = 0
(x1)
solid!bulk!material
E
Stern Layer
(x1) = 0 S
Page 8/23 Mathematical Modeling All Solid State Batteries Boundary Condtions for
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Derivation of Robin boundary conditions for
0
S
x1 = 0
(x1)
solid!bulk!material
E
Stern Layer
(x1) = 0 S
Page 8/23 Mathematical Modeling All Solid State Batteries Boundary Condtions for
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Derivation of Robin boundary conditions for
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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Derivation of Robin boundary conditions for
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
Boundary Condtions for
(Flux) boundary conditions for c
Summary: Equation System and Conclusion
Page 10/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
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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).
Page 10/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
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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]
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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Mathematical description of chemical reactions
Basics:
A redox reaction A B, as a reaction of 1. order is described via
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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Mathematical description of chemical reactions
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.
Page 11/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
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Mathematical description of chemical reactions
Electrochemistry:
Chemical reactions are accelerated due to electric potential differences betweenreacting phases.
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Mathematical description of chemical reactions
Electrochemistry:
Chemical reactions are accelerated due to electric potential differences betweenreacting phases.
Reaction coordinate
Energy
G0f
kf =kBT
hexp
G0fRT
kb =kBT
h
expG0b
RT
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Mathematical description of chemical reactions
Electrochemistry:
Chemical reactions are accelerated due to electric potential differences betweenreacting phases.
Reaction coordinate
Energy
G0f
kf =kBT
hexp
G0fRT
kb =kBT
h
expG0b
RT
Page 11/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
f
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Mathematical description of chemical reactions
Electrochemistry:
Chemical reactions are accelerated due to electric potential differences betweenreacting phases.
Gb
Reaction coordinate
Energy
G
f
G0f
kf =kBT
hexp
GfRT
kb =kBT
h
expGb
RT
Page 11/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
M h i l d i i f h i l i
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Mathematical description of chemical reactions
Electrochemistry:
Chemical reactions are accelerated due to electric potential differences betweenreacting phases.
Gb
Reaction coordinate
Energy
G
f
G0f
kf =kBT
hexp
(G0f )
RT
kb =kBT
h
expGb
RT
Page 11/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
M th ti l d i ti f h i l ti
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Mathematical description of chemical reactions
Electrochemistry:
Chemical reactions are accelerated due to electric potential differences betweenreacting phases.
Gb
Reaction coordinate
Energy
G
f
G0f
kf = kfexp
+
RT
kb =kBT
hexp
(G0b +)
RT
Page 11/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
M th ti l d i ti f h i l ti
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Mathematical description of chemical reactions
Electrochemistry:
Chemical reactions are accelerated due to electric potential differences betweenreacting phases.
Gb
Reaction coordinate
Energy
G
f
G0f
kf = kfexp
+
RT
kb = kbexp RT
Page 11/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
Mathematical description of chemical reactions
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Mathematical description of chemical reactions
Electrochemistry:
Chemical reactions are accelerated due to electric potential differences betweenreacting phases.
dcA
dt
= kf cA exp () + kb cB exp ()
Page 11/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
Mathematical description of chemical reactions
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Mathematical description of chemical reactions
Electrochemistry:
Chemical reactions are accelerated due to electric potential differences betweenreacting phases.
dcA
dt
= kf cA exp () + kb cB exp ()
Main Question:
How to relate chemical reactions to flux boundary conditions?
Page 12/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
Relate (surface) concentration to concentration flux
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Relate (surface) concentration to concentration flux
Everything yet said about chemical reactions was for particles per Volume
molm3
.
Page 12/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
Relate (surface) concentration to concentration flux
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Relate (surface) concentration to concentration flux
Everything yet said about chemical reactions was for particles per Volume
molm3
.
Definition: Surface concentration (in a continuum mechanical sense)
c = c(x)x
dV
mol
m2
Page 12/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
Relate (surface) concentration to concentration flux
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Relate (surface) concentration to concentration flux
Everything yet said about chemical reactions was for particles per Volume
molm3
.
Definition: Surface concentration (in a continuum mechanical sense)
c = c(x)x
dV
mol
m2
Surface reaction:dcA
dt= kf cA + kb cB
Page 12/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
Relate (surface) concentration to concentration flux
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Relate (surface) concentration to concentration flux
Everything yet said about chemical reactions was for particles per Volume
molm3
.
Definition: Surface concentration (in a continuum mechanical sense)
c = c(x)x
dV
mol
m2
Surface reaction:dcA
dt= kf cA + kb cB
A
n
Page 12/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
Relate (surface) concentration to concentration flux
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Relate (surface) concentration to concentration flux
Everything yet said about chemical reactions was for particles per Volume
molm3
.
Definition: Surface concentration (in a continuum mechanical sense)
c = c(x)x
dV
mol
m2
Surface reaction:dcA
dt= kf cA + kb cB
surface reaction
A
nA
n
Page 12/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
Relate (surface) concentration to concentration flux
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Relate (surface) concentration to concentration flux
Everything yet said about chemical reactions was for particles per Volume
molm3
.
Definition: Surface concentration (in a continuum mechanical sense)
c = c(x)x
dV
mol
m2
Surface reaction:dcAdt
= kf cA + kb cB
surface reaction
A
nA
n
Page 12/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
Relate (surface) concentration to concentration flux
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Relate (surface) concentration to concentration flux
Everything yet said about chemical reactions was for particles per Volume
molm3
.
Definition: Surface concentration (in a continuum mechanical sense)
c = c(x)x
dV
mol
m2
Surface reaction:dcA
dt= kf cA + kb cB
Interpretation: Surface reaction as outward flux ofcA.
n A = dcA
dt
x
Page 12/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
Relate (surface) concentration to concentration flux
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( )
Everything yet said about chemical reactions was for particles per Volume
molm3
.
Definition: Surface concentration (in a continuum mechanical sense)
c = c(x)x
dV
mol
m2
Surface reaction:dcA
dt= kf cA + kb cB
Interpretation: Surface reaction as outward flux ofcA.
n A = dcA
dt
x
= kf cA kb cB
Page 12/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
Relate (surface) concentration to concentration flux
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( )
Everything yet said about chemical reactions was for particles per Volume
molm3
.
Definition: Surface concentration (in a continuum mechanical sense)
c = c(x)x
dV
mol
m2
Surface reaction:dcA
dt= kf cA + kb cB
Interpretation: Surface reaction as outward flux ofcA.
n A = dcA
dt
x
= kf cA kb cB = kfdVcAx
kbdVcBx
Page 12/23 Mathematical Modeling All Solid State Batteries (Flux) boundary conditions for c
Relate (surface) concentration to concentration flux
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( )
Everything yet said about chemical reactions was for particles per Volume
molm3
.
Definition: Surface concentration (in a continuum mechanical sense)
c = c(x)x
dV
mol
m2
Surface reaction: dcAdt
= kf cA + kb cB
Interpretation: Surface reaction as outward flux ofcA.
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
Boundary Condtions for
(Flux) boundary conditions for c
Summary: Equation System and Conclusion
Page 14/23 Mathematical Modeling All Solid State Batteries Summary: Equation System and Conclusion
Equation System
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Transport and Potential Equations
Nernst Planck Equation: L1
c = (D1c + D1c) in 1
Poisson Equation: 2eff =1
2(c cAnion) in 1
Diffusion Equation: L2
= (D2) in 2
Page 14/23 Mathematical Modeling All Solid State Batteries Summary: Equation System and Conclusion
Equation System
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Transport and Potential Equations
Nernst Planck Equation: L1
c = (D1c + D1c) in 1
Poisson Equation: 2eff =1
2(c cAnion) in 1
Diffusion Equation: L2
= (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
(())
Page 15/23 Mathematical Modeling All Solid State Batteries Summary: Equation System and Conclusion
Numerical implementation
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Comsol Multiphysics: Some useful facts
Page 15/23 Mathematical Modeling All Solid State Batteries Summary: Equation System and Conclusion
Numerical implementation
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Comsol Multiphysics: Some useful facts
FEM for space discretization
Various methods for time discretization
Possibility to solve coupled PDE systems (Multiphysics)
Page 15/23 Mathematical Modeling All Solid State Batteries Summary: Equation System and Conclusion
Numerical implementation
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Comsol Multiphysics: Some useful facts
FEM for space discretization
Various methods for time discretization
Possibility to solve coupled PDE systems (Multiphysics)
Include own code (C,C++)
Write Matlab scripts using Comsol functions
Easy to use
Page 15/23 Mathematical Modeling All Solid State Batteries Summary: Equation System and Conclusion
Numerical implementation
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Comsol Multiphysics: Some useful facts
FEM for space discretization
Various methods for time discretization
Possibility to solve coupled PDE systems (Multiphysics)
Include own code (C,C++)
Write Matlab scripts using Comsol functions
Easy to use
Example: Time dependent 2D diffusionequation
Page 16/23 Mathematical Modeling All Solid State Batteries Summary: Equation System and Conclusion
Solution: Full coupled PDE system
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Battery thickness: 100nm
Distance electrode/electrolyte: 0.1nm
Page 17/23 Mathematical Modeling All Solid State Batteries Summary: Equation System and Conclusion
Solution: Full coupled PDE system cLi
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Battery thickness: 100nm
Distance electrode/electrolyte: 0.1nm
Page 17/23 Mathematical Modeling All Solid State Batteries Summary: Equation System and Conclusion
Solution: Full coupled PDE system cLi
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Battery thickness: 100nm
Distance electrode/electrolyte: 0.1nm
Page 17/23 Mathematical Modeling All Solid State Batteries Summary: Equation System and Conclusion
Solution: Full coupled PDE system cLi
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Battery thickness: 100nm
Distance electrode/electrolyte: 0.1nm
Page 18/23 Mathematical Modeling All Solid State Batteries Summary: Equation System and Conclusion
Solution: full coupled PDE system cS
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Battery thickness: 100nm
Distance electrode/electrolyte: 0.1nm
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Still (a lot of) problems...
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Page 20/23 Mathematical Modeling All Solid State Batteries Summary: Equation System and Conclusion
Still (a lot of) problems...
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Page 21/23 Mathematical Modeling All Solid State Batteries Summary: Equation System and Conclusion
Still (a lot of) problems...
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Page 22/23 Mathematical Modeling All Solid State Batteries
Summary: Equation System and Conclusion
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 ...
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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