Modeling of ElectrodeModeling of Electrode Materials for Li-Ion Batteries

M Atanasov J -L Barras LM. Atanasov, J. L. Barras, L. Benco, C. Daul, E. Deiss+

University of Fribourg, Switzerland+Paul Scherrer Institute, Villigen, Switzerland

B tt T h l E l tiBattery Technology Evolution

2500Battery Market

2500

NiCd2000 NiMH

Li-Ion

1500 Total

1000

500

01993 1994 1995 1996 1997 1998 1999 2000

Year

T h l i l A li tiTechnological Applications

• Portable PC ’s• Cellular phonesp• Electrical & hybrid cars• High density power storage

t• etc...

H d it k ?How does it work ?

F t f G d B ttFeatures of a Good Battery

• High voltage• High current densityg y• High cyclability : > 1000 cycles• Cheap

E l i l• Ecological• Safe

E i l d E l i l A t M t i lEconomical and Ecological Aspects : Material

Electrode : - Mn2O4 (spinel) Cheap, non-toxic, high energy density- NiO2 (layered) Expensive, non-toxic, high energy density

+O2 ( aye ed) pe s e, o o c, g e e gy de s y

- CoO2 (layered) Expensive, toxic, high energy density- Rutile (layered) Cheap, non-toxic, low energy density- Anatase (cubic) Cheap non-toxic low energy density- Anatase (cubic) Cheap, non-toxic, low energy density- V2O5 (layered) Toxic

Electrode : Graphite (layered) Cheap non toxic high energy densityElectrode : - Graphite (layered) Cheap, non-toxic, high energy density,safety problems

- Rutile (layered) Cheap, non-toxic, low energy density,f t bl

-

no safety problems- Coke (grains) Cheap, non-toxic, high current density,

safety problems

M d liModeling means :Making predictions and/or descriptions of gphenomena based as much as possible on first principles

This yields :• Basis for targetting new experiments (time and

cost saving)• Optimal design• Optimal design• Analysis of experimental and technical difficulties

M th d lMethodology

• Non empirical calculations of periodic structures

– LAPW, tight-binding• Non empirical calculations of clusters

LCAO MO– LCAO-MO• Semi-empirical calculations of peridodic

structuresstructures– Extended Hückel

• Empirical calculations of extended systems– Molecular mechanics and dynamics

• Engineering modelsFinite elements– Finite elements

LAPW (1) ( Li i d A t d Pl W )LAPW (1) ( Linearized Augmented Plane Waves )

The space is devided in two parts :a) non-overlapping spheresb) the interstitial space

Th LAPW f tiThe LAPW wave functionsin the interstitial space :

ϕ =1

eiknrϕkn=

ωe

LAPW (2)LAPW (2)

The LAPW wave functions inside the spheres :

ϕk = Al ul r, El( )+ Bl Ýul r, El( )[ ]∑ Yl r( )ϕknAlmul r, El( )+ Blm Ý u l r, El( )[ ]

lm∑ Ylm r ( )

Boundary conditions : continuity of the wave function and of its first u ct o a d o ts stderivatives

WIEN97 LAPW C l l tiDensity Functional Theory

WIEN97 LAPW CalculationsDensity Functional Theory

LAPW basis set (WIEN97)

GGA corrections : Perdew Burke Ernzerhof 1

Full periodic Boundary Conditions

GGA corrections : Perdew, Burke, Ernzerhof

Full periodic Boundary Conditions

Brillouin zone average by modified tetrahedron scheme (Blöchl et al.2)3Total energy according Weinert et al.3

References

2 P.E. Blöchl et al., Phys. Rev. B 49, 16223 (1994)3 M Weinert et al Phys Rev B 24 864 (1982)

References1 J.P. Perdew et al., Phys. Rev. Let. 77, 3865 (1996)

3 M. Weinert et al., Phys. Rev. B 24, 864 (1982)

C l l ti f th E D itCalculation of the Energy Density

The discharge is considered as a chemical reaction : LiC6 s( ) + MO2 s( ) → 6C s( ) + LiMO2 s( )6( ) 2 ( ) ( ) 2( )

where M = Metal.

At T = 0, the energy density is given byE G Vd T T, = == −0 0Δ

C l l ti f th T t D dCalculation of the Temperature DependanceTotalEWith the temperature dependance, the

Gibbs’ energy is given by :Energy

ΔGT=300 = ΔHT=300 − T ⋅ ΔST=300

But it can be showed that

T ⋅ΔST=300 << ΔH T=300Li positiondistorsion

and so the contribution is given by the ib ti l tvibrational terms

( )ENhvib T

h k T, =

3 υT=0 Equilibrium( )e h k TB −1υ T=0 Equilibrium

Position

Average intercalation voltages forAverage intercalation voltages for M2O4

Average Voltage [V](exp)

CathodeSystem

x parameter for O(exp)

Unit cell a [A](exp)

Unit cell c [A](exp)

Ti2O4

LiTi2O4

Ti :

V2O4V :

Spinels

8.47 Cubic 2.9(3.0)8.45 (8.372) Cubic

0.2664

0.2622 (0.2628)

8 18 Cubic 2 970 2640

Fd3m_

V2O4

LiV2O4

V :

F OF

8.18 Cubic 2.97

8.22 Cubic

0.2640

0.2600

C bi

Mn2O4

LiMn2O4

Mn : 8.134* (8.045) Cubic 3.67(3.9)8.083* (8.247) Cubic

0.2640* (0.2631)

0.2650* (0.2625)

0 2660Fe2O4

LiFe2O4

Fe :

Co2O4

LiCo2O4

Co :

Cubic

... Cubic

7.97 Cubic 4.7

8.04 Cubic

0.2655

0.2640

7.975 0.2660...

...

Ni2O4

LiNi2O4

Ni :

Cu2O4

LiCu2O4

Cu :

8.02 Cubic 3.87

8.12 Cubic

0.2650

0.2645

8.26 Cubic 4.67

8.24 Cubic

0.2655

0.2625

TiO2

LiTiO3

Ti :

Layered

2.99 14.2 1.81

2.93 15.1

0.265

0.255

R3m_

All voltages are calculated (measured) agains metallic lithium anode.*Spin unrestricted calculations

B d St t f M OBand Structure of Mn2O4

fcc Brillouin zone

S i t i t d DOS f M OSpin unrestricted DOS for Mn2O4

S i t i t d DOS f LiM OSpin unrestricted DOS for LiMn2O4

S i t i t d DOS f Li M OSpin unrestricted DOS for Li2Mn2O4(tetragonal distorsion)

El t t ti P t ti l i M OElectrostatic Potential in Mn2O4

In unit cell (008) Hopping Path

Fermi Level Evolution vs. Li IntercalationFermi Level Evolution vs. Li Intercalation

OCV Modeling Using Frozen BandsOCV Modeling Using Frozen Bands

i Li M Ox in LixMn2O4

E i i M d lEngineering Models

S l S t f P ti l Diff ti l E tiSolve Sytem of Partial Differential Equations(Finite Elements)

The following contributions are considered :

• Diffusion of Li in M-oxide grains• Diffusion of Li+ in electrolyte between grains

g

• Diffusion of Li in electrolyte between grains• Electrochemical reaction at grain boundaries• Porosityy• Electrical conduction in electrode• Ion conduction in electrolyte

G f• Grain size distribution of M-oxide

Simulations of Measured PotentialSimulations of Measured Potential Jump Experiments

Adjusted parameters : D = 7.0 10-10 cm2/sk0 =1.3 10-6 cm/s

s/cm

3 ]ns

ity [A

sar

ge d

en

Time [s]

Cha

Time [s]

Illustration of kinetic controlIllustration of kinetic control

Calculated Li+ Concentration in ElectrolyteCalculated Li Concentration in Electrolyte

Hybrid Model of the Li+ Insertion from theHybrid Model of the Li Insertion from the Electrolyte to the Electrode Surface

Li+ Solvent Interaction : In the SolventLi - Solvent Interaction : In the Solvent

E0S l = -qL

2 c (ε -1) (for a spherical cavity)E Solv qLi( )

2R ε(for a spherical cavity)

ε dielectric constant of the solventf

R RSolv(Li+) : radius of the Li+ cavity

E0S l (Li+) = 5 53eV RS l (Li+) = 1 286 ÅE Solv(Li ) 5.53eV RSolv(Li ) 1.286 Å

Li+ Solvent Interaction : At the SurfaceLi - Solvent Interaction : At the Surface

ε 12ESolv(r ) = E0Solv (ε-1)

εqLi

2c1 r

2

Rsolv(Li+)

0 R(1 )α

αr

r : 0 R(1-cos ) = rmax2α

ESSolv(Li+) = E0

Solv - Esolv(rmax) = 21 E0

S l (1+cos )2αE Solv(Li ) E Solv Esolv(rmax) 2 E Solv(1 cos )2

Results

Mn2O4 : ESSolv = 0.50 E0

Solv (16c insertion site) α = 180°

ESSolv = 0.79 E0

Solv (8a insertion site) α = 109.47°

Li+ Crystal Interaction

With the host crystal

Li - Crystal Interaction

Li+ - Crystal Electrostatic Closed shellI t ti E Att ti R l i= +

With the host crystal

Interaction Energy Attraction Repulsion+

Fitti f th ff ti h di t b d t t l l tiFitting of the effective charges according to band structure calculations(vide supra)

With the incoming electron

e- σ - antibonding eg of Mn4+ (Li+ - e-) - coupling

ResultsResults

Conditions of the model :Conditions of the model :Mn2O4 : Surface 010Solvent :Water

8a 16c Energy Barrier (red. charges) 0.16 eV 0.95 eVEnergy Barrier (red. charges) 0.16 eV 0.95 eVLi+ solvation energy -3.27 eV -1.46 eVLi+ - e- interaction -4.32 eV -5.06 eV

1)Li+ - e- coupling : 1) reduction of the bulk energy barrier2) increase of surface energy barrier2) increase of surface energy barrier

ResultsResults

Infinite lattice summation

S i fi i l i iS i i fi it l tti tiSemi-nfinite lattice summationSemi-infinite lattice summation

Infinite lattice summation

Octahedral 16c

Semi-infinite lattice summation

Tetraheral 8a

DFT: Heuristic approach

ρ r( )

pp

X-ray diffractionX-ray diffractionρ( )nuclear positions

ρ r( )∫ dr # electrons# electrons

( )⎛ ⎞

ρ r ( )∫ dr # electrons# electrons

∂ρ Rk( )∂Rk

⎛

⎝ ⎜

⎞

⎠ ⎟

Rk =R0

= −2Zkρ R0( )CuspCusp

HΨ = EΨ H Ψ = EΨ

General TheoryExact energy expression

1 Eel = - 12 ∑

i

∫

φi( r

→1 )∇2φi ( r

→1 )d r

→1

+ ∑A

∫

ZA

|R→

A- r→

1| ρ( r

→1 ) d r

→1

12

⎮⎮⌠

ρ( r→

1)ρ( r→

2) d r

→1 d r

→2 2

⌡⎮

| r→

1 - r→

2 |1 2

E + Exc

Parr R G ;Yang W : Density Functional Theory of Atoms AParr,R.G.;Yang,W : Density Functional Theory of Atoms A

Molecules, Oxford University Press, New York 1989

The Kohn-Sham Equationq

φ φhksφi = εiφi

hKS = - 12∇2 + ∑

ZA → →

KS 2 A |R

→A- r

→1|

⎮⌠ ρ( r

→2)

d→

+

⌡⎮⎮ 2

| r→

1 - r→

2 | d r 2 + VXC

⌡

| r 1 r 2 |

Approximate density functional theories for exchange andApproximate density functional theories for exchange and

X

theories for exchange and correlationtheories for exchange and correlation

XαLocal exchange

Xα : Local exchange functional of the homogeneous electron gas

LDALocal exchange +local correlation

LDA: Local exchange functional + local correlationfunctional of the homogeneous electron gas

local correlation

GGA GGA: Same as LDA + “non-local” gradient correctionsLocal exchange +local correlation +gradient corrections

GGA: Same as LDA non local gradient correctionsto exchange and correlation

3rd Generation

3rd Generation of functionals: Same as GGA + instilation of “exact-exchange” and + 2nd derivatives

of functionals of the density corrections

Practical ImplementationPractical Implementation

1∇ + ( )+

ρ r'( )∫ d ' +V ( )( )

⎡ ⎢

⎤ ⎥ Ψ Ψ ( )SolveSolve −

2∇ + v r( )+

ρ( )r − r'∫ dr' +VXC ρ r( )( )

⎣ ⎢

⎦ ⎥ Ψi = εiΨi r( )Solve

Kohn-Sham eqs

Solve Kohn-Sham eqsSham eqs.Sham eqs.Features:Features:LCAO expansionLCAO expansion: STO, GTO, numerical, plane wavesLCAO expansionLCAO expansion: STO, GTO, numerical, plane waves

Coulomb potentialCoulomb potential: solve Poisson’s eq. or fit ρ(r) to a set of one-center auxilliaryCoulomb potentialCoulomb potential: solve Poisson’s eq. or fit ρ(r) to a set of one-center auxilliarya set of one-center auxilliary functionsa set of one-center auxilliary functions

MatrixelementsMatrixelements: accurate numerical integration in the irreducible wedge of the MatrixelementsMatrixelements: accurate numerical integration in the irreducible wedge of the

Methodology based on Approximate DFT

α1966

αMS-X

MS-X : Make use of partial-waves as basis (37). Relatively fast. Good for ionization potentials and excitation energies (10). Total energies unreliable ( 39). No geometry optimization. Full use of symmetry. H l ti i ti t i (53f) M k f ffi tiApproximate DFT

αDV-X : Make use of numerical atomic orbitals or STO's. Avoids Muffin-tin approximation by fit of density (45a).

1966 Has relativistic extension ( 53f). Make use of muffin-tin approximation (38). Developed by K.H. Johnson (37) .

Avoids Muffin tin approximation by fit of density (45a). Accurate total energies (76d). Relativistic extension (53e). Numerical integration of matrix elements by Diophantine integration (40). Developed by Ellis and Painter (40). Extensive improvements by Delley (D-MOL-program) including new integration scheme (46c) and geometry optimization.

1970

DV-X α

HFS-LCAO : Make use of STO's . Accurate potentials (41). Full use of symmetry. Relativistic extensions (53a,b). Highly vectorized (47). Accurate total energies (49). Geometry optimization (54c). Accurate numerical integration (46b). Many auxiliary property programs Pseudo potentials (52a d)

ADF

FRIMOL 1994-

development in progress

1973 -Many auxiliary property programs . Pseudo potentials (52a,d). Embedding procedures (76h). Energy decomposition scheme (72). Developed by Baerends,Snijders,Ravenek,Vernooijs and te Velde (41,53,47,46d)

development in progressDeMon

1976 -

LCGTO-LSD : Make use of GTO's. Fit of exchange-correlation and Coulomb potential (43). Analytical calculation of matrix elements (48b). Accurate energies. Geometry optimization (54b,h). Strongly vectorized (48b). First developed by Dunlap (43) as well as Sambe and Felton (42). Extensive improvements by Salahub and Andzelm (48b) 1976 - (D-GAUSS-program) as well as Rösch (74a). Also work by Pederson (45e) and Painter (45d)

NUMOL NUMOL : Unique basis-set free program (50a,e). Accurate

1982

NUMOL : Unique basis set free program (50a,e). Accurate numerical integration (46a). Efficient generation of Coulomb potential (50c). Geometry optimization. Developed by Becke (50 ).

Modeling the Intercalation Dynamics ofModeling the Intercalation Dynamics of Li+ :Cl t St d U i M l l DFTCluster Study Using Molecular DFT

Vibrational Modes Involved in VibronicVibrational Modes Involved in Vibronic Coupling of the eg(σ*) Electron

Vibronic Coupling Model of the PolaronVibronic Coupling Model of the Polaron

Energy Contour Diagram of the PolaronicThe image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.

Energy Contour Diagram of the Polaronic Model

Calculated ( ) Electric ConductivityCalculated (-----) Electric Conductivity Using the Polaron Model

Energy Profiles of Li+ Diffusion in BulkEnergy Profiles of Li Diffusion in Bulk Mn2O4

Acknowledgements

Financial Support : Swiss Federal Office for EnergyFinancial Support : Swiss Federal Office for Energy

The Li-Ion Battery Modeling Group :

Erich Deiss,PSI

Claude Daul,Jean-Luc Barras,U i it

Michael Atanasov,Uni ersit

Lubomir Benco,U i i PSIUniversity of

FribourgUniversityof Fribourg

Universityof Fribourg

Universityof Fribourg

Cluster and Intercalation

Band Structure Calculations

Band Structure Calculations

Engineering Models

Project LeaderIntercalation Dynamics

Calculations Calculations Models