Materials for Li-Ion Batteries

Transcription

1 Modeling of Electrode Materials for Li-Ion Batteries M. Atanasov, J. -L. Barras, L. Benco, C. Daul, E. Deiss + University of Fribourg, Switzerland + Paul Scherrer Institute, Villigen, Switzerland

2 Battery Technology Evolution

3 2500 Battery Market NiCd 2000 NiMH Li-Ion 1500 Total Year

4 Technological lapplications Portable PC s Cellular phones Electrical & hybrid cars High density power storage etc...

5 How does it work?

6 Features of a Good Battery High voltage High current density High cyclability : > 1000 cycles Cheap Ecological l Safe

7 Economical and Ecological laspects : Material + - Electrode : - Mn 2 O 4 (spinel) Cheap, non-toxic, high energy density -NiO 2 (layered) ed) Expensive, e, non-toxic, o high energy e density -CoO 2 (layered) Expensive, toxic, high energy density - Rutile (layered) Cheap, non-toxic, low energy density - Anatase (cubic) Cheap, non-toxic, low energy density -V 2 O 5 (layered) Toxic Electrode : - Graphite (layered) Cheap, non-toxic, high energy density, safety problems - Rutile (layered) Cheap, non-toxic, low energy density, no safety problems - Coke (grains) Cheap, non-toxic, high current density, safety problems

8 Modeling means : Making predictions and/or descriptions of phenomena based as much as possible on first principles This yields : Basis for targetting new experiments (time and cost saving) Optimal design Analysis of experimental and technical difficulties

9 Methodology Non empirical calculations of periodic structures LAPW, tight-binding Non empirical calculations of clusters LCAO-MO Semi-empirical calculations of peridodic structures Extended Hückel Empirical calculations of extended systems Molecular mechanics and dynamics Engineering models Finite elements

10 LAPW (1) (Li Linearized i daugmented dplane Waves ) The space is devided in two parts : a) non-overlapping spheres b) the interstitial space The Th LAPW wave functions in the interstitial space : 1 ik nr ϕ kn = e ω

11 LAPW (2) The LAPW wave functions inside the spheres : ϕ k = Ýu kn [ A lm u l ( r, E l )+ B lm u l ( r, E l )] Y lm r ˆr lm ( ) Boundary conditions : continuity of the wave function cto and of its first derivatives

12 WIEN97 LAPW Calculations l Density Functional Theory LAPW basis set (WIEN97) GGA corrections : Perdew, Burke, Ernzerhof 1 Full periodic Boundary Conditions Brillouin zone average by modified tetrahedron scheme (Blöchl et al. 2 ) Total energy according Weinert et al. 3 References 1 J.P. Perdew et al., Phys. Rev. Let. 77, 3865 (1996) 2 P.E. Blöchl et al., Phys. Rev. B 49, (1994) 3 M. Weinert et al., Phys. Rev. B 24, 864 (1982)

13 Calculation l of the Energy Density The discharge is considered as a chemical reaction : LiC 6 s 6( ( ) + MO 2 2( ( s ) 6C ( s ) + LiMO 2 2( ( s ) where M = Metal. At T = 0, the energy density is given by dt, = 0 T= 0 E = ΔG V

14 Calculation l of the Temperature Dependance With the temperature dependance, the Gibbs energy is given by : ΔG T=300 = ΔH T=300 T ΔS T=300 Total Energy But it can be showed that T ΔS T=300 << ΔH T=300 and so the contribution is given by the vibrational terms E vib, T = 3Nh υ ( hυ k B T ) 1 e h Li position distorsion T=0 Equilibrium Position

15 Average intercalation voltages for M 2 O 4 Spinels Cathode System Fd3m _ Unit cell a [A] (exp) Unit cell c [A] (exp) x parameter for OAverage Voltage [V] (exp) (exp) Ti : Ti 2 O 4 LiTi 2 O (8.372) Cubic Cubic (0.2628) 2.9 (3.0) V : V 2 O Cubic LiV 2 O Cubic Mn : Mn 2 O * (8.045) Cubic * (0.2631) 3.67 LiMn 2 O * (8.247) Cubic * (0.2625) (3.9) Fe : Fe 2 O Cubic LiFe 2 O 4... Cubic... Co : Co 2 O Cubic LiCo 2 O Cubic Ni : Ni 2 O Cubic LiNi 2 O Cubic Cu : Cu 2 O Cubic LiCu 2 O Cubic Layered R3m Ti : TiO LiTiO All voltages are calculated (measured) agains metallic lithium anode. *Spin unrestricted calculations

16 Band Structure t of Mn 2 O 4 fcc Brillouin zone

17 Spin unrestricted t DOS for Mn 2 O 4

18 Spin unrestricted t DOS for LiMn 2 O 4

19 S i t i t d DOS f Li M O Spin unrestricted DOS for Li 2 Mn 2 O 4 (tetragonal distorsion)

20 Electrostatic t ti Potential ti in Mn 2 O 4 In unit cell (008) Hopping Path

21 Fermi Level Evolution vs. Li Intercalation

22 OCV Modeling Using Frozen Bands x in Li x Mn 2 O 4

23 Engineering i Models

24 Solve 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 Electrochemical reaction at grain boundaries Porosity Electrical conduction in electrode Ion conduction in electrolyte Grain size distribution of M-oxide

25 Simulations of Measured Potential Jump Experiments Adjusted parameters : D = cm 2 /s k 0 = cm/s Cha arge den nsity [As s/cm 3 ] Time [s]

26 Illustration of kinetic control

27 Calculated Li + Concentration in Electrolyte

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

29 Li + - Solvent Interaction : In the Solvent E 0 2 c (ε -1) Sl=-q Solv ( Li 2R ε (for a spherical cavity) ε R dielectric constant of the solvent R Solv (Li + ) : radius of the Li + cavity E 0 Sl(Li + Solv ) = 5.53eV 53eV R Sl + Solv (Li ) = Å

30 Li + - Solvent Interaction : At the Surface E 1 Solv (r ) = E 0 2 ε Solv (ε-1) q Li2 c 1 r α r R solv (Li + ) r : 0 R(1-cos α ) = r 2 max S Solv(Li + = 0 1 E - = 0 Sl (1+cos α ) E Solv E solv (r max ) 2 E Solv(1 2 ) Results Mn 2 O 4 :E S Solv = 0.50 E 0 Solv (16c insertion site) α = 180 E S Solv = 0.79 E 0 Solv (8a insertion site) α =

31 Li + - Crystal Interaction With the host crystal Li + - Crystal Electrostatic Closed shell Interaction ti Energy = Attractionti + Repulsion Fitting of the effective charges according to band structure t calculations l (vide supra) With the incoming electron e - σ - antibonding e g of Mn 4+ (Li + -e - ) - coupling

32 Results Conditions of the model : Mn 2 O 4 : Surface 010 Solvent :Water 8a 16c Energy Barrier (red. charges) 0.16 ev 0.95 ev Li + solvation energy ev ev Li + -e - interaction ev ev 1)Li + -e - coupling : 1) reduction of the bulk energy barrier 2) increase of surface energy barrier

33 Results Infinite lattice summation Semi-nfinite Semi-infinite i i itlattice summation i Octahedral 16c Infinite lattice summation Semi-infinite lattice summation Tetraheral 8a

34 DFT: Heuristic approach X-ray diffraction ρ( r ) nuclear positions ( ) ρ r dr # electrons Cusp ( ) ρ R k R k = 2Z k ρ R 0 R k =R 0 ( ) ĤΨ H Ψ = EΨ

35 General Theory Exact energy expression 1 Eel = i φ i (r 1 ) 2 φ i (r 1 )d r 1 + ZA A R ρ(r A - r 1 1 ) d r 1 1 ρ( r 1 )ρ( r 2 ) d r 2 d r r 1 -r Exc Parr,R.G.;Yang,W : Density Functional Theory of Atoms A Molecules, Oxford University Press, New York 1989

36 The Kohn-Sham Equation h ks φ i = ε i φ i h = ZA + KS 2 A RA- r 1 + ρ( r 2 ) d d r 2 + V XC r - r 1 2

37 Approximate density functional theories for exchange and correlation Xα Local exchange LDA Local exchange + local correlation GGA Local exchange + local correlation + gradient corrections 3rd Generation of functionals Xα : Local exchange functional of the homogeneous electron gas LDA: Local exchange functional + local correlation functional of the homogeneous electron gas GGA: Same as LDA + non-local gradient corrections to exchange and correlation 3rd Generation of functionals: Same as GGA + instilation of exact-exchange and + 2nd derivatives of the density corrections

38 Practical Implementation Solve Kohn- Sham eqs. Features: LCAO expansion: waves Coulomb potential: aset functions Matrixelements: in the 1 ρ r' + vr ( )+ ρ( ) dr' +V XC ρ ( r ) 2 r r' ( ) Ψ i = ε i Ψ i ( r ) STO, GTO, numerical, plane solve Poisson s eq. or fit ρ(r) to of one-center auxilliary accurate numerical integration irreducible wedge of the

39 Methodology based on Approximate DFT MS-Xα 1966 α 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. Has relativistic i extension 53f). (53f) Make use of muffin-tin approximation (38). Developed by K.H. Johnson (37). DV-X α 1970 α DV-X : Make use of numerical atomic orbitals or STO's. 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. FRIMOL ADF 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). Embedding procedures (76h). Energy decomposition scheme (72). Developed by Baerends,Snijders,Ravenek,Vernooijs and te Velde (41,53,47,46d) development in progress 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 DeMon (54b,h). Strongly vectorized (48b). First developed by Dunlap (43) as well as Sambe and Felton (42). Extensive improvements by Salahub and Andzelm (48b) (D-GAUSS-program) as well as Rösch (74a). Also work by Pederson (45e) and Painter (45d) NUMOL NUMOL : Unique basis-setset free program 50a,e). ( Accurate numerical integration (46a). Efficient generation of Coulomb potential (50c). Geometry optimization. Developed by Becke (50 ). 1982

40 Modeling the Intercalation Dynamics of Li + : Cluster Study Using Molecular l DFT

41 Vibrational Modes Involved in Vibronic Vibrational Modes Involved in Vibronic Coupling of the e g (σ*) Electron

42 Vibronic Coupling Model of the Polaron

43 The 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

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

45 Energy Profiles of Li + Diffusion in Bulk Energy Profiles of Li Diffusion in Bulk Mn 2 O 4

46 Acknowledgements Financial Support : Swiss Federal Office for Energy The Li-Ion Battery Modeling Group : Michael Atanasov, University of Fribourg Cluster and Intercalation Dynamics Jean-Luc Barras, Lubomir Benco, Claude Daul, Erich Deiss, University it University i University of PSI of Fribourg of Fribourg Fribourg Band Structure Band Structure Project Leader Engineering Calculations Calculations Models