Unformatted text preview: The role of tetrahedral site in layered compound Tetrahedral site is face sharing with octahedral site Two cannot be simultaneously occupied. Each (O) site is surounded by three (T) sites within the same layer
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EMA 6446 Solid State Ionics  Rechargeable Batteries The role of the tetrahedral sites 2
EMA 6446 Solid State Ionics  Rechargeable Batteries Create trivacancy around the (T) site C. P. Grey et al, Electrochem. Sol. State Lett. 7, A290 (2004) 7
EMA 6446 Solid State Ionics  Rechargeable Batteries Create trivacancy around the (T) site
No energy barrier for Li to go to the tetrahedral site! Small energy barrier for some M ion to go to the tetrahedral site!
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EMA 6446 Solid State Ionics  Rechargeable Batteries Important to understand structures in details Because the following properties of battery materials are connected to structures a. Diffusion of lithium b. Electronic conductivity c. Phase transformation 9
EMA 6446 Solid State Ionics  Rechargeable Batteries LiFePO4 is 1D diffuser with low activation barrier for Li+ hopping Ea > 2.5 eV Ea > 1 eV Ea 200 to 270meV P Li Fe
10 LiFePO4 EMA 6446 Solid State Ionics  Rechargeable Batteries Lithium Diffusion Path  depend on structure 3D 1D 2D Olivine Layered Spinel Spinel Rate depends lithium ion mobility
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EMA 6446 Solid State Ionics  Rechargeable Batteries LiFePO4 Large band gap Eg = 3.8  4.0 eV1 LiCoO2 smaller band gap Eg = 2.7eV2 1. 2.
EMA 6446 F. Zhou, et al., Solid state communications,132, 181, 2004 M. Catti, et al., Phys Rev B 61, 1795, 2000 12 Solid State Ionics  Rechargeable Batteries Atomistic theory of diffusion in multicomponent solids 13
EMA 6446 Solid State Ionics  Rechargeable Batteries Coarse graining time Diffusion in a crystal
Two levels of time coarse graining A second level of coarse Shorttime coarse graining: graining transition state theory that leads to Fick's law
& ( 'E B # ) = * * exp$ ! kT " % J = ! D"C
GreenKubo  MD simulations  Harmonic approximation Kinetic coefficients derived from fluctuations at equilibrium 16
EMA 6446 Solid State Ionics  Rechargeable Batteries Vineyard, J. Phys. Chem. Solids 3, 121 (1957). Zwanzig, Annu. Rev. Phys. Chem. 16, 67 (1965). Diffusion Important Cases H diffusion in fcc of bcc based alloys Li diffusion in transition metal oxide host O diffusion on Pt(111) surface
In all examples, diffusion occurs on a rigid lattice which is externally imposed by a host or substrate 17
EMA 6446 Solid State Ionics  Rechargeable Batteries Lithium Diffusion in LiCoO2 18
EMA 6446 Solid State Ionics  Rechargeable Batteries Notation
M = number of lattice sites N = number of diffusing atoms vs = volume per lattice site x = N/M C=x/vs 19
EMA 6446 Solid State Ionics  Rechargeable Batteries Fick's first law J Li = "D#CLi
x CLi = vs !
EMA 6446 Solid State Ionics  Rechargeable Batteries ! vs=volume of the host unit cell 20 True driving force is the gradient in chemical potential
J Li = "D#CLi J Li = "L#Li ! ! 21
EMA 6446 Solid State Ionics  Rechargeable Batteries Irreversible thermodynamics: interstitial diffusion of one component J = " L! d D=L dC
J = " D!C
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EMA 6446 Solid State Ionics  Rechargeable Batteries One component diffusion:
KuboGreen relations (linear response statistical mechanics) D = L"! Thermodynamic factor ! "= !C
L= 1 (2d)tMv s kT $ N r '2 & "R (t)) i ) & % i=1 ( Kinetic coefficient # R. Gomer, Rep. Prog. Phys. 53, 917 (1990)/ EMA 6446 A. Van der Ven, G. Ceder,Solid State Ionics  Rechargeable Batteries Handbook of Materials Modeling, chapt. 1.17, Ed. S. Yip, Springer (2005). 23 ! Trajectories
r "R j (t) r "R i (t) ! 1 DJ = (2d)t
EMA 6446 $ N r '2 1& "R i (t)) ) N& % i=1 ( ! # 24
Solid State Ionics  Rechargeable Batteries More common form
~ D = DJ " !
Thermodynamic factor ' $ !% " ~ & kT # (= ! ln x
1 DJ = (2d)t $ N r '2 1& "R i (t)) ) N& % i=1 ( Self diffusion coefficient # R. Gomer, Rep. Prog. Phys. 53, 917 (1990)/ EMA 6446 ! Solid State Ionics  Rechargeable Batteries 25 Common approximation ~ D = D * "!
Thermodynamic factor ' $ !% " ~ & kT # (= ! ln x Tracer diffusion coefficient D* = ("R i (t))
(2d)t 2 R. Gomer, Rep. Prog. Phys. 53, 917 (1990)/ EMA 6446 Solid State Ionics  Rechargeable Batteries 26 Expressions for ideal system
Assume noninteracting Li ions: ideal solution Normalized free energy of LixM Lix M g(x) = k B T ( x ln x + (1" x ) ln(1" x )) ! ! 27
EMA 6446 Solid State Ionics  Rechargeable Batteries Expressions for ideal system
Assume noninteracting Li ions: ideal solution Normalized free energy of LixM Lix M g(x) = k B T ( x ln x + (1" x ) ln(1" x )) ! "g Li = "x ! !
EMA 6446 Solid State Ionics  Rechargeable Batteries 28 Expressions for ideal system
Assume noninteracting Li ions: ideal solution "g Li = "x
# x & Li = kB T ln% ( $ 1" x ' Lix M !
! ! 29
EMA 6446 Solid State Ionics  Rechargeable Batteries Expressions for ideal system
Assume noninteracting Li ions: ideal solution "g Li = "x
# x & Li = kB T ln% ( $ 1" x ' Lix M !
! $ Li ' #& ) ~ = % kT ( " #ln x
EMA 6446 ! 30
Solid State Ionics  Rechargeable Batteries Expressions for ideal system
Assume noninteracting Li ions: ideal solution $ Li ' #& ) ~ = % kT ( " #ln x
~ = 1 " 1# x Lix M ! ! ! 31
EMA 6446 Solid State Ionics  Rechargeable Batteries Thermodynamic factor for an ideal system
Lix M ! g(x) = k B T ( x ln x + (1" x ) ln(1" x ))
EMA 6446 # x & Li = kB T ln% ( $ 1" x '
Solid State Ionics  Rechargeable Batteries ~ = 1 " 1# x 32 ! ! Kinetic factor for an ideal system
General expression
1 DJ = (2d)t $ N r '2 1& "R i (t)) ) N& % i=1 ( # Noninteracting interstitial species with constant hop frequency ! DJ = (1" x ) #a $
EMA 6446 2 & $%E B ) " = # exp( + kB T * '
* ! Solid State Ionics  Rechargeable Batteries 33 Kinetic factor for an ideal system DJ = (1" x ) #a $
Probability to be next to a vacancy for noninteracting diffusers 2 ! EMA 6446 x~0 x~0.5
Solid State Ionics  Rechargeable Batteries x=1 34 Kinetic factor for an ideal system DJ = (1" x ) #a $
Geometric factor Hop distance 2 " =1 ! For: linear (1dim) lattice square (2dim) lattice cubic (3dim) lattice For: triangular (2dim) lattice For: fcc (3dim) lattice For: bcc (3dim) lattice
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Solid State Ionics  Rechargeable Batteries "=32 "=2 "=4 3
EMA 6446 Kinetic factor for ideal system
DJ = (1" x ) #a $
2dimensional triangular lattice 2 !
!
EMA 6446 a = 3 "10#8 cm " * = 1012 Hz
"E B = 300meV Solid State Ionics  Rechargeable Batteries ! T = 300K
36 ! Overall diffusion coefficient
~ D = DJ " !
~ = 1 " 1# x DJ = (1" x ) #a $ 2 D = "a # 2 ! ! 37
EMA 6446 Solid State Ionics  Rechargeable Batteries Non ideal System: LixCoO2 ~ D = DJ " ! Diffusion coefficient at 300 K Thermodynamic factor EMA 6446 A. Van der Ven, G. Ceder, M. Asta, P.D. Tepesch, Phys Rev. B 64 (2001) 064112 38
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 Fall '08
 Wachsman
 Thermodynamics, Lithiumion battery, Solid State Ionics, Rechargeable Batteries Vineyard

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