Transcript ТЕРМОДИНАМИКА И КИНЕТИКА ПРОЦЕССА ИНТЕР

Cooperation: Prof. Dr. J. Kortus
Cooperation: Prof. Dr. H.J. Seifert
Thermodynamics and Kinetics of
Processes of the Intercalation/DeIntercalation in Submicron Particles of
Cathode Li-ions Batteries
С.Н. Поляков
S.N. Polyakov, J. Kortus, H.J. Seifert
Bauman MHTU, January 26-28, 2011, Moscow
World production
S.N. Polyakov, J. Kortus, H.J. Seifert
МГТУ им.Баумана, 26-28 января, 2011, Москва
Lithium-ion battery
Cathodes
Electrode material
Average potential
difference
Specific capacity
Specific energy
LiCoO2
3.7 V
140 mA·h/g
0.518 kW·h/kg
LiMn2O4
4.0 V
100 mA·h/g
0.400 kW·h/kg
LiNiO2
3.5 V
180 mA·h/g
0.630 kW·h/kg
LiFePO4
3.3 V
150 mA·h/g
0.495 kW·h/kg
Li2FePO4F
3.6 V
115 mA·h/g
0.414 kW·h/kg
LiCo1/3Ni1/3Mn1/3O2 3.6 V
160 mA·h/g
0.576 kW·h/kg
Li(LiaNixMnyCoz)O2
220 mA·h/g
0.920 kW·h/kg
4.2 V
S.N. Polyakov, J. Kortus, H.J. Seifert
Bauman MHTU, January 26-28, 2011, Moscow
Comparison of the gravimetric and volumetric energy
densities of various rechargeable battery systems*
*) A. Manthiram,
Lithium batteries,
Edited by
Gholam-Abbas Nazri,
USA, Springer, 2009.
S.N. Polyakov, J. Kortus, H.J. Seifert
Bauman MHTU, January 26-28, 2011, Moscow
Schematic illustration of the charge/discharge in
lithium-ion cell
Charge (de-intercalation)
Discharge (intercalation)
Li
O
c
+
Li

Li + e + Mn2O4
intercalation,discharge

LiMn2O4
deintercalation,charge
Mn
LiMn2O4 C6  Li1x Mn2O4  Lix C6
S.N. Polyakov, J. Kortus, H.J. Seifert
Bauman MHTU, January 26-28, 2011, Moscow
Problem of the cyclability
(Material degradation)
Cycling crumbling
(chemical corrosion) [**]
Crack after
cycling [*]
Optimal distribution of size of
the cathode particles
Cyclability data for LiMn2O4 cathode[***]
S.N. Polyakov, J. Kortus, H.J. Seifert
[*] J. of Power Sources 140 (2005) 125-128
[**] J. of Power Sources 143 (2005) 203-211
[***] Solid State Ionics 167 (2004) 237-242
Bauman MHTU, January 26-28, 2011, Moscow
Thermodynamic
Larch-Cahn-Theory
V
d (V )  d (V0 )
dG  sdT  V ' ij d ij   Li V ' dcLi
V0
Maxwell’s relations
 ij
c Li
 Li

 ij
The chemical strain tenzor for cubic symmetry
(2)
ij 
 ij
cLi
For small deformations
Cubic cell LiMn2O4
(1)
ij   Li  ij
 ii  
Partial molar volume of Li in the host lattice
  (cLi , T )
  3
Li ( ij , xLi )  Li (0, xLi )   h
 h  (11   22   33 ) / 3
S.N. Polyakov, J. Kortus, H.J. Seifert
(3)
- hydrostatic stress
Bauman MHTU, January 26-28, 2011, Moscow
(4)
Kinetics of the Li-ion in Electrode
Porous electrode
Li-ion flux density, Onsager Theory
J Li = c Li M Li μLi
Particle
Binder
M Li
Electrolyte
- Li-ion mobility
Li-ion mobility, Larch-Cahn-Theory
Kinetic model for one particle
J Li = c Li M Li μLi 0,cLi   Ωσ h 
c Li
= DLi c Li  c Li M Li Ωσ h 
t
DLi = cLi M Li

μLi
cLi

1
ε ij = 1+ ν σ ij  νσ kk + α cLi  c0 δij
E
S.N. Polyakov, J. Kortus, H.J. Seifert
(5)
(6)
(7)
Bauman MHTU, January 26-28, 2011, Moscow
Diffusion
Spherical particle
c Li
1
= 2  r 2 DLi c Li  c Li M Li Ω σ h  (8)
t
r

J   DLi

cLi
j
(r0 , t ) 
r
F
(9)
The Butler-Volmer equation
  1  β Fu 
 βFu 
j = j0 exp
  exp

 RT 
  RT 
j0=Fkcl1 β cθ1 β csβ
- exchange flux density
u = U a U0
S.N. Polyakov, J. Kortus, H.J. Seifert
Bauman MHTU, January 26-28, 2011, Moscow
(10)
Open-circuit potential
U 0 (cLi , T )  
 LiMn O (cLi , T )   Li (T )
2
4
(11)
F
Experiment [*]
Calphad-Method [**]
U0
U 0=4 ,19829+0 ,056661t anh 14,5546x+8,60942

 0 ,0754790 ,998432 x 
 0 ,157123exp 0 ,04738x 
 0 ,42465
+0 ,810239exp 40x+5.355

 1,90111
x
cLi
c0
[**] J. Electrochem. Soc., 143, 1890 (1996)
S.N. Polyakov, J. Kortus, H.J. Seifert
[*] Solid State Ionics, 69 (1994) 59.
Bauman MHTU, January 26-28, 2011, Moscow
Intercalation/De-Intercalation stress
in a spherical particle
Charge (contraction/expansion)
dσ r 2
+ σ r σ t  = 0
dr r
(12)
σ t (r = 0 ) = σ r (r = 0 ) σ r (r = r0 ) = 0
1
εr = σ r  2νσ t + α cLi  c0  (13)
E
Discharge (expansion/contraction)
1
εt = σ t  νσ t + σ r + αcLi  c0  (14)
E
E - elastic modulus
ν - Poisson’s ratio
S.N. Polyakov, J. Kortus, H.J. Seifert
Bauman MHTU, January 26-28, 2011, Moscow
Analysis of the diffusion by stresses in
a spherical particle
E
2 E
 p(r0 )  p(r ) (15) t (r )  1   2 p(r0 )  p(r )  c  c0  (16)
 r (r ) 
1 
r
2E
1
3 p(r0 )  c  c0  (17) p(r )  3  cLi  c0 r 2dr
h (r )  r  2t  / 3 
31   
r 0
σ h
2α Li E c Li
(r) = 
r
31  ν  r
(5)
c Li
J Li = c Li M Li μ Li = D + c Li θ 
r
S.N. Polyakov, J. Kortus, H.J. Seifert
2ΩLi2 E
θ=
M Li
91  ν 
(18)
Bauman MHTU, January 26-28, 2011, Moscow
Numerical procedure
cin+1  cin 1
= 2
Δt
ri
2
i+1 / 2 i+1 / 2
D
r
cmn+1  cmn+11
 kc 
=  e  cmn+1
Δx
 D 
cin++11  cin+1
cin+1  cin+11
2
 Di 1/ 2 ri 1/ 2
Δr
Δr
i = 1,2...m  1
Δr
  c
β
0
c
c0n+1 = c1n+1

n+1 1 β
m
  uin+1( 1  β) 
 uin+1 β  
exp
  exp

RT

 RT  
 
 c n+1 
uin+1 = U 0 + vtn  U  i 
 c0 
n+1
 ηin+1 β  
β m1
Δr   kc e  β
  ηi ( 1  β) 
1 β 
f(x)= x 

exp

+
 x  c0  x  exp
 





1  α m1   D 
RT
 

 RT   1  α m1
f ' (x)= 1 
Δr d
Φ(x)
1  α m1 dx
where
n+1
 ηin+1 β  
 kc e  β
  ηi ( 1  β) 
1 β 
Φ(x) = 
x  c0  x  exp
  exp

RT
 
 D 

 RT  
S.N. Polyakov, J. Kortus, H.J. Seifert
(19)
(20)
xi+1 = xi 
f(xi )
f ' (xi )
Bauman MHTU, January 26-28, 2011, Moscow
Material properties of LiMn2O4 and parameters for the
lithium intercalation reaction
U a (V )
OCP
Deintercalation
Intercalation
U a (V )
c Li
c0
D(cm2 s 1 )
1013  109

0.5
cl (moldm3 )
1.0
r0 ( m)
0.1-10.0
c0 (mol/ m3 )
2.29 104
k (cm5 / 2 s 1mol1/ 2 )
0.00019
(m3 / mol)
E (GPa)
v
S.N. Polyakov, J. Kortus, H.J. Seifert
3.497106
10
0.3
Bauman MHTU, January 26-28, 2011, Moscow
Stress and Li-concentration
Deintercalation
S.N. Polyakov, J. Kortus, H.J. Seifert
Intercalation
Bauman MHTU, January 26-28, 2011, Moscow
Hydrostatic stress (deintercalation/intercalation)
S.N. Polyakov, J. Kortus, H.J. Seifert
Bauman MHTU, January 26-28, 2011, Moscow
Hydrostatic stress
(deintercalation/intercalation)
S.N. Polyakov, J. Kortus, H.J. Seifert
Bauman MHTU, January 26-28, 2011, Moscow
Stress in a submicron particle
τ r,t  = c  c0 = At r 2 + Bt 
A = c/r t, r0  = i/ 2r0 DF
j = ir0 /c0 DF 


0.2Ωc0 E
σ r (x)=
j 1  x 2 (21)
31  ν 
0.2Ωc0 E
σ t x  =
j 1  2x2  (22)
31  ν 
0.2Ωc0 E
σ h x  =
j 3  5x2 (23)
91  ν 
v  1m V / s
r0  1m

x

r
r0
r
r0
S.N. Polyakov, J. Kortus, H.J. Seifert
Bauman MHTU, January 26-28, 2011, Moscow
Stress in a 10μm particle
v  1m V / s
r0  10m
x
S.N. Polyakov, J. Kortus, H.J. Seifert
r
r0
Bauman MHTU, January 26-28, 2011, Moscow
Hydrostatic stress in a particle
r = 10μm
(deintercalation, v = 1 μV/s)
Dangerous zone
Deintercalation
S.N. Polyakov, J. Kortus, H.J. Seifert
Intercalation
Bauman MHTU, January 26-28, 2011, Moscow
Extreme values of hydrostatic stresses on the particle
surface[*]
Deintercalation
Hydrostatic stress for various scan
rates in a particle of 2 μm radius
Intercalation
С. Н. Поляков, ПЖТФ, 2010,
Vol. 36, No. 24, pp. 25–32.
[* ]
S.N. Polyakov, J. Kortus, H.J. Seifert
Bauman MHTU, January 26-28, 2011, Moscow
Numerical simulation (diffusion)
r0  10m, v  1mV s.
Deintercalation
r0  0.5m, v  1mV s.
Intercalation
S.N. Polyakov, J. Kortus, H.J. Seifert
Bauman MHTU, January 26-28, 2011, Moscow
Numerical simulation
(current density )
r0  10m, v  1mV / s
r0  0.5m, v  1mV / s
Deintercalation
Intercalation
S.N. Polyakov, J. Kortus, H.J. Seifert
Bauman MHTU, January 26-28, 2011, Moscow
Analysis of the current density in submicron particles
y/xτ,1+ω1  yτ,1
1 β
y τ,1 χ=0
β
(24)
χ=exp1  β Fη / RT   exp βFη/RT
τ=tD/r0
2
x=r/r0
y/xτ,0=0
y=c/c0
ω=r0 / D/ 1/ kc
1
l
ω0
S.N. Polyakov, J. Kortus, H.J. Seifert

 y0,τ   yτ,1  0
Bauman MHTU, January 26-28, 2011, Moscow
Current density in submicron particles
 c/t dv= dc/dtdv=dc dt dv=V dc/dt
0
V0
V0
V0

 divD  gradc dv= D  gradc ds
V0
S0

J=  1 S0  D  gradc ds
S0
V0 /S0 dc/dt+J=0
S.N. Polyakov, J. Kortus, H.J. Seifert
(25)
Bauman MHTU, January 26-28, 2011, Moscow
Current density in submicron particles
S.N. Polyakov, J. Kortus, H.J. Seifert
Bauman MHTU, January 26-28, 2011, Moscow
Conclusions
• The effect of stresses and deformations in a cathode material (LiMn2O4) is
taken into account using the Larche–Cahn thermo-chemical theory.
• Equations for calculating kinetics of mechanical stresses in submicron
particles were derived.
• The Li-ion current density dependence of the particle size and of the ID
rate was obtained for a cathode material.
• A kinetic equation for the current density in the absence of diffusion
polarization was derived; it was shown that diffusion polarization
decreased for submicron particles.
• The influence of a particle size on the maximum Li-ion current density was
evaluated.
S.N. Polyakov, J. Kortus, H.J. Seifert
Bauman MHTU, January 26-28, 2011, Moscow
I thank you for your attention
Большое спасибо за внимание!
S.N. Polyakov, J. Kortus, H.J. Seifert
Bauman MHTU, January 26-28, 2011, Moscow