Daniil Itkis Lomonosov Moscow State University - elch.chem.msu.ru

Model
D = 2RT Development
efft + − 1 f
1 +
F ln c Li 8
Basic equations.— The mathematical Model description Development of Li-air bateries
should include accurate models for the lithium-ion and oxygen
iffusion absolute inside the temperature. cell, theInelectron our simulations, conductivity f / of ln cthe Li iscarbon
approximated
Li 2 to O 2 zero. formation and deposition at the cathode, the po-
athode, the
osity change inside the cathode, and rate equations for Reaction 1
nd the Li-ion formation Li → Li + +e − . In this work, we used the
heory
ude.
of concentrated solutions 10 to model the Li + and oxygen difusion
and drift in the anode protective layer APL, separator, and
Model Development
athode electrolyte. 3.0 We also assumed a binary monovalent electroyte
and no convection. The electrostatic potential of Li ions Li is
2.8
ssumed to satisfy the following drift-diffusion equation
O2 c O2
ref
=2k
r¯pc O2
e 1−F/RT c − e −F/RT c if LC x
where
Basic
R =
equations.—
8.314 J/mol KThe is themathematical universal gas constant
description
and T is theof Li-air bat-
First theoretical models
0 otherwise
clude.
C
iffusion and reaction rate at the cathode. The formation of Li 2 O 2 at
e cathode is modeled similarly to the one developed by Sandhu et
. 6 In the second section, we present the numerical algorithm and
mple simulation results. Then, we discuss about possible aproaches
to improve the specific capacity SC of the cathode electeries
should include accurate models for the lithium-ion and oxygen
diffusion inside the cell, the electron conductivity of the carbon
where = 0.5, k is a reaction rate constant, c
ode and the energy density of Li-air batteries, after which we con-
O2
cathode, the Li 2 O 2 formation and deposition at the cathode, normalization the porosity
change inside the cathode, Journal of and The Electrochemical rate equations Society, for 157 12 Reaction IfA1287-A1295 we take1into 2010 consideration the electrical A1291 resistivity
parameter, and c is the overpotential a
which is assumed to deposit uniformly on the inner
and the Li-ion formation Li → Li + +e − . In this work, we
0.8 pores,
usedthe theoverpotential can be written as
theory of concentrated solutions 10 to model the Li
Basic equations.— The mathematical description of Li-air + and oxygen diffusion
and drift in the anode protective layer APL, separator, and
2
batries
should include accurate models for the lithium-ion and oxygen
c = Li − − U c − V Li2 O
cathode electrolyte.
0.6
iffusion inside the 2.6 We also assumed a binary monovalent electro-
· cell, the electron conductivity of the carbon = Li − − U c − R C eff Li + D ln c Li − R C =0 2
Li2 O 2
r¯p,0
0
ln
I=0.1mA/cm 2
lyte and no convection. The electrostatic potential of Li ions
thode, the Li 2 O 2 formation and deposition at the cathode, the posity
change inside the cathode, and rate equations for Reaction 1
Li is
here eff is the effective electric conductivity of the electrolyte, D
lectrolyte d the Li-ion which formation is equal toLi the→concentration Li + +e − . of InLi this + , and work, R C iswe theused the
eory xygenofconversion concentrated rate, solutions which is 10 equal to model to zerothe in Li the + APL and oxygen and difsion
and
eparator and
drift
is positive
in the anode
in the cathode.
protective
The
layer
concentration
APL, separator,
of Li +
and
atisfies
thode 10
electrolyte. We also assumed a binary monovalent electrote
and no convection. The electrostatic potential of Li ions Li is
sumed to satisfy the following drift-diffusion equation
Voltage (V)
2.4
assumed to satisfy the following drift-diffusion (e) equation where V Li2 O 2
is the voltage drop across Li 2 O 2 , Li2
0.4
2.2
(a)(b)
(c) (d)
trical resistivity 0capacity of Li 2 O 2 , and U c is the equilibrium
2.9
· 0.05 mA/cm 2
Reaction 1 at the cathode.
2.8 eff Li +
(e) D ln c Li − R C =0 2 1/4 capacity
2.0
2.7
(d) 0.1 mA/cm 2
Equations 2-4, 2/4 capacity 9, 10, and 12 represent a system of
(c)
where eff is the effective 2.6 electric conductivity 0.2 mA/cm 2 of the electrolyte, 0.2 ential equations
(b)
D 3/4that capacity should be subject to boundary a
2.5
1.8
(a)
is the diffusional conductivity, c 0.5 mA/cm 2
ditions and should 4/4 capacity be solved self-consistently to
2.4
0.0 0.1 0.2 0.3 0.4 0.5 Li is the concentration of the lithium
lithium-ion and oxygen concentrations, the electrosta
electrolyte
Specific capacity (mAh/g C
)
1mA/cm
c which is equal to the concentration 2
of Li + Li
, and
1.6
0.0 and
R C the
is the
porosity at each location inside the electroche
oxygen= conversion · D0 Li,eff 200 rate, c Li 400 which − 1 − 600 t+
800 1000 1200
0.0 function of 0.2 time. The 0.4 initial 0.6 conditions0.8
t
F
is R C equal − I Li · t +
to zero3
F
in the APL and
as well as
Distance (mm)
separator Figure 2. Color and is online positive Modeling Specific inof capacity the oxygen cathode. (mAh/g diffusion C
) The and concentration Li 2 O 2 formation
in the porous carbon cathode.
presented in the next two
conditions of Li + for the one-dimensional cell simulated in
subsections.
s the diffusional conductivity, c Li is the concentration of the lithium
Voltage (V)
here is the porosity, D Li,eff is the effective electrolyte diffusion
oefficient, t + is the transference number, F = 96,487 C/mol is
satisfies 10 · eff Li + D ln c Li − R C =0 2
Figure 4. Color online Cell voltage as a function of the SC for different
discharge currents.
here araday’s eff constant, is the c effective and Li I Li electric = − eff conductivity Li − D of the electrolyte, D
the diffusional conductivity, = · D c Li Li,eff is the c Li − 1 ln − c t+ Li
of lithium
t
F R is the
C − I Li · t +
lectrolyte density current. The oxygen concentration satisfies the 3
ollowing
ectrolytediffusion which
equation
is equal to the concentration of Li + F
, and R C is the
xygenwhere conversion is therate, porosity, whichDis Li,eff equal is the to effective zero in the electrolyte APL andiffusion
parator coefficient, and is positive t + in the cathode. The concentration of Li +
tisfies Faraday’s 10 constant, and I Li = − eff Li − D ln c Li is the
maximum limit given by Eq. 27 because, for narrow cathodes, the one should either increase the diffusion coefficient of O 2 to make it
oxygen can diffuse completely throughout the whole cathode volume.
The02 cell Feb voltage 2011 decreases to 195.208.208.29. with decreasing Redistribution thickness of the subject side of to the ECS cathode license near the or copyright; separator to efficiently see http://www.ecsdl.org/t
fill in the pores
comparable to that of Li ions or increase the reaction rate at the left
Downloaded c
cathode O2
electrode. This is because in batteries with smaller cathode of the cathode with Li 2 O 2 .
thickness, there
= is is the less· surface
Dtransference O2 ,eff areaavailable c O2 −and number, R C
the overpotential F = atP.Andrei 4 96,487 The dependence et C/mol al. // of J. is the Electrochem. reaction rate as a function Soc. of 157 the position (2010)
t
2F
the cathode as expressed by Eq. 14 becomes higher. The SC in Fig. inside the cell is represented in Fig. 7 for different states of discharge.
While the battery discharges, the reaction rate decreases 23
4 and 5 as well as in other figures in this section is expressed in
mAh/g , where the mass includes only the mass of the carbon in the deep inside the cathode and increases at the surface of the cathode.
here D O ,eff is the effective diffusion constant of the oxygen.
Porosity
Figure 6. Color online Local dependence of the porosity when the battery
is fully charged 0 capacity, partly charged 1/4, 2/4, and 3/4 capacity, and
nearly completely discharged 4/4 capacity.
ref =