Mathematical Model of RuO2/Carbon
Composite Electrode for Supercapacitors
by
Hansung Kim and Branko N. Popov
Department of Chemical Engineering
Center for Electrochemical Engineering
University of South Carolina
Department of Chemical Engineering
University of South Carolina
Review of previous models for supercapacitors
based on pseudocapacitance
•
•
•
C. Lin, J.A. Ritter, B.N. Popov and R.E. White, J. Electrochem. Soc., 146 3169 (1999)
– RuO2 electrode with one dimension
– Particle size effect on the performance
– Surface reaction
– Constant electrolyte concentration
C. Lin, B.N. Popov and H.J. Ploehn, J. Electrochem. Soc., 149 A167 (2002)
– RuO2/Carbon composite electrode with one dimension
– Particle size and porosity effect on the performance
– Electrolyte concentration changes with discharge rate and time
– Surface reaction
The approach of this study by H. Kim and B.N. Popov
– RuO2/Carbon composite electrode with pseudo two dimension
– Bulk reaction by considering proton diffusion for each particle
– Constant power discharge study
– Optimization of carbon and RuO2 content in the electrode
Department of Chemical Engineering
University of South Carolina
Objectives of the modeling study
• Development of general model to expect the
performance based on operating parameters
• Effect of particle size of active oxide on the performance
• Effect of porosity on the rate capability
• Optimization of the ratio between carbon and RuO2
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Schematic diagram of supercapacitors and
reaction mechanism
Negative
electrode
Positive
electrode
Separator
H0.8 RuO2
Current
Collector
Carbon
Electrolyte
1M H2SO4
x
0
L
Ls
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Faradaic reaction of ruthenium oxide
• Positive electrode
Discharge:
H 0.8 RuO2  xH 2O  H 0.8 RuO2  xH 2O  H   e
Charge:
H 0.8 RuO2  xH 2O  H   e  H 0.8 RuO2  xH 2O
• Equilibrium potential (V vs. SCE)
H 0.3 RuO2  xH 2O
:1V
H 0.8 RuO2  xH 2O
: 0.5 V
H1.3 RuO2  xH 2O
:0V
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Assumptions
• Porous electrode theory.
• Double layer capacitance per area (Cd) is constant for
carbon and RuO2.
• Diffusion coefficients are assumed to be independent of the
concentration variation.
• Side reactions and temperature variation are neglected.
• Transport in electrolyte phase is modeled by using the
concentrated solution theory.
• The exchange current density is constant.
• Transference number and activity coefficient are constant.
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Model description: Basic equations and parameters
• Variables
C Concentration of electrolyte
1 Solid phase potential
 2 Solution phase potential
C s Concentration in solid
• Total current
i2
 (1   2 )
 S d Cd
 Sf jf
x
t
• Sd (cm2/cm3): Specific surface area for double layer capacitance per unit volume
6 xRu (1   )
S d  Sc  S f  SC C xC (1   ) 
d Ru
• Sf (cm2/cm3): Specific surface area for pseudocapacitance per unit volume
6 xRu (1   )
Sf 
d Ru
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• jf (A/cm2): Faradaic current by pseudocapacitance
j f  i0 {exp[ a (1   2  U1 ) F / RT ]  exp[ c (1   2  U1 ) F / RT ]}
• U1 (V vs. SCE): Equilibrium potential
U 1  2V0 (
M RuO
2
 RuO
Cs  1.3)
V0: 0.5V
2
• Solid phase current density
• Conservation of charge
1
i1  
x
I  i1  i2
0
i1 i2

x x
• Effective diffusivity and conductivity
D  D0 0.5
k p  k p 0 1.5
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Material balance on the electrolyte using
concentration solution theory
Porous electrode
C
   d (ln C 0 )  C  ajn (1  t 0 )
i2 t 0



  
D1 


t x  
d (ln C )  x 
v
z F x
C
 2C si
 21 (1  t 0 )

 D 2 

t
x
nF
x 2
v
Separator part
C
   d (ln C 0 )  C 
i2 t 0



D1 
  
t x  
d (ln C )  x  z F x
Ce
 2C
0.5
  D0
t
x
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The variation of potential in the separator and the
porous electrode
Porous electrode
 2  P RT
(ln f  ) s
t0 (ln C )
i2  I   P

(1 
)(

)
x
F
(ln Ce ) nv z v
x
1
 2  P RT s
t0 (ln C )
I  i1  I  
  P

(

)
x
x
F
nv z v
x
Separator part
 2  P RT
(ln f  ) s
t0 (ln C )
i2  I   P

(1 
)(

)
x
F
(ln Ce ) nv z v
x
 2  P RT s
t0 (ln C )
i2  I   P

(

)
x
F
nv z v
x
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Boundary and Initial conditions
B.C.
At x = 0 : (current collector of positive electrode)
C
0
x
 2
0
x
1
x
I cell  i1  
At x = Le: (interface between separator and electrode)
 s1.5 D0
C
C
  1.5 D0
x sep
x elec
1
0
x
 s1.5 Kp 0
 2
 2
  1.5 Kp 0
x sep
x elec
At x = 2Le+Ls : (current collector of negative electrode)
C
0
x
I.C.
At t = 0, C = C0 ,
1
x
I cell  i1  
1 positive 
M RuO
2
 RuO
2
0.3
2  0
1negative 
M RuO
2
 RuO
1.3
2
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A mass balance of spherical particle of ruthenium oxide
  2Cs 2 Cs 
Cs
 Ds  2 

t
r

r

r


r=0:
Cs
0
r
r = Rs :
jf
C s

r
Ds F
B.C
j f  i0 {exp[ a (1   2  U1 ) F / RT ]  exp[ c (1   2  U1 ) F / RT ]}
U 1  2V0 (
M RuO
2
 RuO
Cs  1.3)
2
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Parameters used in the model
• Fixed values
– Thickness: 100m for electrode,
25  m for separator
– Exchange current density: 10-5 A/cm2
– Double layer : 210-5 F/cm2
– Sigma: 103 S/cm
– K0: 0.8 S/cm
– Density: 2.5 g/cm3, 0.9 g/cm3
– D: 1.8  10-5 cm2/s
– Ds: 10-11 cm2/s
– Transference number: 0.814
– Porosity of separator: 0.7
– Concentration of electrolyte: 1M H2SO4
• Variable values
– Particle size of RuO2
– Porosity of electrodes
– The ratio between
RuO2 and carbon
– Discharge current
density
– Discharge power
density
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Porosity of the electrode as a function of the mass
fraction of RuO2
0.5
Pore volume base (VBP2000=0.93cm3/g,VRuO2=0.019cm3/g)
Packing theory
Porosity
0.4
0.3
Packing theory (
0.2
Pore volume base (VVulcan XC-72=0.38cm3/g,VRuO2=0.019cm3/g)
0.1
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Mass fraction of RuO2
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Effect of the diffusion coefficient of proton in the solid particle on
the capacitance at the constant current discharge of 30 mA/cm2
40wt% RuO2 ,Porosity: 0.214, Particle size: 5nm
1.0
Cell potential (V)
0.8
0.6
1.010-11 cm2/s
105 F/g
0.4
1.010-16 cm2/s
59 F/g
0.2
0.0
0
10
20
30
40
50
60
70
Discharge time (s)
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Discharged energy density curves at the constant power
discharge of 50w/kg for different particle sizes of RuO2
1.0
5nm
15nm
60nm
200nm
Cell potential (V)
0.8
0.6
0.4
0.2
0.0
0
2
4
6
8
Discharged energy density (Wh/kg)
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Discharged energy density curves at the constant power
discharge of 4kw/kg for different particle sizes of RuO2
1.0
5nm
15nm
60nm
200nm
Cell potential (V)
0.8
0.6
0.4
0.2
0.0
0
1
2
3
4
5
6
7
Discharged energy density (Wh/kg)
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Local utilization of RuO2 at the interface of separator as a
function of particle size at different discharge rates
.
100
50 W/L
4000 W/L
95
Local utilization (%)
90
85
80
75
70
65
60
0
50
100
150
200
250
Particle size of RuO2 (nm)
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Dimensionless parameter, Sc (diffusion in the solid/discharge
time), as a function of particle size of RuO2
102
200nm
10
1
100nm
50nm
Sc (dimensionless)
100
10-1
15nm
10-2
Rs2 I
Sc 
Ds F (1   )Ct c
10-3
10-4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Discharge current density (A/cm2)
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Electrochemical performance of the RuO2/carbon composite
electrode (60wt% RuO2) with respect to constant current discharge
1.0
30 mA/cm2
100 mA/cm2
500 mA/cm2
1000 mA/cm2
Cell potential (V)
0.8
0.6
Rs: 50nm
: 0.181
0.4
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
2
Discharge density (C/cm )
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Electrolyte concentration distribution of the cell at the end of
discharge with different current densites
Concentration of electrolyte (mol/L)
2.0
1.5
30 mA/cm2
1.0
100 mA/cm2
200 mA/cm2
0.5
500 mA/cm2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
Dimensionless distance
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Potential distribution in the electrolyte at the end of discharge
at different current densities
30 mA/cm2
0.00
100 mA/cm2
Electrolyte potential drop (V)
-0.02
-0.04
-0.06
500 mA/cm2
-0.08
-0.10
-0.12
1000 mA/cm2
-0.14
-0.16
-0.18
1.0
0.8
0.6
0.4
0.2
0.0
Dimensionless distance
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Potential distribution in the electrolyte at the end of discharge
at the different porosities of electrode
Electrolyte potential drop (V)
0.0
 : 0..35
 : 0.24
-0.1
 : 0.15
-0.2
-0.3
RuO2 ratio: 60wt%
Particle size: 50nm
Current density: 1A/cm2
-0.4
 : 0.09
-0.5
1.0
0.8
0.6
0.4
0.2
0.0
Dimensionless distance
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Discharge density as a function of RuO2 content,
particle size and porosity of electrodes at 1.5A/cm2
1.6
100 nm
5 nm
0.126
0.129
0.220
2
Discharged charge density (C/cm )
1.4
1.2
0.111
1.0
0.181
Porosity
0.09
0.8
0.097
0.247
0.6
0.214
0.4
0.04 0.06
0.237
0.2
0.0
0
20
40
60
80
100
Weight percent of RuO2 (%)
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Ragone plot for RuO2/carbon composite electrode containing
different Ru loading using a colloidal method
16
100% RuO2, : 0.037, Rs: 200 nm
14
80% RuO2, : 0.097, Rs: 200 nm
Energy density (Wh/Kg)
12
60% RuO2, : 0.124, Rs: 100 nm
10
8
40% RuO2, : 0.205, Rs: 3 nm
6
4
2
0
70
200
500
1000
2000
3500 5000
Power density (W/Kg)
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Conclusions
• The general model was developed successfully to expect the performance
of oxide/carbon composite electrode based on porosity, particle size, the
content of RuO2 in the electrode.
• It was found that porosity and particle size have a tremendous effect on
the performance especially at high rate discharge.
• With increasing the discharge rate, transportation of electrolyte imposes
the limitation on the performance by increasing solution potential drop.
• With increasing the particle size of RuO2, since the diffusion process in
the solid particle is a limiting step, the discharge stops before the RuO2
particle has fully been utilized.
• Increasing porosity decreased the electrolyte deviation and solution
potential drop. After the porosity increases up to about 0.15, the particle
size is important to get a high performance until the discharge rate of
1.5A/cm2
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