Hansung Kim and Branko N. Popov

1
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
2
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

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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
  

4
Schematic diagram of supercapacitors and reaction
mechanism
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Faradaic reaction of ruthenium oxide

• Positive electrode

Discharge
Charge

• Equilibrium potential (V vs. SCE)

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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

Concentration of electrolyte
Solid phase potential
Solution phase potential
Concentration in solid

• Total current

• Sd (cm2/cm3) Specific surface area for double
  layer capacitance per unit volume

• Sf (cm2/cm3) Specific surface area for
  pseudocapacitance per unit volume

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• jf (A/cm2) Faradaic current by
  pseudocapacitance

• U1 (V vs. SCE) Equilibrium potential

V0 0.5V

• Conservation of charge

• Solid phase current density

• Effective diffusivity and conductivity

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Material balance on the electrolyte using
concentration solution theory
Porous electrode
Separator part
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The variation of potential in the separator and
the porous electrode
Porous electrode
Separator part
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Boundary and Initial conditions
B.C.
At x 0 (current collector of positive
electrode)
At x Le (interface between separator and
electrode)
At x 2LeLs (current collector of negative
electrode)
I.C.
At t 0, C C0 ,
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A mass balance of spherical particle of ruthenium
oxide
B.C
r 0
r Rs
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Parameters used in the model

• Variable values
• Particle size of RuO2
• Porosity of electrodes
• The ratio between RuO2 and carbon
• Discharge current density
• Discharge power density

• 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

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Porosity of the electrode as a function of the
mass fraction of RuO2
Packing theory
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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/cm240wt
RuO2 ,Porosity 0.214, Particle size 5nm
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Discharged energy density curves at the constant
power discharge of 50w/kg for different particle
sizes of RuO2
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Discharged energy density curves at the constant
power discharge of 4kw/kg for different particle
sizes of RuO2
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Local utilization of RuO2 at the interface of
separator as a function of particle size at
different discharge rates.
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Dimensionless parameter, Sc (diffusion in the
solid/discharge time), as a function of particle
size of RuO2
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Electrochemical performance of the RuO2/carbon
composite electrode (60wt RuO2) with respect to
constant current discharge
Rs 50nm ? 0.181
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Electrolyte concentration distribution of the
cell at the end of discharge with different
current densites
30 mA/cm2
100 mA/cm2
200 mA/cm2
500 mA/cm2
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Potential distribution in the electrolyte at the
end of discharge at different current densities
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Potential distribution in the electrolyte at the
end of discharge at the different porosities of
electrode
? 0..35
? 0.24
? 0.15
RuO2 ratio 60wt Particle size 50nm Current
density 1A/cm2
? 0.09
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Discharge density as a function of RuO2 content,
particle size and porosity of electrodes at
1.5A/cm2
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Ragone plot for RuO2/carbon composite electrode
containing different Ru loading using a colloidal
method
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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