by Hansung Kim and Branko N. Popov Department of Chemical Engineering

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

3. 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. Negative electrode Positive electrode Separator Carbon Current Collector Electrolyte 1M H2SO4 x 0 L Ls Schematic diagram of supercapacitors and reaction mechanism

5. : 1 V : 0.5 V : 0 V Faradaic reaction of ruthenium oxide • Positive electrode Discharge: Charge: • Equilibrium potential (V vs. SCE)

6. 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.

7. 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

8. 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

9. Material balance on the electrolyte using concentration solution theory Porous electrode Separator part

10. The variation of potential in the separator and the porous electrode Porous electrode Separator part

11. Boundary and Initial conditions B.C. At x = 0 : (current collector of positive electrode) At x = Le: (interface between separator and electrode) At x = 2Le+Ls : (current collector of negative electrode) I.C. At t = 0, C = C0 ,

12. A mass balance of spherical particle of ruthenium oxide B.C r = 0 : r = Rs :

13. 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 Parameters used in the model

14. Porosity of the electrode as a function of the mass fraction of RuO2 Packing theory

15. 1.010-11 cm2/s 105 F/g 1.010-16 cm2/s 59 F/g 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

16. Discharged energy density curves at the constant power discharge of 50w/kg for different particle sizes of RuO2

17. Discharged energy density curves at the constant power discharge of 4kw/kg for different particle sizes of RuO2

18. Local utilization of RuO2 at the interface of separator as a function of particle size at different discharge rates.

19. Dimensionless parameter, Sc (diffusion in the solid/discharge time), as a function of particle size of RuO2

20. Electrochemical performance of the RuO2/carbon composite electrode (60wt% RuO2) with respect to constant current discharge Rs: 50nm : 0.181

21. 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

22. Potential distribution in the electrolyte at the end of discharge at different current densities

23. 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

24. Discharge density as a function of RuO2 content, particle size and porosity of electrodes at 1.5A/cm2

25. 25 nm TEM image of RuO2·nH2O/carbon composite electrode (40 wt% Ru)

26. 3 m 3 m SEM images of RuO2.nH2O/carbon composite electrode (80 wt% Ru) (60 wt% Ru )

27. Specific capacitance of RuO2·nH2O as a function of Ru loading

28. Ragone plot for RuO2/carbon composite electrode containing different Ru loading

29. Ragone plot for RuO2/carbon composite electrode containing different Ru loading using a colloidal method

30. 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