1. Department of Chemical Engineering University of South Carolina by Hansung Kim and Branko N. Popov Department of Chemical Engineering Center for Electrochemical Engineering University of South Carolina Mathematical Model of RuO 2 /Carbon Composite Electrode for Supercapacitors

2. 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) –RuO 2 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) –RuO 2 /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 –RuO 2 /Carbon composite electrode with pseudo two dimension –Bulk reaction by considering proton diffusion for each particle –Constant power discharge study –Optimization of carbon and RuO 2 content in the electrode

3. 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 RuO 2

4. Department of Chemical Engineering University of South Carolina Schematic diagram of supercapacitors and reaction mechanism Current Collector Negative electrode Positive electrode Separator 0 x LLs Carbon Electrolyte 1M H 2 SO 4

5. Department of Chemical Engineering University of South Carolina Faradaic reaction of ruthenium oxide Positive electrode Discharge: Charge: Equilibrium potential (V vs. SCE) : 1 V : 0.5 V : 0 V

6. Department of Chemical Engineering University of South Carolina Assumptions Porous electrode theory. Double layer capacitance per area (C d ) is constant for carbon and RuO 2. 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. Department of Chemical Engineering University of South Carolina Model description: Basic equations and parameters Total current S d (cm 2 /cm 3 ): Specific surface area for double layer capacitance per unit volume S f (cm 2 /cm 3 ): Specific surface area for pseudocapacitance per unit volume Variables Concentration of electrolyte Solid phase potential Solution phase potential Concentration in solid

8. Department of Chemical Engineering University of South Carolina j f (A/cm 2 ): Faradaic current by pseudocapacitance U 1 (V vs. SCE): Equilibrium potential V 0 : 0.5V Solid phase current density Conservation of charge Effective diffusivity and conductivity

9. Department of Chemical Engineering University of South Carolina Material balance on the electrolyte using concentration solution theory Porous electrode Separator part

10. Department of Chemical Engineering University of South Carolina The variation of potential in the separator and the porous electrode Porous electrode Separator part

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

12. Department of Chemical Engineering University of South Carolina A mass balance of spherical particle of ruthenium oxide B.C r = 0 : r = Rs :

13. Department of Chemical Engineering University of South Carolina Parameters used in the model Fixed values –Thickness: 100  m for electrode, 25  m for separator –Exchange current density: 10 -5 A/cm 2 –Double layer : 2  10 -5 F/cm2 –Sigma: 10 3 S/cm –K 0 : 0.8 S/cm –Density: 2.5 g/cm 3, 0.9 g/cm 3 –D: 1.8  10-5 cm 2 /s –Ds: 10 -11 cm 2 /s –Transference number: 0.814 –Porosity of separator: 0.7 –Concentration of electrolyte: 1M H 2 SO 4 Variable values –Particle size of RuO 2 –Porosity of electrodes –The ratio between RuO 2 and carbon –Discharge current density –Discharge power density

14. Department of Chemical Engineering University of South Carolina Porosity of the electrode as a function of the mass fraction of RuO 2 Packing theory

15. Department of Chemical Engineering University of South Carolina Effect of the diffusion coefficient of proton in the solid particle on the capacitance at the constant current discharge of 30 mA/cm 2 40wt% RuO 2,Porosity: 0.214, Particle size: 5nm 105 F/g 59 F/g 1.0  10 -11 cm 2 /s 1.0  10 -16 cm 2 /s

16. Department of Chemical Engineering University of South Carolina Discharged energy density curves at the constant power discharge of 50w/kg for different particle sizes of RuO 2

17. Department of Chemical Engineering University of South Carolina Discharged energy density curves at the constant power discharge of 4kw/kg for different particle sizes of RuO 2

18. Department of Chemical Engineering University of South Carolina Local utilization of RuO 2 at the interface of separator as a function of particle size at different discharge rates.

19. Department of Chemical Engineering University of South Carolina Dimensionless parameter, Sc (diffusion in the solid/discharge time), as a function of particle size of RuO 2

20. Department of Chemical Engineering University of South Carolina Electrochemical performance of the RuO 2 /carbon composite electrode (60wt% RuO 2 ) with respect to constant current discharge Rs: 50nm  : 0.181

21. Department of Chemical Engineering University of South Carolina Electrolyte concentration distribution of the cell at the end of discharge with different current densites 500 mA/cm 2 200 mA/cm 2 100 mA/cm 2 30 mA/cm 2

22. Department of Chemical Engineering University of South Carolina Potential distribution in the electrolyte at the end of discharge at different current densities

23. Department of Chemical Engineering University of South Carolina Potential distribution in the electrolyte at the end of discharge at the different porosities of electrode  : 0..35  : 0.24  : 0.15  : 0.09 RuO 2 ratio: 60wt% Particle size: 50nm Current density: 1A/cm 2

24. Department of Chemical Engineering University of South Carolina Discharge density as a function of RuO 2 content, particle size and porosity of electrodes at 1.5A/cm 2

25. Department of Chemical Engineering University of South Carolina TEM image of RuO 2 ·nH 2 O/carbon composite electrode (40 wt% Ru) 25 nm

26. Department of Chemical Engineering University of South Carolina 3  m SEM images of RuO 2.nH 2 O/carbon composite electrode (60 wt% Ru ) (80 wt% Ru)

27. Department of Chemical Engineering University of South Carolina Specific capacitance of RuO 2 ·nH 2 O as a function of Ru loading

28. Department of Chemical Engineering University of South Carolina Ragone plot for RuO 2 /carbon composite electrode containing different Ru loading

29. Department of Chemical Engineering University of South Carolina Ragone plot for RuO 2 /carbon composite electrode containing different Ru loading using a colloidal method

30. Department of Chemical Engineering University of South Carolina Conclusions The general model was developed successfully to expect the performance of oxide/carbon composite electrode based on porosity, particle size, the content of RuO 2 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 RuO 2, since the diffusion process in the solid particle is a limiting step, the discharge stops before the RuO 2 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/cm 2