Mathematical Model of RuO 2 /Carbon Composite Electrode for Supercapacitors

Transcription

1 Mathematical Model of RuO /Carbon Composite Electrode for Supercapacitors by Hansung Kim and Branko N. Popov Center for Electrochemical Engineering

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., (1999 RuO 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 ( RuO /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 /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 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 RuO

4 Schematic diagram of supercapacitors and reaction mechanism Negative electrode Separator Positive electrode Current Collector H.8 RuO Carbon Electrolyte 1M H SO 4 x L Ls

5 Faradaic reaction of ruthenium oxide Positive electrode Discharge: Charge: H.8 δ RuO xh O H.8RuO xh O δh δe H.8RuO xh O δh δe H.8 δ RuO xh O Equilibrium potential (V vs. SCE H.3RuO xh O H.8RuO xh O H1.3RuO xh O : 1 V :.5 V : V

6 Assumptions Porous electrode theory. Double layer capacitance per area (C d is constant for carbon and RuO. 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 C Concentration of electrolyte Φ 1 Solid phase potential Φ Solution phase potential Concentration in solid C s Total current i Φ Φ = SdC ( 1 d S f j f t S d (cm /cm 3 : Specific surface area for double layer capacitance per unit volume 6xRu (1 ε Sd = Sc S f = SC ρc xc (1 ε d S f (cm /cm 3 : Specific surface area for pseudocapacitance per unit volume 6xRu (1 ε S f = d Ru Ru

8 j f (A/cm : Faradaic current by pseudocapacitance j f = i{exp[ α a ( Φ1 Φ U1 F / RT ] exp[ αc ( Φ1 Φ U1 F / RT ]} U 1 (V vs. SCE: Equilibrium potential M U1 = V ( RuO Cs 1.3 ρ RuO Solid phase current density i Φ1 = 1 σ Effective diffusivity and conductivity.5 D = D ε 1.5 k p = k p ε V :.5V Conservation of charge I = i1 i i1 i =

9 Material balance on the electrolyte using concentration solution theory x t F z i v t aj x C C d C d D x t C n = (1 (ln (ln 1 ν ε ε Porous electrode Φ = v t x nf s x C D t C i (1 1 σ ε ε Separator part x t F z i x C C d C d D x t C = (ln (ln 1 ν ε ε x C D t C e =.5 ε

10 The variation of potential in the separator and the porous electrode Porous electrode i = I i 1 I = κ Separator part i i = = I P Φ Φ = I σ = κ I P = κ Φ P Φ 1 κ PRT F = κ P κ PRT F κ PRT F (ln f (1 (lnc Φ e κ PRT F (ln f (1 (lnc s ( nv e s ( nv t z v (lnc ± s ( nv s ( nv t z v (lnc t z v (lnc ± t z v (lnc

11 B.C. Boundary and Initial conditions At x = : (current collector of positive electrode C = I cell Φ1 = i1 = σ Φ = At x = Le: (interface between separator and electrode ε 1.5 s D C sep = ε 1.5 D C elec Φ 1 = ε 1.5 s Kp 1.5 = ε Kp sep elec Φ Φ At x = LeLs : (current collector of negative electrode C = I.C. At t =, C = C, I cell Φ1 = i = 1 σ Φ = M M RuO RuO Φ =.3 Φ = positive 1negative ρ ρ RuO RuO

12 A mass balance of spherical particle of ruthenium oxide C t C r s s = Ds r C r s j f B.C r = : r = Rs : C s r Cs = r = = i{exp[ α a ( Φ1 Φ U1 F / RT ] exp[ αc ( Φ1 Φ U1 F / RT ]} j D s f F M U1 = V ( RuO Cs 1.3 ρ RuO

13 Parameters used in the model Fixed values Variable values Thickness: 1µm for electrode, 5 µ m for separator Exchange current density: 1-5 A/cm Double layer : 1-5 F/cm Sigma: 1 3 S/cm K :.8 S/cm Density:.5 g/cm 3,.9 g/cm 3 D: cm /s Ds: 1-11 cm /s Transference number:.814 Porosity of separator:.7 Concentration of electrolyte: 1M H SO 4 Particle size of RuO Porosity of electrodes The ratio between RuO and carbon Discharge current density Discharge power density

14 Porosity of the electrode as a function of the mass fraction of RuO.5.4 Pore volume base (V BP =.93cm 3 /g,v RuO =.19cm 3 /g Packing theory Porosity.3 Packing theory (α : Pore volume base (V Vulcan XC-7 =.38cm 3 /g,v RuO =.19cm 3 /g Mass fraction of RuO

15 Effect of the diffusion coefficient of proton in the solid particle on the capacitance at the constant current discharge of 3 ma/cm 4wt% RuO,Porosity:.14, Particle size: 5nm 1..8 Cell potential (V cm /s 15 F/g cm /s 59 F/g Discharge time (s

16 Discharged energy density curves at the constant power discharge of 5w/kg for different particle sizes of RuO nm 15nm 6nm nm Cell potential (V Discharged energy density (Wh/kg

17 Discharged energy density curves at the constant power discharge of 4kw/kg for different particle sizes of RuO nm 15nm 6nm nm Cell potential (V Discharged energy density (Wh/kg

18 Local utilization of RuO at the interface of separator as a function of particle size at different discharge rates W/L 4 W/L 9 Local utilization (% Particle size of RuO (nm

19 Dimensionless parameter, Sc (diffusion in the solid/discharge time, as a function of particle size of RuO nm 1nm Sc (dimensionless nm 15nm 1-3 Sc = Rs I D F(1 ε C δ s t c Discharge current density (A/cm

20 Electrochemical performance of the RuO /carbon composite electrode (6wt% RuO with respect to constant current discharge ma/cm 1 ma/cm 5 ma/cm 1 ma/cm Cell potential (V.6.4 Rs: 5nm ε: Discharge density (C/cm

21 Electrolyte concentration distribution of the cell at the end of discharge with different current densites. Concentration of electrolyte (mol/l ma/cm 1 ma/cm ma/cm 5 ma/cm Dimensionless distance

22 Potential distribution in the electrolyte at the end of discharge at different current densities. 3 ma/cm -. 1 ma/cm Electrolyte potential drop (V ma/cm 1 ma/cm Dimensionless distance..

23 Potential distribution in the electrolyte at the end of discharge at the different porosities of electrode Electrolyte potential drop (V RuO ratio: 6wt% Particle size: 5nm Current density: 1A/cm ε :..35 ε :.4 ε :.15 ε : Dimensionless distance..

24 Discharge density as a function of RuO content, particle size and porosity of electrodes at 1.5A/cm nm 5 nm Discharged charge density (C/cm Porosity Weight percent of RuO (%

25 Ragone plot for RuO /carbon composite electrode containing different Ru loading using a colloidal method % RuO, ε:.37, Rs: nm 1 8% RuO, ε:.97, Rs: nm Energy density (Wh/Kg % RuO, ε:.5, Rs: 3 nm 6% RuO, ε:.14, Rs: 1 nm Power density (W/Kg

26 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 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, since the diffusion process in the solid particle is a limiting step, the discharge stops before the RuO particle has fully been utilized. Increasing porosity decreased the electrolyte deviation and solution potential drop. After the porosity increases up to about.15, the particle size is important to get a high performance until the discharge rate of 1.5A/cm