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

Department of Chemical EngineeringUniversity of South Carolina

by Hansung Kim and Branko N. Popov

Department of Chemical EngineeringCenter for Electrochemical Engineering

University of South Carolina

Mathematical Model of RuO2/Carbon Composite Electrode for Supercapacitors


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


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


Schematic diagram of supercapacitors and reaction mechanism

Current Collector

Negativeelectrode

PositiveelectrodeSeparator

0

x

L Ls

28.0 RuOH Carbon

Electrolyte1M H2SO4


Faradaic reaction of ruthenium oxide

• Positive electrode

Discharge:

Charge:

eHOxHRuOHOxHRuOH 228.0228.0

OxHRuOHeHOxHRuOH 228.0228.0

• Equilibrium potential (V vs. SCE)

OxHRuOH 223.0

OxHRuOH 228.0

OxHRuOH 223.1

: 1 V

: 0.5 V

: 0 V


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.


Model description: Basic equations and parameters

• Total current

ffdd jSt

CSx

i

)( 212

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

Ru

RuCCCfcd d

xxSSSS

)1(6)1(

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

Ru

Ruf d

xS

)1(6

• Variables

C

sC

1

2

Concentration of electrolyteSolid phase potentialSolution phase potentialConcentration in solid


]}/)(exp[]/)({exp[ 1211210 RTFURTFUij caf

• jf (A/cm2): Faradaic current by pseudocapacitance

• U1 (V vs. SCE): Equilibrium potential

)3.1(22

2

01 s

RuO

RuO CM

VU V0: 0.5V

• Solid phase current density

xi

11

• Conservation of charge

x

i

x

i

iiI

21

21

0

• Effective diffusivity and conductivity 5.0

0DD 5.1

0pp kk


Material balance on the electrolyte using concentration solution theory

x

t

Fz

i

v

taj

x

C

Cd

CdD

xt

C n

02

00 )1(

)(ln

)(ln1

v

t

xnF

s

x

CD

t

C i )1( 0

21

2

2

2

x

CD

t

Ce

2

0

5.0

Porous electrode

Separator part

x

t

Fz

i

x

C

Cd

CdD

xt

C

02

0

)(ln

)(ln1


The variation of potential in the separator and the porous electrode

Porous electrode

Separator part

x

C

vz

t

nv

s

C

f

F

RT

xIi

e

PP

)(ln))(

)(ln

)(ln1(

02

2

x

C

vz

t

nv

s

F

RT

xIi P

P

)(ln)(

02

2

x

C

vz

t

nv

s

C

f

F

RT

xIi

e

PP

)(ln))(

)(ln

)(ln1(

02

2

x

C

vz

t

nv

s

F

RT

xxIiI P

P

)(ln)(

021

1


Boundary and Initial conditionsB.C.

At x = 0 : (current collector of positive electrode)

02 x

0

x

Cx

iIcell 1

1

At x = Le: (interface between separator and electrode)

elecseps x

CD

x

CD

0

5.1

0

5.1 01 x

At t = 0, C = C0 ,

I.C.

elecseps x

Kpx

Kp

2

0

5.120

5.1

At x = 2Le+Ls : (current collector of negative electrode)

0

x

Cx

iIcell 1

1 02

3.02

2

1

RuO

RuO

positive

M

3.1

2

2

1

RuO

RuO

negative

M


A mass balance of spherical particle of ruthenium oxide

r

C

rr

CD

t

C sss

s 22

2

0

r

Cs

FD

j

r

C

s

fs

B.Cr = 0 :

r = Rs :

]}/)(exp[]/)({exp[ 1211210 RTFURTFUij caf

)3.1(22

2

01 s

RuO

RuO CM

VU


Parameters used in the model

• Fixed values

– Thickness: 100m for electrode, 25 m for separator– Exchange current density: 10-5 A/cm2

– Double layer : 210-5 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


Porosity of the electrode as a function of the mass fraction of RuO2

Mass fraction of RuO2

0.0 0.2 0.4 0.6 0.8 1.0

Por

osity

0.0

0.1

0.2

0.3

0.4

0.5

Packing theory (

Pore volume base (VVulcan XC-72=0.38cm3/g,VRuO2=0.019cm3/g)

Pore volume base (VBP2000=0.93cm3/g,VRuO2=0.019cm3/g)

Packing theory


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

105 F/g

59 F/g

1.010-11 cm2/s

1.010-16 cm2/s


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


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


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


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

cts

s

CFD

IRSc

)1(

2


Electrochemical performance of the RuO2/carbon composite electrode (60wt% RuO2) with respect to constant current discharge

Rs: 50nm: 0.181


Electrolyte concentration distribution of the cell at the end of discharge with different current densites

500 mA/cm2

200 mA/cm2

100 mA/cm2

30 mA/cm2


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


Potential distribution in the electrolyte at the end of discharge at the different porosities of electrode

: 0..35

: 0.24

: 0.15

: 0.09RuO2 ratio: 60wt%Particle size: 50nmCurrent density: 1A/cm2


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


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

25 nm


3 m 3 m

SEM images of RuO2.nH2O/carbon composite electrode

(60 wt% Ru ) (80 wt% Ru)


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

Weight Percent of Ru (%)

10 20 30 40 50 60 70 80 90

Spe

cific

Cap

acita

nce

(F/g

of R

uO2)

600

650

700

750

800

850

900


Ragone plot for RuO2/carbon composite electrode containing

different Ru loading

Power density (W/Kg)

70 200 500 1000 2000 30004000

En

erg

y d

en

sity

(Wh

/Kg

)

0

10

20

30

40

50

60

20wt% Ru40wt% Ru60wt% Ru80wt% Ru


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


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

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

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