Chapter 9. Supercapacitors

Introduction to Chemical Convergence for Energy & Environment

Chapter 9. Supercapacitors

Spring Semester, 2012

Kookheon Char

Batteries

(Chapt. 7) Fuel Cells (Chapt. 8)

Supercapacitors (Chapt. 9)

Electrochemical Energy Storages

Background on Conventional Capacitors

 Schematic of Essential Components  Fundamental Equations

d C = ε

_r

ε

₀

A

2

2 1 CV E =

ESR P V

4

2 max

=

C: Capacitance

ε_r : Dielectric Constant of Medium (Electrolyte)

ε₀: Dielectric Constant of Vacuum A: Surface Area

d: Electrode Distance E: Energy

V: Voltage P: Power

ESR: Equivalent Series Resistance

 Capacitance proportional to electrode

surface area

 Energy proportional to capacitance

 Maximum power limited by internal

resistances (ESR) of a capacitor

Negative

Electrode Positive

Electrode

Applied Voltage d

Electric Field

+

−

+ −

−

+

−

+

−

+

−

+

−

+ +

− −

+

−

+ +

− −

+

−

+

Store Energy as Electric Potential between Two Electrodes and a Dielectric Layer

Comparison of Capacitor Technologies: Electrostatic Capacitor

Electrostatic Capacitors

 Materials

- Electrodes: metals

- Dielectric materials: metal films, ceramic, glass, mica

 Capacitance - pF ~ μF

- capacitance is constant

 Voltage

V E

εA Ed dz Qd

A Edz Q

V = ∫

0^d

= ∫

0^d

ε = =

d E = V d

+ Dielectric -

Layer

ε

r

A: area

dt dV(t) dt C

dq(t) I(t) = =

where E: Electric Field, Q: Charge

V C = Q

 Current

where q=CV, C: constant (capacitance)

nm) 1000

(d ≈

Electrolytic Capacitors

V E

 Materials

- Anode: Aluminum (or Tantalum) - Dielectric: Aluminum Oxide

- Cathode: Electrolyte & Aluminum

 High Capacitance - μF ~ mF

- High capacitance arises from high

dielectric constant and extremely small, uniform thickness of metallic

oxide (d: 10 ~ 100 nm)

- Capacitance of electrolyte is not a constant (i.e., electric double layer)

+

Dielectric

-

(aluminum oxide)

d ∆ d

Aluminum

ε

 Potential (V) Profile in the Diffuse Layer Electrolyte

Comparison of Capacitor Technologies: Electrolytic Capacitor

χ

=

−

κ

κ

_K

s 0 s

) e T 4 / ZeV tanh(

) T 4 / ZeV

tanh(

²

1

0 2 2 0

T V V

e Z n

K 2  

 





κ

= ⋅

,

Electrochemical Capacitors (Supercapacitors)

 Materials

- Electrodes: Activated carbon (porous) - Dielectric: Electric double layer

- Separator: Electric barrier (ex. glass paper) - Electrolyte: Sulfuric acid

 Ultrahigh capacitance - ~ F

- Capacitance is directly proportional to the surface areas of plates.

(Surface area: 500 to 2500 m

²/g )

- Charge separation distance (d): < 10 Å

+

+ -

- - - - - -

- - -

- - + + +

+ + +

+ + + + +

- - - - - - - -

- - + - + +

+ + + + + + +

Separator Electrode

(Current Collector)

Activated Carbon

Electrolyte

+ + + + + + + + + - - - - - - -

- -

+ + - - - + - - +

- - - - -

+ + +

+

Charged Discharged

Electrolyte

Comparison of Capacitor Technologies: Electrochemical Capacitor

Part I. Basics of Supercapacitors

What are Supercapacitors?

Also known as ultracapacitor or electrochemical capacitor

 Very high capacitance (even ~ 1000 F)

- Molecularely thin dielectrics

- High surface area & nanostructured electrodes

 High power density

 Virtually unlimited number of charge-discharge cycles

 No toxic substances as in conventional batteries

Electrode with electrolyte Electrode with electrolyte Separator

Current Collector Current Collector

 Cross-Section of a Typical Supercapacitor

 Ragone Plot of Energy Systems

• Activated carbon electrode

• Current collectors (positive and negative)

• Micro-porous separator

• Spiral-wound or prismatic

• Aqueous or non-aqueous electrolytes

Super- capacitor

M. Winter et al., Chem. Rev. (2004)

Batteries vs Supercapacitors

 EDLC (Electrical Double Layer Capacitor) research is largely focused on increasing their energy performance and widening the temperature limits into the range where batteries cannot operate

Properties

Batteries Supercapacitors

Storage Mechanism Chemical Physical

Power Limitation Reaction kinetics &

mass transport Separator ionic conductivity

Energy Limitation Electrode mass Electrode surface area

Output Voltage Constant value Sloping value

(State-of-Charge (SOC) known precisely) Charge Rate Reaction kinetics &

mass transport Very high, same as discharge rate Cycle Life Limitations Physical stability &

chemical reversibility

No Limitation by electrochemical kinetics (not swollen in active material)

Life Limitation Thermodynamic stability No Limitation

Solvent Limitation Decomposition of electrolyte at high voltage or high temperature

No limitation of solvents

 with high power performance at low temperatures (down to – 40 °C)

Li-Ion Batteries vs Supercapacitors

R. Miller et al., ECS Interface (2008)

* Time for discharge and charge of usable total energy stored in devices

** Power capability of battery for short duration partial discharge at 90% efficiency Characteristics State-of-the-Art

Li-ion Batteries Supercapacitors

*Charge Time ~ 3-5 min ~ 1 sec

*Discharge Time ~ 3-5 min ~ 1 sec

Cycle Life < 5,000 @ 1C rate > 500,000

Specific Energy (Wh/kg) 70 - 100 5

Specific Power

(kW/kg) **0.5 - 1 5 - 10

Cycle Efficiency (%) < 50% to > 90% < 75 to > 95%

Cost/Wh $ 1 - 2/Wh $10 - 20/Wh

Cost/kW $75 - 150/kW $25 - 50/kW

Taxonomy of Supercapacitors

Supercapacitors

Pseudo- Capacitors

Electrochemical

Double Layer Capacitors

Hybrid Capacitors

Carbon Nanotubes Activated

Carbons Conducting

Polymers Metal Oxides

Asymmetric

Hybrids Battery-Type Hybrids Composite

Hybrids

Carbon Aerogels

0 200 400 600 800 1000 1200 1400

AC CNT

MPC C₆₀ CAG NRC

PANI/AC

PEDT/AC PIThi

PFDT

PPy/AC MPFPT

DAAQ PAn/CNT

P3MT PEDT

PAn

RuO₂/PAPPA

RuO₂(sol-gel) RuO₂(ED) RuO₂(sol-gel) RuO₂/AC RuO₂/AC RuO₂(ESD) RuO₂/CB

RuO₂/CNT RuO₂/AC RuO₂/CXG RuO₂/MPC RuO₂/AC

NiO SnO₂ / Fe₃O₄ MnO_x

NiO/RuO₂

MnO₂

NiO

Ti/RuO_x/Co₃O₄

MnO₂

Ir_0.3Mn_0.7O₂

MnO₂

Sp eci fi c Ca pa ci ta nce / F g

-1

Carbons Polymers Metal Oxides RuO

₂

MPC: meso-porous carbon CAG: carbon aerogel NRC: nitrogen-rich carbon

PFPT: poly(fluorophenylthiophene) P3MT: poly(3-methylthiophene) PIThi: poly(isothianaphthene)

CXG: carbon xerogel CB: carbon black

PAn: polyaniline

( Literatures 2002~2006 )

ED: electrochemical deposition ESD: electrostatic spray deposition

Proposed Materials for Supercapacitors

Pseudo Capacitors

EDLCs

Part II. Electrochemical Double Layer Capacitors

V

^-

V

⁺

Q

_m^-

Q

_m⁺

Electrode Electrode

Electrolyte

+ + + + + - -

- - -

- - - - - +

+ + + +

Overview of Electrochemical Double Layer Capacitors (EDLCs)

 There is no transfer of charge, non-Faradaic

 Two carbon-based electrodes, aqueous or organic electrolyte

 Electrode is made from porous nanostructures– activated carbon, nanotubes, aerogels

 Schematic of ELDC

C

⁺

C

^-

R

_el

R

⁺_rx

R

^-_rx

1 1 1

C

_t

= C + C

+ −

+ +

=

+

V C Q

^m

C Q V

− m−

=

−

: Store the charge electrostatically using the reversible adsorption of ions of electrolyte onto active materials

 Simplified Electric Circuit

Models for Double Layer Structure

 Stern Model

 Helmholtz Model  Gouy-Chapman Model

– No Debye layer (ρ = 0) – Voltage jump at interface

(“dipole layer”)

– Constant differential capacitance

– Finite Debye length (“double layer”) – Parabolic differential

capacitance

– Linear voltage adjacent to electrode

– Parabolic differential capacitance with “wings”

– IHP : Inner Helmholtz Layer – OHP : Outer Helmholtz Layer

Conway BE., “Electrochemical Supercapacitors – Scientific Fundamentals and Technological Applications”, Kluwer Academic/Plenum (1999).

d dV

C

_d

dQ ε

^r

ε

⁰

=

=

_^



 



−

= kT

eV exp Z

] i [ ] i [

s 0 i

D H

d

C

1 C

1 C

1 = +

]0

i

[ : bulk ion concentration Zi : charge of ions

Vs : potential in solution

V I

∆V

1

E. Frackowiak et al., Carbon (2001)

where Ι : current

(dV/dt) : potential scan rate C : double layer capacitance

 Typical Charge/Discharge Voltammogram

1 - ideal capacitor

2 - capacitor with resistivity

3 - capacitor with carbon material 4 - influence of redox reactions

∆V - voltage delay I_c - capacitive current

dt C dV

I = ×

Sign of current is immediately reversed upon the reversal of potential sweep

Current is independent of potential (i.e., purely electrostatic)

Cyclic Voltammogram 1

* Capacitive Current: current (or current density) flowing through an electrochemical cell (It only causes accumulation (or removal) of electrical charges (not chemical reaction) on the electrode and in the electrolyte solution near the electrode)

V I

∆V

 Typical Charge/Discharge Voltammogram

E. Frackowiak et al., Carbon (2001)

R I = V

2

, A

R = ρ l Cyclic Voltammogram 2

1 - ideal capacitor

2 - capacitor with resistivity

3 - capacitor with carbon material 4 - influence of redox reactions

∆V - voltage delay I_c - capacitive current

: resistance,

ρ

: resistivity

R

 Resistivity is an inherent property of material

 Unit of resistivity is ohm-meter

V I

∆V

1 - ideal capacitor

2 - capacitor with resistivity

3 - capacitor with carbon material (EDLCs) 4 - influence of redox reactions

∆V - voltage delay I_c - capacitive current

 Typical Charge/Discharge Voltammogram

E. Frackowiak et al., Carbon (2001)

3

Cyclic Voltammogram 3

• Special oxidation of carbon for increasing the surface functionality (through chemical treatment, electrochemical polarization)

• Formation of carbon/conducting polymers

composites by electropolymerization of a suitable monomer (aniline or pyrrole)

• Insertion of electroactive particles of transition metals oxides such RuO2, TiO2, Cr2O3, MnO2, Co2O3 into the carbon materials

Enhancement of specific capacitance for cabon materials by:

V I

∆V

Cyclic Voltammogram 4

 Typical Charge/Discharge Voltammogram 4

E. Frackowiak et al., Carbon (2001)

1 - ideal capacitor

2 - capacitor with resistivity

3 - capacitor with carbon material 4 - influence of redox reactions (pseudocapacitor)

∆V - voltage delay I_c - capacitive current

* Voltage Delay (∆V): time interval at the start of a discharge during which the working voltage of a cell is below its steady value

: scan rate (dV/dt)

: surface concentration of reactant

C

1

s

I

pc

RT s VnF

C K

RT VnF

C K RT

nF

I q ⋅

− +

= −

₂

1 1

1 1 1

)}

/ (

exp {

) /

( exp

RT s nF I

_pc

= q ⋅

4

1

q

1 : amount of charge to form a monolayer

Strategy for High Energy Density

2

2 ) 1

( E CV

Energy =

Capacitance (C) Voltage (V)

• Surface area & pore size control

• Internal & external resistance ↓

• Wettability ↑

• Introduce of hetero-atoms

• Electrode density ↑

• Asymmetric cell (Hybrid capacitor)

• Electrochemical activation

• Electrolyte

- Aqueous : 1.2 V - Organic : over 2.5 V

• Purification and additives

• Protective coating of the electrode

• Asymmetric cell (Hybrid capacitor)

Porous Carbon Materials

Carbons

Category Representative Materials

Porous Carbons

Activated Carbons

steam-activated carbons alkali-activated carbons

Template Carbons Carbon Aerogel, Xeogels

LiF-Activated Carbons

Nanocarbons

CNTs（single-walled & multi-walled）

C60s

Carbon Nanohorns

* Graphenes (most recently)

 Activated Carbon

Electrode Material (Activated Carbon I)

 Large surface area (500 ~ 2500 m

²

/g)

 Optimized pore structure

 Low resistance

 Electrochemically stable

・Rechargeable at high temperature

・Wettable with solvents

Activated Carbon

Conducting Agents

+

_Binder

Adsorped ions (anions, cations)

Solvents

Micro pores (< 2 nm) Meso pores (2 ~ 50 nm) Macro pores

( > 50 nm)

Activation

CO

₂

CO

 Gas Activation  Activation by Reagents

 Alkaline metal (KOH)

Zinc chloride, Phosphate, etc.

 Steam

C+H₂O→CO+H₂ C+2H₂O→CO₂+2H₂

 Carbon dioxide

C+CO₂→2CO

 Oxygen

C+O₂→CO₂ 2C+O₂→2CO

4C+K₂SO₄→K₂S+4CO

for high surface area

Pore

Electrode Material (Activated Carbon II)

Porous Carbon Materials (Graphene)

 Chemically Modified Graphene (CMG) Particle Electrode Surface

 Cyclic Voltammogram

Nearly Rctangular in Shape

 Good charge propagation within the electrodes

 Insensitive to varying voltage scan rates (short and equal diffusion path length of ions in electrolyte)

R. Ruoff et al., Nano Lett. (2008)

 Measured conductivity of these CMG materials (∼ 2 × 10

²

S/m) closely approaches that of pristine graphite

 Synthesis process: suspension of graphene oxide (GO) sheets in water  reduction using

hydrazine hydrate

 During reduction, individual “graphene” sheets

agglomerate into particles of approx 15-25 μm in

diameter (surface area of CMG agglomerates: 705

m

²

/g)

O. Barbieri et al., Carbon (2005).

micro pore

meso pore macro pore

d < 2 nm

2 < d < 50 nm d > 50 nm

活性炭の細孔と容量の限界

Gravimetric capacitance / F g-1

Specific surface area S

_BET

/ m

²

g

^-1

 Capacitance vs Specific Surface Area

0 0 1000 2000 3000

20 40 60 80 100 120

Pore Distribution Control & Capacitance Limitations

High Surface Area NOT Necessarily Lead to High Capacitance over S

_BET

> 1700 ~ 2000 m

²

g

^-1

Optimum

Surface Area

Pore Size & Pore Size Distribution

Sub-Micro

Pore (< 0.5 nm) Micro Pore

(0.5 ~ 2 nm) Meso Pore

(2 ~ 50 nm) Macro Pore (> 50 nm)

 Pores with different size have different time constants

 R

_i

and C

_i

can be treated as resistance and capacitance of pores with a certain pore size ; R

_i

C

_i

which gives the unit of time, RC = (V/I)(I∙t/V)= t (time)

Indicating how fast pores of a certain size can be accessed

 A well balanced micro- or mesoporosity was needed to maximize capacitance

Transmission Line Equivalent Circuit Model

Hydrated Ion

H. Shi, Electrochim. Acta (1995)

E. Frackowiak et al., Electrochim. Acta (2005)

• Zones I & II : Electric Wire-in-Cylinder Capacitor in Solvent-Free Electrolyte

• Zone III : Electric Double-Cylinder Capacitor (EDCC)

• Zone IV : Planar Electric Double Layer Capacitor (EDLC)

Current Issues for Carbon Materials

 Specific Capacitance as a Function of Pore Size



 



= 

0 0

ln /

a b b A

C

ε

^r

ε

 

 





−

=

d b b b A

C ^r

ln

/ ε ε

⁰

Y. Gogotsi et al., Science (2006)

J. Huang et al., Angew. Chem. Int. Ed. (2008)

a₀ : effective size of unsolvated ion b: pore radius

d: distance of approach of ion to carbon surface

A, B: templated mesoporous carbons; C: activated mesoporous carbons; D, F: microporous CDCs ; E: microporous activated carbons

 Partial or complete removal of the

solvation shell and increased confinement of ions leads to increased capacitance.

• High surface area and double layer of charge allows for much higher energy densities than conventional capacitors, with comparable power densities

• No chemical or structural change during charge storage up to 10

⁶

cycles for supercaps compared with 10

³

cycles for batteries

• Work in extreme temperature and very safe

• Nanostructured carbon materials are relatively cheap and fabrication techniques are well developed

Advantages:

• Cannot match energy densities of mid-level batteries Disadvantages:

Advantages & Disadvantages of EDLCs

Part III. Pseudocapacitors

Overview of Pseudocapacitors

• Charge transfer through surface Faradaic, redox

reactions

• Similar to EDLCs, but electrodes are made from

metal-oxides or conducting polymers

- Generally, lower power densities than EDLCs

- Cycle life can be limited by mechanical stress caused during the reduction-oxidation reactions - Negatively charged conducting polymer electrodes are not very efficient

- Best metal-oxide electrodes are very expensive and require aqueous electrolytes, implying lower voltage

• Advantages:

• Disadvantages

Current Voltage

Current Voltage

- Electrolyte ions diffuse into pores and undergo fast, reversible surface reactions

- Relationship between charge and potential gives rise to a pseudocapacitance

- Can achieve very high capacitances & energies

High surface area and fast Faradaic reactions allow for higher energy densities than EDLCs (Hydrous Ruthenium Oxide can achieve extraordinary capacitance)

+

-

Ru O O (IV) Ru OH

O (III) Ru OH

OH

(II)

_charge

discharge

charge

discharge

Ru O O (IV) Ru OH

O (III) Ru OH

OH (II)

Redox of RuO ₂

Stable Redox Reaction of RuO ₂

V

Ca pa ci ta nc e

Part IV. Hybrid Capacitors

Hybrid Capacitors

 Hypothetical Energy-Power Behavior

Technologies must be decoupled to effectively exploit the combination

Specific Energy

Specific Power

Technology 1 Combination of Tech 1 & 2 Technology 2

 Symmetric

Double layer

+

- - - - - + + + + +

_

ele ct ro ly te

Q

Lower Limit Upper Limit

Potential / V

+ -

Q

Lower Limit

Upper Limit +

- Double layer

+ + + + ele ct ro ly te

_ +

Battery Electrode

 Asymmetric

Potential / V

−

∞ +

= C

C

_t

1 1

1

Doubling capacitance of carbon electrode over symmetric device (battery)

(capacitor)

Design of Hybrid Capacitors & Advantages



Charging-Discharging Principle of a Hybrid Capacitor

“Pre-Doping” System:

Doping anode with Li-metal before charging-discharging

a cell

• Doubling capacitance of carbon electrode over symmetric device

• Higher operating voltage than symmetric device

• Voltage self-balance in series strings

• Very high specific energy and energy density demonstrated

• Response times of 2 to 100 seconds typical

Part V. Needs and Applications of Supercapacitors

Supercapacitor Technology Needs

 Increase cell operating voltage to: > 4.0 V with RC < 1 s, high cycle life electrode/electrolyte system

 Use lower cost design — exploit anomalous capacitance observed in asymmetric aqueous electrolyte ECs

 Use electrolyte additive to reduce drying costs and control other impurities

• Lower Cost Cells

• Longer Life Cells

• Higher Cycle Efficiency Cells

 Well-sealed cells always fail with package rupture (except valved caps)

 Use electrolyte additive to prevent or control gas generation

 Devise more effective ways for removing impurities

 Carbon composite electrode may obviate current collector in asymmetrics

 Higher conductivity electrolyte

 Thinner, more open separator

 Resistances need to be reduced everywhere

Many Potential Applications

Emergency Power Back-Up

Peak Power Load Leveling

& Energy Saving Line-Drop

Protection

HONDA “FCX Concept”

 Startup and Acceleration

Ultra-capacitor assists the fuel cell stack to achieve crisp, responsive performance

 Deceleration

Ultra-capacitor recovers the energy released during deceleration and stores it along with power from the fuel cell stack

 Stopped

Auto idle stop system shuts off output from fuel cell stack to reduce fuel consumption. Electricity required to operate the other components is supplied by the ultra-capacitor

http://world.honda.com/FuelCell/FCX/overview/

Ultracapacitor Emergency Power Module for Wind Turbines

http://www.rell.com/resources/

With no moving parts, ultracapacitors provide a simple, solid state, highly reliable solution to buffer short-term mismatches between power available and power required.

Examples

Support Material 1: What is Capacitance?

• Capacitance (C): capability for charge storage per voltage (unit: farads) * Capacity: used in battery terminology to indicate the extent of Faradaic charge storage (coulombs or watt-hours)

 Q = ε

_r

ε

₀

V/d

where ε

_r

is the electrolyte dielectric constant, ε

₀

is the dielectric constant of the vacuum, V is the voltage, and d is the electrode distance

 C/A = Q/V, C = ε

_r

ε

_o

A / d where A is the surface area

2 Q 2

0

2 CV 1 C

Q 2 dq 1 C

q = =

∫

 Capacitance (C) = Q/V

 dW = (q / C)dq, W

_charging

(E)=

(Moving a small element of charge dq from one plate to the other against the potential difference V = q/C requires the work dW)

Support Material 2. Equivalent Series Resistance & P

_max

• Equivalent Series Resistance (ESR) : all physical devices (such as capacitors or inductors) constructed of materials with finite electrical resistance,

implying that physical components contain some resistance in addition to their other properties

Non-Ideal Capacitor with Series Resistor

) ESR (

I V V =

_i

−

•

) ESR (

I IV

P =

_i

−

²

(this is a maximum when, ) dP / dI = 0 = V

_i

− 2 I ( ESR ) (where, V

_i

= initial potential)

) ESR (

4 / V I

V P

2 / V )

ESR ) (

ESR (

2 V V

) ESR (

I V V

) ESR (

2 / V I

2 i p

. max p

. max max

i i

i i

p . max

i p

. max

=

×

=

∴

=

−

=

−

=

=

∴

Support Material 3. Potential Sweep Methods

 Linear Sweep Voltammetry (LSV)

 Cyclic Voltammetry (CV)

: current at a working electrode is measured while the potential between the working electrode and a reference electrode is swept linearly in time

* Voltammetry

: An electrochemical measuring technique used for electrochemical analysis or for the determination of the kinetics and mechanism of electrode reactions.

: a linear-sweep voltammetry with the scan continued in the reverse direction at the end of the first scan, this cycle can be repeated a number of times.

: voltage cycle is defined by four parameters

 the maximum and minimum voltages, the starting

voltage, and the initial direction of scan (to more + or

to more – voltage values).

Support Material 4. Physics of Double Layer

 Diffuse layer of charge carriers at the interface

) exp(

] [ ]

[

⁰

kT eV i Z

i

s

−

i

=

 Electrolytic Capacitor

 The total charge per unit volume in the diffuse layer

e Z i x

Q

_i

∑

i

= [ ] )

(

According to Poisson’s Equation,

) exp(

] [

⁰

0 2

2

kT eV Z Z

e i dx

V

d

_i ^s

i i

s

−

− ⋅

= _{ε ε} ∑

Boundary conditions

= 0

∞

= sx

V = 0

∞ x= s

dx

, dV

[i]: Bulk Ion concentration Z_i: Charge of Ions

e: Charge of Electron

V^s: Potential in the Solution

ε : Electrolyte Dielectric Constant ε₀: Dielectric Constant of the Vacuum

 Potential (V) Profile in the Diffuse Layer

χ

=

−

κ

κ

_K

s 0 s

) e T 4 / ZeV tanh(

) T 4 / ZeV

tanh(

²

1

0 2 2 0

T V V

e Z n

K 2  

 





κ

= ⋅

,

) ) 1 exp( ( )

exp(

) 1

(

₁

1

1

RT

k VnF RT

C VnF nF k

I

_net

= − θ β −

₋

θ − ⁻ β

: transfer coefficient

) /

exp(

1

) / exp(

1 1

1 1

RT VnF

C K

RT VnF

C K

= + θ

at Equilibrium Condition (I_net=0)

s RT RT VnF

nF RT

VnF C

K

C K dt

d ⋅

−

= + exp( / )

)}

/ (

exp

{

_{1 1} ²

1

θ

1

: scan rate

RT s VnF

C K

RT VnF

C K RT

nF q dt

q d

I ⋅

− +

= −

=

∴

₂

1 1

1 1 1

1

{ exp ( / )}

) / (

θ exp

Peak Current of Pseudocapacitor

0

&

0

₂

2

=

= dt

d dt

dI θ

RT s nF I

_pc

= q ⋅

4

1

Support Material 5. Peak Current of Pseudocapacitor

θ

β

1 1 1

= k / k

₋

K

dt s = dV θ

d q Idt

dQ = =

₁ : amount of charge

to form a monolayer : surface coverage

q

1

dt q d

I θ

=

1

∴

Butler-Volmer equation By redox rxn at the surface

(1)

(2)

(3)

(4)

(5)

by eqn.(1) nF

I₀

) 1

1(

1 θ

ν = Ck − rxn rate