MODELING OF AN ELECTROCHEMICAL CELL

by

Jin Hyun Chang

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Graduate Department of Electrical and Computer Engineering

University of Toronto

© Copyright by Jin Hyun Chang 2009

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Modeling of an Electrochemical Cell

Jin Hyun Chang

Master of Applied Science

Department of Electrical and Computer Engineering University of Toronto

2009

Abstract

This thesis explores a rigorous approach to model the behaviour of an electrochemical cell.

A simple planar electrochemical cell consisting of stainless steel electrodes separated by a sulfuric

acid electrolyte layer is modeled from first principles. The model is a dynamic model and is valid

under constant temperature conditions. The dynamic model is based on the Poisson-Nernst-Planck

electrodiffusion theory and physical attributes such as the impact of nonlinear polarization, the

stoichiometric reactions of the electrolyte and changes to the transport coefficients are investigated

in stages. The system of partial differential equations has been solved using a finite element software

package. The simulation results are compared with experimental results and discrepancies are

discussed. The results suggest that the existing theory is not adequate in explaining the physics in the

immediate vicinity of the electrode/electrolyte interface even though the general experimental and

simulation results are in qualitative agreement with each other.

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Acknowledgement

I would like to give my deepest gratitude to my thesis supervisors, Professor Francis

Dawson and Professor Keryn Lian, for their support, encouragement, and patience throughout the

program. This thesis could not have been completed without their help and guidance.

I am grateful for my parents for their endless love and support. You have been extremely

supportive with all of my decisions and I could not have been able to pursue my dreams without you.

My best friend Jason, who passed away in a tragic car accident on July 13th, 2009, we have

always been a great team and we will always be. I will miss you. Rest in peace.

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Table of Contents

Abstract ............................................................................................................................................................... ii

Acknowledgement ............................................................................................................................................ iii

Table of Contents ............................................................................................................................................. iv

List of Symbols ................................................................................................................................................. vi

List of Figures ................................................................................................................................................. viii

List of Tables ..................................................................................................................................................... xi

Chapter 1 Introduction ............................................................................................................................... 1

1.1 Thesis Objective ............................................................................................................................ 9

1.2 Thesis Outline ................................................................................................................................ 9

Chapter 2 Introduction to Electrochemistry ......................................................................................... 11

2.1 Fundamental Concepts and Terminology ............................................................................... 11

2.2 Key Areas for Electrochemical Analysis .................................................................................. 20

2.1.1 Transport Phenomena ................................................................................................... 20

2.1.2 Thermodynamics ............................................................................................................ 22

2.1.3 Kinetics ............................................................................................................................. 25

2.1.4 Poisson's Equation ......................................................................................................... 26

Chapter 3 Overview of Electrochemical Capacitors ............................................................................ 29

3.1 Electrochemical Capacitor Technologies ................................................................................ 29

3.2 Construction of Electrochemical Capacitors .......................................................................... 31

3.3 Review of Existing Models ........................................................................................................ 32

3.1.1 Empirical Models ............................................................................................................ 33

3.1.2 Dissipative Transmission Line Model ......................................................................... 35

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3.1.3 Dynamic Models ............................................................................................................. 35

Chapter 4 Model Development .............................................................................................................. 37

4.1 Characterization of the Electrode ............................................................................................. 38

4.2 Properties of the Electrolyte Solution ...................................................................................... 43

4.3 Characterization of the Electrolyte ........................................................................................... 47

4.4 Dielectric Polarization of the Electrolyte ................................................................................ 50

4.5 Electrolyte Reactions .................................................................................................................. 52

4.6 Coefficients as a Function of Concentration .......................................................................... 55

4.7 Scaling of Variables ..................................................................................................................... 57

Chapter 5 Results and Discussion .......................................................................................................... 60

5.1 Experimental Settings and Results ............................................................................................ 61

5.2 Comparison of Experimental & Simulation Voltage Curves................................................ 67

5.3 Comparison of Electric Field & Concentration ..................................................................... 72

Chapter 6 Conclusions ............................................................................................................................. 80

References ......................................................................................................................................................... 85

Appendix A Properties of Sulfuric Acid ................................................................................................ 88

A.1 Conductivity of Sulfuric Acid .................................................................................................... 88

A.2 Concentrations & Transport Coefficients of Sulfuric Acid .................................................. 98

Appendix B Dielectric Polarization of Water ....................................................................................... 99

Appendix C Calculation of Forward/Reverse Reaction Rates ......................................................... 104

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List of Symbols

Symbol Meaning

α Apparent transfer coefficient

0ε Vacuum permittivity (8.8541878 F/m)

rε Relative permittivity (or dielectric constant)

iγ Activity coefficient of species i

iµ Electrochemical potential of species i [J/mol]

iµ Chemical potential of species i [J/mol]

fρ Free charge density [C/m3]

Φ Electrostatic potential [V]

sη Surface overpotential [V]

χ Electric susceptibility

ia Activity of species i

ic Concentration of species i [mol/m3]

D Electric displacement field [C/m2]

iD Diffusion coefficient (diffusivity) of species i [m2/sec]

E Electric field intensity [V/m]

equiE Equilibrium potential [V]

F Faraday constant (96485.3399 C/mol)

G Gibbs free energy [J]

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0i Exchange current density [A/m2]

k Boltzmann’s constant ( 231.38065 10 J K−× )

fk Forward reaction rate

rk Reverse reaction rate

iu Mobility of species i [m2·mol/J·sec]

,i migrationN Flux density of species i due to migration [mol/m2·sec]

,i diffusionN Flux density of species i due to diffusion [mol/m2·sec]

,i convectionN Flux density of species i due to convection [mol/m2·sec]

iN

Total net flux density of species i [mol/m2·sec]

P Polarization density [C/m2]

q Elementary charge ( 191.602176 10 C−× ).

R Universal gas constant (8.314472 J/K·mol)

iR Reaction rate of the species i [mol/m3·sec]

is Stoichiometric coefficient of species i

T Absolute temperature [K]

v Bulk velocity of the electrolyte [m/sec]

,i migrationv Velocity of species i due to migration [m/sec]

iz Charge number of species i

viii

List of Figures

Figure 1.1 - Ragone plot [1] .............................................................................................................................. 2

Figure 1.2 - Comparison with conventional capacitors ................................................................................ 4

Figure 1.3 - Efficiency and lifetime properties of energy storage devices [1] ............................................ 7

Figure 2.1 - Simple structure of electrochemical system ............................................................................ 11

Figure 2.2 - Stern model of electric double layer ......................................................................................... 16

Figure 3.1 – A schematic of electrical double layer capacitor .................................................................... 32

Figure 3.2 - Equivalent electrical (left) and thermal (right) circuit by Zubieta [11, 12] .......................... 33

Figure 3.3 - Electro-thermal equivalent circuit model by Rafik [13] ........................................................ 34

Figure 3.4 - Transmission line equivalent electrical circuit model [15, 16] .............................................. 35

Figure 4.1 – Flat structure of electrochemical capacitor ............................................................................ 39

Figure 4.2 - Gaussian surface around perfect conductor ........................................................................... 41

Figure 4.3 - Conductivity VS concentration of sulfuric acid at 25 °C ...................................................... 44

Figure 4.4 - Polarization density VS electric field for water ....................................................................... 52

Figure 5.1 - Schematic of the experimental conditions .............................................................................. 61

Figure 5.2 – Optical image of the stainless steel electrode surface ........................................................... 63

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Figure 5.3 - 3-D contour map of the stainless steel electrode surface ..................................................... 63

Figure 5.4 - Experimentally measured voltage drop VS time for 0.1 M H2SO4 at different

temperatures ...................................................................................................................................................... 65

Figure 5.5 – Experimentally measured peak voltage at 55 sec VS temperature for 0.1 M H2SO4 ....... 65

Figure 5.6 - Experimentally measured voltage drop VS time for 1 M H2SO4 at 25 ˚C ......................... 66

Figure 5.7 – Simulated voltage drop VS time for Poisson/Nernst-Planck model ................................. 67

Figure 5.8 – Simulated voltage drop VS time for reaction chemistry model .......................................... 68

Figure 5.9 – Simulated voltage drop VS time for dynamic coefficient model ........................................ 68

Figure 5.10 – Simulated voltage drop VS time for dynamic coefficient/reaction model ...................... 69

Figure 5.11 – Comparison of simulated/experimental peak voltages at 55 sec VS temperature for 0.1

M H2SO4 ............................................................................................................................................................ 69

Figure 5.12 – Simulated/experimental voltage drop VS time for 0.1 M H2SO4 system at 25 ˚C ......... 71

Figure 5.13 – Simulated with ACF/experimental voltage drop VS time for 0.1 M H2SO4 system at 25

˚C ........................................................................................................................................................................ 72

Figure 5.14 – Simulated electric fields VS distance from the left electrode at 25 ˚C ............................. 73

Figure 5.15 – Simulated electric field VS distance from the right electrode at 25 ˚C ............................ 73

Figure 5.16 – Simulated H3O+ concentration VS distance from the left electrode at 25 ˚C ................. 74

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Figure 5.17 – Simulated H3O+ concentration VS distance from the right electrode at 25 ˚C .............. 75

Figure 5.18 – Simulated SO42- concentration VS distance from the left electrode at 25 ˚C .................. 75

Figure 5.19 – Simulated SO42- concentration VS distance from the right electrode at 25 ˚C ............... 76

Figure 5.20 – Simulated HSO4- concentration VS distance from the left electrode with at 25 ˚C ....... 76

Figure 5.21 – Simulated HSO4- concentration VS distance from the right electrode at 25 ˚C ............. 77

Figure 5.22 – Simulated OH- concentration VS distance from the left electrode at 25 ˚C ................... 77

Figure 5.23 – Simulated OH- concentration VS distance from the right electrode at 25 ˚C ................ 78

Figure B.1 - Differential dielectric constant VS electric field ................................................................. 101

Figure B.2 - Polarization density VS electric field .................................................................................... 102

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List of Tables

Table 3.1 - Comparison of aqueous and non-aqueous electrolytes .......................................................... 31

Table 4.1 - Forward and reverse reaction rate constants of sulfuric acid at 25 °C ................................. 45

Table 4.2 – Concentration of species for 0.1 M sulfuric acid at 25 °C .................................................... 46

Table 4.3 - Dielectric constant of water [23] ................................................................................................ 50

Table 4.4 - Forward and reverse reaction rates of 0.1 M sulfuric acid at various temperatures ........... 55

Table 5.1 – Types of electrostatic models developed ................................................................................. 60

Table 5.2 - Experimental Setup ...................................................................................................................... 61

Table 5.3 - Surface parameters of stainless steel electrodes ....................................................................... 63

Table A.1 - Conductivity of sulfuric acid (conductivity values given in mho) [22] ................................ 97

Table A.2 - Concentration of species in H2SO4 [mol/L] ............................................................................ 98

Table A.3 – Activity coefficient of species in H2SO4 .................................................................................. 98

Table A.4 – Mobility of species in H2SO4 [s·mol/kg] ................................................................................. 98

Table A.5 – Self diffusivity of species in H2SO4 [m2/s] .............................................................................. 98

Table B.1 - Differential dielectric polarization of water [24] .................................................................. 103

1

Chapter 1 Introduction

As a consequence of elevated oil prices and an effort to reduce the carbon dioxide emissions,

both alternative energy sources and energy storage devices are receiving an unforeseen level of

attention. The electrical energy generated from renewable sources, such as wind and solar, offers

an enormous potential to fulfill the next generation energy requirements. However, these renewable

energy sources alone are not viable for power grid applications, where reliable, fluctuation free

electricity is necessary at all times. Therefore, efficient electrical energy storage is essential for load

levelling and peak shaving in order to allow large-scale renewable energy sources to be integrated

with the existing power grid system.

A rising demand for hybrid electric vehicles and all-electric vehicles have also put pressure on

manufacturers of batteries to find more efficient means of storing electrical energy. The main

criteria for wider adaption of both types of vehicles are the higher energy storage capacity for longer

driving distance per charge, and longer lifetime and higher reliability due to the high replacement

costs. Therefore, further performance improvements of the energy storage devices are the key issue

for such vehicles to be accepted by the market.

Many types of energy storage devices exist including various types of batteries, capacitors,

electrochemical capacitors and fuel cells. They exhibit different combinations of specific power

and specific energy1

1 Specific power and specific energy refer to power and energy per unit volume or mass, respectively. The terms are often used interchangeably with power density and energy density.

as shown in Figure 1.1 and as a result, their suitability for an application can

vary as well. Capacitors and batteries have been the most widely used energy storage devices in a

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variety of applications for an extended period of time and thus, are often used as a reference when

other types of technology are evaluated. They cover the two ends of the spectrum for specific power

and specific energy profiles as there are currently no energy devices that exhibit both high specific

power and high specific energy.

Figure 1.1 - Ragone plot2

An electrochemical capacitor is a device that bridges the properties between traditional

capacitors and batteries, both in its performance and the underlying physics to store energy – it is

categorized as a capacitor because (1) its electrical characteristics resembles traditional capacitors

rather than the batteries, (2) it allows charges of equal and opposite polarity to accumulate on either

side of an electrode/electrolyte interface (a dipole) rather than allowing this dipole layer to be

produced by means of electron-transfer chemical reactions across the electrode/electrolyte interface.

Because electrochemical capacitors have a significantly higher capacitance value compared to

[1]

2 A Ragone plot is the most widely used for performance comparison of energy storage devices.

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conventional capacitors, they are often referred to as supercapacitors or ultracapacitors. This high

capacitance translates to moderate specific energy with high specific power, which no other existing

technologies have. As the result, the electrochemical capacitor is often conceived as a competing

technology that supplements or in some cases, can replace batteries, and its performance is most

often compared with that of batteries. In simple words, electrochemical capacitors exhibit few orders

of magnitude higher specific power and lower specific energy (around 10 per cent) than batteries.

Therefore, they can supplement batteries in applications with high peak power demand or can

replace batteries in the cases where unnecessarily large batteries are used to provide short electric

pulses.

The most prominent characteristic of an electrochemical capacitor that distinguishes it from

other types of capacitors is its much higher capacitance (up to thousands of Farads). This can be

easily explained by comparing the construction of different types of capacitors, as depicted in Figure

1.2 and by the simple equation for capacitance,

ACd

ε= (1-1)

, where ε is permittivity, A is the area of the electrode, and d is the separation distance

between the layers of charges of opposing polarity. The key differences between a conventional

capacitor and a supercapacitor, as shown in Figure 1.2, are much higher surface area and smaller

charge separation distance on the scale of atomic dimensions. The electrochemical capacitors have a

much higher surface area since they utilize a porous electrode structure. The charge separation

distance, on the other hand, is much smaller than other capacitors because it employs a mechanism

called the electric double layer. The electric double layer refers to a formation of two layers of

opposite charge at the interface between the electrode and the electrolyte; one in the electrode

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(electrons or holes) and the other in the electrolyte (cations or anions), when the electrode is

immersed in an electrolyte solution. This double layer has a separation distance in the nanometer

range, which is much smaller than that of other types of capacitors where the separation distance

conventionally refers to the distance between the two electrodes. The combination of orders of

magnitude higher surface area and smaller charge separation distance gives rise to the much higher

value of capacitance for electrochemical capacitors.

Figure 1.2 - Comparison with conventional capacitors

Unlike an electrostatic capacitor in which the two current collector plates are separated by a

dielectric material to increase the energy stored, electrolytic capacitors employ an electric double

layer across a thin insulating oxide film at the interface which is electrochemically deposited on the

surface of the anode. This metal oxide (i.e. aluminum oxide, niobium oxide, tantalum pentoxide),

however, is much thicker (several hundreds of angstroms) than the thickness of the double layer in

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electrochemical capacitors (about 3 angstroms), which contributes to the lower capacitance per unit

area for the electrolytic capacitors, in contrast to supercapacitors [2]. Although the electrolyte in the

electrolytic capacitors gives rise to its high capacitance by acting as a reagent for the oxide that forms

on the anode and forming an intimate contact with the oxide and the cathode, the electrolyte's high

resistance (relative to metal) is the main source of power loss within the device.

The simplified equivalent circuit model for various capacitor technologies as shown in Fig.

1.2 illustrate the origins of losses. Electrolytic capacitors have two primary loss mechanisms

represented by resistances; a series resistance due to the resistance of the electrolyte and a parallel

resistance due to the leakage and polarization loss of the thin oxide layer. The simplified model is

more complicated for the electrochemical capacitor; the resistors on the left and the right ends

account for the resistance of the carbon electrodes, the parallel capacitor-resistor configuration

describes the electric double layer capacitance and the potential-dependent resistance (the potential

dependent resistance accounts for transfer of charge across an interface due to a chemical reaction at

the interface), and the resistor in the middle comes from the combined resistance of the electrolyte

and the separator3

In addition to the high capacitance, electrochemical capacitors have higher efficiency and

longer lifetime compared to the competing technologies as shown in Figure 1.3. The main reason

for the higher efficiency and longer lifetime originates from the high electrolyte conductivity and the

electrochemical capacitors' mechanism for storing charges, respectively. Therefore, the properties of

.

3 All electrochemical capacitors have their electrolyte solution soaked in porous separator except for the electrochemical capacitors with a solid electrolyte, which are still in the research stage and are not commercially available. Separators, which are porous, prevent the two electrodes from shorting while allowing ions in the electrolyte to move around as freely as possible. Thus, the series resistance value is influenced by combined properties of the electrolyte and separator, and the respective impact of each depends on their volume fractions.

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the electrochemical capacitor remain virtually unchanged as long as the temperature is maintained

constant since chemical activity depends on temperature.

Efficiency and lifetime are critical parameters especially for large scale applications where

the replacement cost is high. However, some precautions must be taken to achieve high efficiency

and long lifetime. For example, the maximum input/output current the device can pass is governed

by the pore network structure of the electrode.

An attempt to push the current level beyond the maximum rate that charges can move

in/out of the pores of the electrode will cause a decrease in the efficiency of the device. This can be

explained with the following example. Consider a pore with large internal volume and a small

opening. The charges in the pore are confined and only a small number of ionic charges can flow

through the opening at a time hence the constriction generated by the pore makes the ionic charges

within a pore relatively immobile. Consequently, the effective conductivity of the electrolyte within

the pore channels is much lower than the conductivity in the bulk electrolyte and thus additional

electrical losses are contributed by the electrolyte motion within pores. At higher currents, the

additional resistive heating, if applied for longer periods of time, will promote localized heating

which leads to unwanted chemical reactions that occur at the electrode/electrolyte interface.

Certain types of loading profiles will generate sufficient internal heat to cause

decomposition of the solvents in the electrolyte solution resulting in gas [3]. This chemical process

leads to a dramatic reduction in the lifetime (acceleration factor for degradation is about 2 for a

temperature increase of 10 °C [4]). The most significant characteristic of the degradation is the

increase of the equivalent series resistance (by a factor of 1.5 – 5 while a 10 % decrease in

capacitance is observed [5]). This implies that the thermal degradation of the electrolyte consumes

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charge carriers through irreversible chemical reactions, which accelerates further heating of the

device through higher resistive heating (in effect, a thermal runaway process).

Figure 1.3 - Efficiency and lifetime properties of energy storage devices [1]

Future efforts to improving lifetime should focus on the following activities: (1) explore

ways of choosing materials to improve chemical stability, (2) develop models that give information

on internal temperature and its impact on lifetime, and (3) develop models to predict lifetime under

different loading profiles. However, these three activities face a common problem; they require a

dynamic model which couples the electrical and thermal fields in a rigorous fashion. This model

does not currently exist.

A complete physical understanding of an electrochemical capacitors requires an

interdisciplinary effort: materials science for characterizing the material properties, physical and

quantum chemistry for quantifying the reaction mechanisms and the ionic behaviour at the

molecular level near the electrode-electrolyte interface, semiconductor physics for characterizing the

electronic behaviour of the porous semiconducting electrodes, thermodynamics for understanding

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the influence of the elevated temperature, and heat transfer for determining the temperature

distribution in the capacitors. However, not all the required information is available for the

development of a suitable dynamic model. For example, ionic behaviour near the interface at the

atomic and molecular level and how ion transport occurs inside small pores of the size comparable

to the size of the ions are poorly understood. The absence of specific information as well as the

interdisciplinary nature of this problem is a major reason why no good dynamic electro-thermal

models for electrochemical capacitors exist currently.

Even one simple charge-discharge cycle of an electrochemical capacitor requires knowledge

regarding the influence of device geometry, electronic and ionic contributions to electric potential,

and chemical reactions as a function of temperature and electric field. In order to analyze and

characterize such a device, it is necessary to start with a simple model of the system and then

increase the model complexity in stages.

The modeling process consists of two stages: (1) development of a dynamic electrical model

that properly incorporates the impact of temperature and charge distribution assuming the

temperature is held constant, (2) development of a dynamic thermal model that can be used to

predict temperature distribution of the capacitor assuming the existence of an appropriate electrical

model. A full dynamic model is one that couples the electrical and thermal models. The dynamic

model should demonstrate the nonlinear behaviour of an electrochemical capacitor as a function of

voltage level (state of charge), temperature, and charge/discharge rate.

At this point in time, the electrical model of an electrochemical capacitor is not well

understood and the existing models fail to capture the necessary physics even under simplifying

assumptions where tests are performed at a constant temperature. Porous structures is not well

understood hence it is prudent to begin a study based on a primitive electrochemical cell model

9

where the electrodes are planar and chemical reactions occurring at the electrode/electrolyte

interface are neglected (an ideal supercapacitor should not exhibit reactions at the

electrode/electrolyte interface).

1.1 Thesis Objective

The objective of this thesis is to make a contribution to the modeling of the electrochemical

capacitors by starting with a simplest system. This simple system will be used to determine what is

not understood and the areas that need to be investigated further. The model presented in this thesis

assumes a flat plate electrode structure with no temperature coupling and no chemical reactions

occurring across the electrode/electrolyte interface. The simplified model includes the fundamental

properties such as ion transport, nonlinear polarization of the electrolyte, and stoichiometric

reactions in the bulk electrolyte solution and near the interface with the electrode. Hence the impact

of each physical attribute can be investigated and this sets the basis for introducing additional details

such as the impact of geometry, temperature, and chemical reactions, for the development of a

dynamic electro-thermal model in the future.

1.2 Thesis Outline

Chapter 2 introduces the fundamentals of electrochemistry including a description of the

terminology, fundamental theories, and concepts (i.e. electrochemistry and governing equations).

Chapter 3 presents the general overview of available electrochemical capacitor technologies and the

review of existing models. Chapter 4 summarizes the steps taken to develop a one dimensional

electrostatic model of a primitive electrochemical cell constructed of stainless steel electrodes and a

sulfuric acid electrolyte. Chapter 5 provides the simulation results for each of the steps discussed in

Chapter 4 and compares them with the experimental results. An analysis of discrepancies between

the simulation and experimental results is provided. Chapter 6 concludes the thesis by discussing

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the limitations of the developed model and future steps that need to be taken to further develop the

dynamic electro-thermal model.

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Chapter 2 Introduction to Electrochemistry

One of the key characteristics of electrochemistry comes from the fact that it involves the

interaction between transport phenomena, thermodynamics, and kinetics. This, in fact, is a major

challenge for researchers. Due to the complex and highly interdisciplinary nature of the field, this

chapter will introduce basic terminology and fundamental concepts that are necessary for the

analysis of electrochemical capacitors prior to describing the problem.

2.1 Fundamental Concepts and Terminology

An electrochemical system contains at least two electrodes separated by an electrolyte. The

external electric circuit is connected to electrodes, and the electrolyte acts as a medium that bridges

the two electrodes via ionic conduction. A simple illustration of such system is shown in Figure 2.1.

Figure 2.1 - Simple structure of electrochemical system

In electrochemical systems, an electrode refers to a material in which electrons (and holes, for

the case of semiconductors) are the charge carriers. Electrodes can be made from a metal (i.e. Pt, Pb,

Au), a semiconductor (i.e. Si), or other material with good electronic conductivity such as conducting

polymers and carbon. An oxide film (i.e. RuO2, IrO2, Co3O4) is also used as a type of electrode

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material due to its enhanced capability of generating electrons at the interface with an electrolyte,

though electrochemical reactions.

An electrochemical reaction is a type of chemical reaction in which the transfer of electrons

between molecules takes place across an interfacial zone. The electrolyte refers to a material in which

ions are the charge carriers. An ion with a positive net charge is referred to as a cation and an ion with

a negative net charge is referred to as an anion. However, an electrolyte consists of neutral molecules

as well as ions. The term species is often used to refer to all molecules constituting the electrolyte

without limiting the discussion to the charged species, or ions.

In order to understand the electrochemical reaction process, one often uses the term

reduction/oxidation (redox) reaction, which is a synonym for electrochemical reaction. As the term

suggests, redox reaction has two parts: reduction reaction and oxidation reaction. In a general sense,

an oxidation reaction refers to a reaction that involves an increase in the oxidation state (i.e. loss of

electrons from the reactant) and a reduction reaction refers to a reaction that involves a decrease in

oxidation state (i.e. gain of electrons of the reactant). However, in a strict sense, not all changes in

the oxidation state leads to electron transfer, while all electron transfer leads to a change in oxidation

state. Therefore, it is preferred to use the term Faradaic redox reaction to distinguish between these

two cases, where Faradaic refers to the case where electron transfer between reacting species takes

place (non-Faradaic refers to the case where there is no electron transfer). Continuing with the

same analogy, an electrode where a reduction reaction takes place is called a cathode, while an

electrode where an oxidation reaction takes place is called an anode. However, a distinction must be

made between an electrochemical reaction and a chemical redox reaction. Electrochemical reactions

have reduction occurring at one electrode and the oxidation occurring at the other. The exception is

corrosion where one electrode acts as a cathode and an anode, which means oxidation and reduction

13

occur on the same electrode surface but displaced with respect to each other in a planar sense. In

contrast, chemical reactions have reduction and oxidation occurring at the same physical location.

Electrochemical reactions always take place at the interface between the electrode and electrolyte, a

zone where the electric fields are high enough to impact the reaction chemistry, while chemical

reactions are not limited to the reactions at the interface.

The chemical (and electrochemical) reactions are also categorized by their reversibility and

the reaction rates. For a simple reaction with two reactants and two products,

A B C D+ +

(2-1)

, the forward reaction with reactants A and B produces the products C and D. In a reversible

reaction, the mechanism in one direction is exactly the reverse of the mechanism in the other

direction. For the reaction in equation (2-1), the reverse reaction has reactants C and D to produce

products A and B. Although all reactions are reversible to some extent (i.e. there is always a finite

contribution of a reverse reaction), some reactions consume nearly all of the reactants to form

products and are considered as a complete reaction. This type of reaction is referred to as an

irreversible reaction.

On the other hand, the rate of reactions categorizes the chemical reactions as well: fast

reactions and slow reactions. They are relative terms and there is no reference reaction rate which

differentiates between fast and slow. A fast reaction refers to a reaction that responds quickly to an

external force so that there is no hysteresis (or a time delay to force or suppress a reaction), and a

slow reaction is the type of reaction that causes hysteresis in the system.

As electrode and electrolyte have different charge carriers, their conductivities are given

separate names - electronic conductivity for electrode and ionic conductivity for electrolyte. The separate

14

terms only serve to distinguish the conducting mechanisms but their units are identical, S/m or

S/cm. There, however, are some exceptions. Mixed conductors, for example, have both electrons

and ions as their charge carriers, but this type is not used for electrochemical capacitors and further

discussion is omitted. Conductivity, both electronic and ionic, only provides a macroscopic sense of

the material property without providing insight about the conducting mechanism of the medium.

The transport parameters such as mobility and diffusivity are used for more detailed

discussions on the material properties. Mobility is defined as an average drift velocity of charged

particles (i.e. ions and electrons) divided by the electric field strength assuming linearity is valid.

Diffusivity (or diffusion coefficient), on the other hand, is defined as a proportionality constant

which relates the flux of species to its concentration gradient. Both mobility and diffusivity are

functions of temperature, pressure, and local concentration. Their dependencies on each other and

all of the variables are not adequately understood especially when the system is operating under a

highly nonlinear condition (i.e. at the interface).

The most crucial region of all electrochemical systems is the interface region between the

electrode and the electrolyte. This region is where most of the energy is stored in both a chemical (i.e.

Faradaic reactions) and electrostatic (i.e. accumulation of charges) form and is therefore of great

interest to investigate in order to characterize any electrochemical system. However, many of the

rules and characteristics of the materials that govern the properties of the electrolyte in the bulk do

not hold, or are altered by a significant amount, in the interfacial region. One simple example of this

problem is adsorption. Adsorption refers to a phenomenon in which there is an increase in

concentration of dissolved substances near the interface due to the surface forces. This surface force

comes from chemical bond between the materials on other side of the interface (uncompleted bonds

of the last layer of atoms are filled with the atoms of the adjacent material).

15

Electrochemical capacitor technology, in fact, is built around two key interfacial phenomena:

electric (or electrochemical) double layer and pseudocapacitance. The electric double layer and the

pseudocapacitance refer to the charge accumulation layers near the interface and the electric field

dependent Faradaic reaction that creates charges of opposite polarity on each side of the interface,

respectively. One should note, however, that although distinctions are made between the two

phenomena, electrical double layer and pseudocapacitance are not physically separable in an actual

system. Therefore, the overall performance of any type of electrochemical capacitor comes from a

combined contribution of both electrical double layer and pseudocapacitance.

Electric Double Layer

The name, electrical double layer, comes from the historic understanding of the

phenomenon in which it was thought that there is one layer of charged particles on one side of the

interface (i.e. electrode) while another layer of oppositely charged particle on the other side (i.e.

electrolyte). This simple model has been modified by a number of scientists over the years and the

most widely accepted model, the Stern model, takes on a more complex form. A simple illustration

of the Stern model of an electric double layer is shown in Figure 2.2.

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Figure 2.2 - Stern model of electric double layer

In a general sense, the electric double layer can be thought of as the result of the

arrangement of species adopting compromised positions under the conflicting influence of two

materials, the electrode and the electrolyte. Therefore, the properties of the species in the close

vicinity of the interface region differ from that of the same species in the bulk. The electric double

layer can arise in the following ways: through the electron-transfer reactions at the boundary to form

the electric potential difference across the interfacial zone, charging the electrode using an external

source to promote an equal and opposite charge to develop on the electrolyte side of the boundary,

and an electrostatic4

4 The term ‘electrostatic’ is used to refer to all cases where there is no Faradaic reaction in which the net charge of each domain (i.e. electrode, electrolyte) is conserved. This accounts for the processes involving movements of charged species within a domain and association/dissociation of the species in an electrolyte (or generation/recombination of electrons and holes in case of an electrode).

formation of a double layer without electron-transfer reactions or an external

source. Therefore, it is difficult to categorize the electric double layer as being either Faradaic or

17

non-Faradaic as it depends on the types of materials for the electrode and the electrolyte as well as

other conditions such as temperature, input current density, electric field, etc.

There are a few types of electrolytes in solid or liquid form (aqueous, non-aqueous). Figure

2.2 represents the case where the electrolyte is an aqueous solution (solution in which the solvent is

water) since it is the most widely used and also forms the basis of the interface analysis. For

simplicity, the discussion regarding the role of ions within the double layer will be limited to a

negatively charged electrode immersed in an aqueous electrolyte solution. The first molecular layer

(usually referred to as a monolayer) is mainly composed of water due to its abundance and strong

influence of the electric dipoles of water molecules (water molecules are neutral and the electric

dipoles allow them to get attracted to the charged surface). Even a very slight potential difference

between electrode and electrolyte causes electrostatic interactions among constituting species within

the electrolyte. Water molecules reorient themselves (to match the net charge of the electrode) and

become the first ones to be attracted to the surface to form the first molecular layer. However, ions

can also inhabit the monolayer due to chemical adsorption or electrostatic attraction similar to the

way the water dipoles are attracted to the surface and this phenomenon is called specific adsorption.

Before continuing the discussion on the roles of ions in a double layer, it should be noted

that all ions in an aqueous system are in their solvated form, which means that the ions have water

molecules surrounding them due to the electrostatic force between the ions (its net charge) and the

dipolar structure of the solvent, water. The level of solvation is dictated by the size and net charge of

ions; ions with smaller size and strong net charge attract more water molecules around them than the

ones with larger size and weak net charge. Cations are usually much smaller than anions and thus, are

more strongly solvated than anions. For example, a proton, which is a hydrogen atom without an

electron, is one of the most common cations in an aqueous solution and is the ion having the

18

smallest size. Therefore, more strongly solvated cations have less of a chance to get specifically

adsorbed due to the cancelling out of electrostatic attraction and physical interference by layers of

water molecules surrounding them. One may assume that a negatively charged electrode would only

attract cations and positive ends of water dipoles. However, one often finds anions in the monolayer

of negatively charged electrodes. This happens because the chemical attractive force is greater than

the electrostatic repulsive force. The locus of such specifically adsorbed ions is called the Inner

Helmholtz Plane (IHP). In fact, the case where cations are specifically adsorbed to a positively charged

electrode has not been observed [6]. It should be noted that if the electrode is made sufficiently

negative so that the electrostatic repulsive force can overcome the chemical attractive force, the

specifically adsorbed ions would be driven off the monolayer.

The region or layer further out from the IHP is occupied by electrostatically attracted

cations in their solvated form. The electrostatic force is not usually strong enough to push aside the

water dipole adjacent to the surface and thus, the locus of the first layer of cations, or Outer Helmholtz

Plane (OHP), is further away from the surface.

The thickness of both the IHP and the OHP are not constant; it is dictated by the

concentration of electrolyte, types of cations and anions, electric field, temperature, and electrode

surface. The concentration of cations gradually decreases as one gets further away from the OHP to

a bulk concentration value and this transition region is called the diffuse layer. As mentioned above,

the species near the boundary is under the conflicting influence of the properties of the electrode

and the electrolyte. The net influence of the electrode decreases as the distance from the interface

becomes larger (in the direction toward the bulk electrolyte) and at a sufficiently large distance from

the electrode/electrolyte interface, the concentration of cations and other characteristics become

that of the bulk electrolyte.

19

Pseudocapacitance

Another concept of importance, especially for electrochemical capacitors, is

pseudocapacitance. As the name suggests, it describes a phenomenon in which its electrical

characteristics is synonymous with a capacitor (i.e. I ∝ dV/dt). However, the origin of the

pseudocapacitance differs from the origin of the double layer capacitance. The pseudocapacitance is

Faradaic in nature, involving the passage of charges across the double layer. It also differs from the

Faradaic redox reactions for the case of batteries, where (ideally) constant electrode potential arises.

Pseudocapacitance is the result of a Faradaic charge transfer process in which the passage of

charges depends on potential difference across the interfacial zone. There are only a few cases where

the pseudocapacitance arises. Conway [2] described the following examples of pseudocapacitance: (1)

two-dimensional deposition of adatom5 arrays on the electrode surface, (2) a redox process with the

electrode potential as a function of the log of the ratio of reductant converted to oxidant (or vice-

versa), and (3) a chemisorption6

Pseudocapacitance has a greater potential for generating a higher energy density than the

electrical double layer. The double layer capacitance depends solely on a large surface area for the

high energy density, and the physical limit of the energy density is determined by the maximum

of anions at electrode interfaces with a potential-dependent partial

Faradaic charge transfer. Regardless of the origins of the pseudocapacitance, the mechanism which

leads to its existence is a fast and highly-reversible Faradaic process that behaves as a capacitance due

to the dependence of the rate of the process on the electric field at the interface.

5 The term used in surface chemistry to describe an atom that sits on a crystal surface that is often considered as the opposite of vacancy. The word is a contraction of adsorbed atom. 6 A type of adsorption where the driving force is strong interactions between the particles and surface such as chemical bonds of an ionic or covalent nature.

20

effective surface area7

2.2 Key Areas for Electrochemical Analysis

. On the other hand, pseudocapacitance which arises from a fast and highly

reversible Faradaic reaction has much more potential to increase the energy density. Conway made a

comparison in his book [7] that double layer capacitance and pseudocapacitance can store 0.17 –

0.20 electrons and 2.5 electrons per atom of accessible surface, respectively. This leads to capacitance

for the pseudocapacitance phenomenon being 10 to 100 times larger than that of the double layer

capacitance, assuming the same weight and volume.

The previous section provided an overview of electrochemical systems in general. However,

an analysis of such systems requires an understanding of specific areas of electrochemistry. This

section highlights the key areas for fundamental electrochemical analysis.

2.1.1 Transport Phenomena

The following three types of transport phenomena exist in an electrochemical system:

migration, diffusion and convection. It is important to understand each type because all of the

electrochemical systems involve more than one type of transport process; the species move in

response to the electric field (migration), concentration gradients (diffusion), and the motion of the

bulk fluid (convection). Each type of transport mechanism is discussed separately below, but it

should be noted that one type of transport often influences the others (i.e. migration under the

presence of an electric field can cause concentration gradients).

7 The term effective surface area is used to emphasize the limiting condition of the maximum surface area. Under some extreme conditions, such as pores smaller than the size of ions and the case where the structure is very porous to a point where a large number of ions in pores could cause a “traffic jam” to prevent fast availability of ions, a further increase in surface area does not increase the capacitance value.

21

Migration

In the presence of the electric field, charged species (ions) experience a force exerted on

them causing them to move. The direction of the ions movement depends on their polarity; anions

move in the direction of the electric field and cations move in a direction opposing the electric field.

The velocity at which the ions travel in the presence of the electric field depends on the field

strength and the net charge of ions and is given by

,i migration i iv z u F= − ∇Φ (2-2)

, where iz is the charge number of species i (net charge of the species as multiples of the charge

of a single electron), F is Faraday's constant (96485.3399 C/mol), Φ is the electric potential in

volts, and iu is the mobility of species i in m2· mol/J· sec. The mobility is a proportionality

factor that relates the velocity of ions to the electric field. The flux density of species i due to

migration, ,i migrationN , can be calculated by multiplying its concentration by the velocity and is given

by

,i migration i i iN z u Fc= − ∇Φ (2-3)

, where ic is the concentration of species i in mol/m3.

Diffusion

In the case where the concentration of the species is not uniform, the concentration

gradient acts as a driving force to make the species move by the process of diffusion. The cause of

the non-uniform concentration can vary. Introducing solutes in solvent (i.e. dissolve salt in water) as

well as migration in the presence of an electric field can be reasons for a difference in concentration

at different points in space. The diffusion acts to make the concentration uniform and the flux

22

density due to diffusion, ,i diffusionN , is defined as

,i diffusion i iN D c= − ∇ (2-4)

, where iD is the diffusion coefficient of species i in m2/sec.

Convection

Some electrochemical devices such as fuel cells experience the bulk movement of a fluid

called convection. The flux density of species for convection depends on its concentration and the

bulk velocity as given by

,i convection iN c v= . (2-5)

Net Transport of Species

The net flux density of a species is the combination of the three transport mechanisms

described in equations (2-3) to (2-5):

i i i i i i iN z u Fc D c c v= − ∇Φ − ∇ +

. (2-6)

For the case of the electrochemical capacitors, the system does not involve any bulk motion of the

fluid and thus, migration and diffusion are the two types of transport that need to be considered.

2.1.2 Thermodynamics

A thermodynamic measure of the stored energy in electrochemistry takes a combination of

both a chemical and electrical form, referred to as the electrochemical potential, µ . In a simplest

description, it is the sum of electrostatic potential and chemical potential as shown in equation (2-7).

i i iz Fµ µ= + Φ (2-7)

23

, where iµ is the chemical potential of species i , iz is the charge number of species i , F is

Faraday's constant, and Φ is the electrostatic potential in volts. Another commonly used form to

measure the energy is the Gibbs free energy, which is a measure of the thermodynamic potential (i.e.

amount of work that can be extracted from a system) in an isothermal (constant temperature) and

isobaric (constant pressure) system. It was mentioned in a previous section that every

electrochemical system undergoes a reaction (i.e. pseudocapacitance for electrochemical capacitors).

For the case of a two electrode system, there are two half-cell reactions, one at each electrode.

These half-cell reactions, presuming that a reaction occurs at the interface, will result in a change in

energy and it can be measured by the change in Gibbs free energy, G∆ , given by

1 2

i i i ii ielectrode electrode

G s sµ µ ∆ = − ∑ ∑ (2-8)

, where G is the Gibbs free energy, iµ is the electrochemical potential of species i , and is is the

stoichiometric coefficient of species i . The stoichiometric coefficient is a measure of degree to

which a species is participating in a reaction. For example, the anodic reaction (oxidation) in an

oxygen-hydrogen fuel cell

22 4 4H H e+ −→ + (2-9)

has 2

2Hs = and 4Hs + = − , with the convention of writing electrochemical reactions in the form

izi ii

s M ne−→←∑ (2-10)

, where n is the number of electrons transferred in the reaction [8].

In equation (2-8), the polarity of G∆ shows the spontaneity of the reaction in a defined

arbitrary direction; electrode 1 and 2 are defined as positive and negative, respectively. Following the

same convention, negative G∆ means that the reaction in a positive direction requires negative

24

energy and thus, the reaction is spontaneous. On the other hand, if G∆ is positive, a minimum

energy of G∆ is required for the reaction to take place. This, in fact, defines the distinction

between the two categories of electrochemical cells: galvanic cells and electrolytic cells. Galvanic cells

have a negative G∆ in which the work is produced spontaneously through the reaction and

electrolytic cells have a positive G∆ in which work is needed to drive the reaction. The most

familiar case for galvanic and electrolytic types of cells is the discharging and charging of batteries,

respectively.

Let us consider a case where we apply an electric potential difference across an

electrochemical system. The condition of the reaction described by equation (2-8) is now altered

due to the introduction of a bias potential difference8

equiE

. Depending on the polarity and magnitude of

the applied potential difference, it can stop the spontaneous reactions, force the reaction to occur in

a non-spontaneous system, or change the direction and rate of the reactions. In a case where the

magnitude of the applied potential difference is set to a point where there is no net current flow

through the spontaneous system (i.e. galvanic), the applied potential difference gets a special name,

an equilibrium potential, 9

equiG nFE∆ = −

. The equilibrium potential matches the net electrochemical potential

of the cell and is related to the Gibbs energy by

. (2-11)

The equilibrium potential is thus a function of the nature of the electrode and electrolyte properties

and the temperature. It should be noted that the equilibrium potential, in spite of its name, is the

8 Electrochemical potential is composed of both a chemical and electric potential. An applied electric potential difference will alter the electrochemical potentials at electrodes 1 and 2, which forces G∆ to differ from the original value. 9 In an electrolytic cell, the equilibrium potential is the magnitude of the applied potential difference where the reaction just begins to start. This definition is based on an electrolyte with infinite conductivity, so that the voltage drop due to a concentration potential and ohmic resistance is equal to zero.

25

potential difference relative to a reference point (i.e. hydrogen electrode is a universal reference

electrode for the potential difference measurement) and should not be confused with an absolute

potential. Even in the case where there is no net current in the system in equilibrium, there still exist

reactions caused by random collisions among species due to their thermal movements [8]. However,

the rate of forward and reverse direction of reaction at equilibrium is matched and thus, there is no

net flow of charges.

2.1.3 Kinetics

As mentioned before, a finite amount of current flows in both directions through the

interface even under equilibrium conditions. There is a special term given for the current density

under conditions of an equilibrium potential: the exchange current density, 0i . However, in the

presence of the applied potential difference, the forward and reverse reaction rates change and as a

result, a net current flows though the system. A proportion of the potential applied across the

interface that forces the electrochemical reaction to occur is referred to as the surface overpotential,

sη . The surface overpotential can be considered as the measure of deviation of the potential

difference of the electrode from its equilibrium potential [9], which sets a reference potential

difference at which no net reaction occurs, and is given by

s electrode equiE Eη = − . (2-12)

One of the most fundamental equations in electrochemistry that relates the rate of reaction

to the surface overpotential is referred to as the Butler-Volmer equation. It takes the form in which

the reactions in both directions, positive (anodic) and negative (cathodic), are considered.

0 exp expa cs sF Fi i

RT RTα αη η

= − − (2-13)

26

, where i is the current density, R is the universal gas constant, T is the absolute temperature in

Kelvin, and aα and cα are referred to as the apparent transfer coefficients. This equation applies

at both electrode interfaces. The system often exhibits a non-symmetrical response to the polarity of

the applied potential and favours one direction of reaction over the other. The apparent transfer

coefficients are the terms that scale the sensitivities in both directions and usually range between 0.2

and 2 [8]. A few things can be noted from the Butler-Volmer equation: (1) a positive surface

overpotential produces a positive current, (2) the system with high exchange current density

responds much more sensitively to the surface overpotential, and (3) either the cathodic or the

anodic current vanish exponentially as the surface overpotential becomes larger and thus only one of

the terms plays a role in the overall current density at higher values of surface overpotential.

Tafel later made an observation that the system responds exponentially to the large value of

sη and one of the terms in the Bultler-Volmer equation is negligible, and the overall current density

is given by either

0 exp a sFi i

RTα η =

for a sF RTα η >> (2-14)

or

0 exp c sFi i

RTα η = − −

for c sF RTα η

27

Gauss’s law for displacement fields given by

fD ρ∇⋅ = (2-16)

, where D is the electric displacement field in C/m2 and fρ is the free charge density in C/m3.

The electric displacement field is related to the electric field intensity, E , by

0D E Pε= + (2-17)

, where 0ε is the vacuum permittivity (8.8541878 F/m) and P is the polarization density in C/m2.

The electric field intensity is related to the electric potential by

E = −∇Φ (2-18)

and substituting equation (2-18) into (2-17) and (2-16) will result in

0 0( ) ( ) fD E P Pε ε ρ∇ ⋅ = ∇ ⋅ + = ∇ ⋅ − ∇Φ + = . (2-19)

The equation (2-19) relates the electric potential to the free charge density. The free charge

density in an electrochemical system is a distribution of ions in the electrolyte or the electron (or

hole) distributions in the electrode. However, attention must be given to the units. The

concentration of ions in the electrolyte is often represented in mol/L and is not in SI units as the left

hand side is. Therefore, the free charge density in the electrochemical systems is defined as

( )f i iF z cρ = ∑ (2-20)

, where F is Faraday’s constant in C/mol to convert the concentrations in mol/m3 to C/m3. iz

and ic are the charge number and the concentration of the species i , respectively. Finally,

substituting equation (2-20) into (2-19) gives Poisson’s equation

28

( )0( ) i iP F z cε∇ ⋅ − ∇Φ + = ∑ (2-21)

, and this equation takes a form where ic is in mol/m3, Φ is in V, and P is in C/m2.

For a special case where the material is lossless, homogeneous, and isotropic, the electric

field intensity and the electric displacement field are related to the permittivity of the material as

follows

( )0 0 0 01 rD E P E E E Eε ε χ ε χ ε ε= + = + = + = (2-22)

, where χ is the electric susceptibility and rε is a the relative permittivity. The resulting Poisson’s

equation for a lossless, linear isotropic type of material is

( )0( )r i iF z cε ε−∇ ⋅ ∇Φ = ∑ . (2-23)

29

Chapter 3 Overview of Electrochemical Capacitors

The fundamental concepts and terminologies of electrochemistry were reviewed in Chapter

2. In this chapter, the existing types of electrochemical capacitors and the construction details of the

most common types of capacitors are introduced. The last part of the chapter is devoted to

reviewing the existing models of electrochemical capacitors such as the types of models, their

limitations and the underlying assumptions used in developing the models.

3.1 Electrochemical Capacitor Technologies

Electrochemical capacitors are generally categorized according to the technology used for

storing energy. The two main electrochemical capacitor technologies are the Electrical Double Layer

Capacitor (EDLC) and the pseudocapacitor. As the names suggest, EDLC and pseudocapacitor

employ electrical double layer and pseudocapacitance, respectively, and the fundamentals of both

were explained in the previous chapter.

The main difference between the two types of supercapacitors is the type of electrode being

used; EDLCs use various types of carbon (i.e. activated carbon, carbon nanotubes, nanofiber) and

pseudocapacitors use metal oxides or conducting polymers. The main advantage of the

pseudocapacitors comes from the higher capacitance per unit size and volume as a result of fast and

highly reversible Faradaic reactions, which occur at the electrode/electrolyte interface. This allows

more electrons to be stored in a surface atom. However, the pseudocapacitors have the following

disadvantages: electrode material is expensive, lifetime is shorter due to the origins of

pseudocapacitance which is described next. Although the Faradaic reactions of the pseudocapacitors

are highly reversible, they are not indefinitely repeatable like the (ideal) electrical double layer [10].

30

Perhaps it is for these reasons why the pseudocapacitor is not widely available in the market and why

it remains a research topic.

Although a distinction is made with respect to the origins of the capacitance, both EDLCs

and pseudocapacitors exhibit their capacitance as the combination of both electrical double layer and

the pseudocapacitance. The carbon materials in EDLCs have, in very small amounts, Faradaic

reactivity of surface oxygen-functionalities. The reactivity depends on the preparation method of the

carbon material, but around 1 to 5 per cent of the total capacitance comes from pseudocapacitance.

On the other hand, pseudocapacitors have around 5 to 10 per cent of the total capacitance as a

double layer capacitance as the result of the formation of a double layer across the accessible

interfacial areas [2]. Furthermore, the pseudocapacitance and the double layer capacitance are

coupled to each other and the analysis of both types of capacitance is necessary for the development

of an adequate model, regardless of the type of electrochemical capacitor to be investigated.

Electrochemical capacitors are also classified by the types of electrolyte used: aqueous and

non-aqueous. The aqueous electrolyte has the main advantage of low electric resistance because of

its small and mobile ions. It is a critical parameter especially for the high power electrochemical

capacitors since a small increase in resistance can lead to significant power loss and thus, elevates the

internal temperature which promotes undesirable, non-reversible chemical reactions that are fatal for

the device. However, aqueous based electrochemical capacitors suffer from a low operation voltage

due to the small electrochemical stability window of water molecules. In other words, water

molecules decompose into hydrogen and oxygen gas at relatively low voltage levels (this voltage

refers to the total voltage across both double layers of the device) referred to as the hydrogen

evolution overpotential and oxygen evolution overpotential, respectively. The voltage window for

the ELDCs that employ an aqueous electrolyte is about 1.4 V because it is influenced primarily by

31

the decomposition potential of water (1.23 V at 298 K) and other kinetics factors [2]. This low

decomposition voltage is the main drawback of the aqueous electrolyte. Aqueous and non-aqueous

electrolytes are compared in Table 3.1.

Aqueous Non-aqueous Inexpensive Expensive Small, polar solvent10 Large, non-polar organic solvent Many small ions per unit surface area Few large ions per unit surface area Ions are mobile → low ionic resistance Ions are not mobile → high ionic resistance Low voltage operation window High voltage operation window Corrosive, but not explosive Toxic, explosive

Table 3.1 - Comparison of aqueous and non-aqueous electrolytes

3.2 Construction of Electrochemical Capacitors

The basic electrochemical capacitor consists of two porous electrodes attached to current

collectors, and separated by an ion permeable separator soaked in electrolyte. While the current

collector’s main purpose is to collect current from the electrodes with a minimum resistance (thus

highly conductive material is used), the electrode, separator and electrolyte constitute the core of the

electrochemical capacitor. The separator can be thought of as a paper with many tiny holes. Its main

goal is to prevent the two electrodes from touching each other (to avoid an electrical short) while

allowing ions to flow freely (through the holes). The electrolyte, on the other hand, is usually in

liquid form and contains charged ions that move back and forth during a charge/discharge cycle to

ionically conduct current. The electrodes are porous so their effective surface area is very large.

Figure 3.1 illustrates the construction of a common EDLC. The most widely used type of

electrode is activated carbon because of its very high surface area to weight and volume ratio, high

10 Solutes dissolve much better in solvents of similar molecular types. Therefore, aqueous electrolytes have small and polar ions while non-aqueous electrolytes have large and non-polar ions. Smaller and more polar ions result in a larger number of charges per unit area, which results in a higher capacitance.

32

conductivity and low cost. However, other types of electrodes, such as carbon nanotubes, have

been suggested due to their potential to have higher surface area and better performance such as a

faster transfer of ions through a nanotube because of the unimpeded path (straight columns).

Figure 3.1 – A schematic of electrical double layer capacitor

3.3 Review of Existing Models

A number of models have been proposed to describe the nonlinear behavior of

electrochemical capacitors, and the approach taken in developing each model differs from the others.

Due to the difference in approaches and underlying assumptions, the different models exhibit

different limitations.

This section categorizes the existing models according to the following methods of

characterization employed in the literature: empirical, dissipative transmission line, and dynamic

models. Every model discussed in this section shares one commonality; the time evolution of the

temperature and its effects on the coefficient values in the underlying equations are neglected. This

results in a loss of accuracy of the model after an extended period of operating time. In summary,

there are no models that are widely accepted by the industry to date due to the limitations and the

non-universality of the models.

33

3.1.1 Empirical Models

One of the first empirical models that captured the long term variation of capacitance was

developed by Zubieta. The electrical and thermal equivalent circuit models of the electrochemical

capacitor were developed separately in 1997 and 1999, respectively. The models do not capture the

underlying physics, but the equivalent circuits were developed intuitively to match what is observed

during the supercapacitor’s operation. The empirical dynamic electrical and empirical static thermal

equivalent circuit models are illustrated in Figure 3.2.

Figure 3.2 - Equivalent electrical (left) and thermal (right) circuit by Zubieta [11, 12]

The empirical dynamic electrical model in Figure 3.2 (left) is often referred to as a 3 branch

model. The branch on the left is an immediate branch with a time constant in the order of a few

seconds, the middle branch is a delayed branch with a time constant in the range of minutes and the

branch on the right is a long term branch with a time constant larger than ten minutes. This model

assumes that the immediate branch capacitance is voltage dependent and the long-term branch only

accounts for 30 minutes at the most. This model has major limitations in that it cannot capture the

voltage dependence as well as the electrical behaviour (i.e. voltage versus current) over a period of

several hours or days. The empirical static thermal model in Figure 3.2 (right) was also developed

intuitively based on the physical construction of the capacitor. The thermal model accounted for the

34

power dissipation within the connectors, case and the electrolyte using three heat sources (two at the

terminal and one inside the device) and paths for heat dissipation (one through the case and a path

through each connector). The model assumes only the static thermal response and not the dynamic

thermal performance.

Based on the Zubieta model, some improvements were made by Rafik et al. [13] in 2007 to

obtain better agreement between simulations and experiments for the long term temporal voltage

versus current behaviour of the supercapacitor. The approach taken was a form of curve-fitting

and the introduction of circuit components with empirical justifications. The model is depicted in

Figure 3.3. Circuit 1 is introduced to account for the temperature dependence of the electrolyte

resistance in the low frequency range, and circuit 2 is introduced to increase the precision of the

model over a 1 to 10 Hz frequency range. Circuit 3 is the slight modification of the Zubieta model,

where leakage current and the internal charge distribution are accounted for. This model does not

account for the thermal coupling in a rigorous way and was developed by making experimental

measurements at different ambient temperatures using a controlled climatic chamber.

Figure 3.3 - Electro-thermal equivalent circuit model by Rafik [13]

35

3.1.2 Dissipative Transmission Line Model

In 1963, de Levie first treated the capacitance in porous electrodes as a dissipative

transmission line [14]. The model treats each pore as a dissipative transmission line and models a

distributed double layer capacitance and distributed electrolyte resistance that extends into the depth

of the pore. This model assumes a straight, cylindrical pore of uniform diameter and a perfectly

conducting electrode to estimate the double layer capacitive effects. Two embodiments of the

dissipative transmission line model are shown in Figure 3.4.

Figure 3.4 - Transmission line equivalent electrical circuit model [15, 16]

3.1.3 Dynamic Models

There are a number of existing models developed on the basis of the underlying physics of

the electrochemical capacitors. Many groups have taken different approaches to predict the

behaviour of electrochemical capacitors and there are a few groups [17-21] who focused on the

fundamental behaviour of the electrical double layer and the pseudocapacitance. However, the

existing models base their calculations on the assumed values for the parameters such as the

36

exchange current density, diffusion coefficients, and dielectric constant. The values were chosen

using approximations so that the models provide simulation results in a close agreement with the

experimental measurements of the electrochemical capacitors. The coefficient values they assumed,

however, are functions of the changing environmental conditions such as the temperature, the local

concentration of species and the electric field. Therefore, the models provide the approximate

predictions using the averaged parameter values (i.e. the arbitrarily chosen values lie somewhere in

between two extreme values obtained from experimental tests). This limits the further inclusion of

other parameters (i.e. heat generation, temporal evolution of the device voltage versus current under

different loading profiles, etc.) and cannot be used to predict the ageing process or to couple the

thermal effects.

There also are some cases where the models are oversimplified and the useful information

about the molecular and ionic behaviour is lost. Some models grouped the ionic species into two

categories, cations and anions, in the equations and calculated the diffusion coefficient and mobility

of the two groups. Another example is the use of an arbitrary built-in capacitance value to represent

the capacitance associated with the electrical double layer. These types of simplifications prevent the

further coupling of the local concentration of species to electrochemical reactions and the

thermodynamics as well as the understanding of such phenomena at the molecular level. There are

no models that successfully capture the essential properties of the physics that can be used to couple

parameters such as the temperature to predict the long term behaviour of the device.

37

Chapter 4 Model Development

A dynamic model exhibits cross-coupling between many variables and multiple feedback

loops within the equations describing the physical system. Therefore, the development of the model

and the analysis thereof become very complex and challenging. The approach taken is to make

simplifying assumptions in the initial model and then adding in complexities in a step-by-step

manner. This process leads to an improved understanding of which physical mechanisms are

important to model and what aspects of the model need refinement.

This chapter outlines the formulation of each step taken to develop the dynamic model

assuming constant temperature operation and no temperature coupling. Temperature coupling can

be incorporated in a rigorous fashion in the future. The system under study consists of two flat

stainless steel plates separated by a sulfuric acid electrolyte solution. No chemical reactions are

presumed to occur at the electrolyte/electrode interface.

The flat electrode surface is assumed throughout the analysis to neglect geometric details

deviating from a planar geometry. The premature introduction of geometric effects will complicate

the analysis of the system considering that the details of the simplest possible structure, that is the

one that we are considering, is still poorly understood. The movement of all species (charged or

neutral) are assumed to move in a straight line between the plates and fringing effects at the plate

boundaries can be ignored (plate separation distance is much smaller than the plate dimensions).

Hence the model is in effect a one-dimensional problem. The system as a whole can be modeled

using a series of coupled partial differential equations.

38

Te system of partial differential equations is solved using COMSOL, a multiphysics

commercial finite element software package. Most of the energy in electrochemical capacitors is

stored in close vicinity of the interface therefore the size of the mesh close to the interface should be

in the sub-micrometre range. The FEM software requires the size of the mesh to be much smaller

than the sub-micrometer range (i.e. sub-nanometre range) in order to capture the changes in this

region.

4.1 Characterization of the Electrode

The main criterion for the electrode materials in electrochemical capacitors is the high

conductivity for minimum ohmic resistance. The resistance value of the electrode in electrochemical

capacitors is orders of magnitude lower than that of the electrolyte since electrons in a good

conductor are more mobile than ions in an electrolyte. As the ohmic resistance is negligible and

electrochemical reactions are ignored, the electrode is assumed to be a perfect conductor (i.e. the

electric field in the conductor is equal to zero and all charges reside within a finite distance of the

surface; the Thomas-Fermi screening distance).

The geometry and external excitation conditions for the electrochemical capacitor studied in

this thesis are shown in Fig. 4.1. The separator is omitted in the diagram and in the analysis since its

impact on the results can be neglected to first order.

39

Figure 4.1 – Flat structure of electrochemical capacitor

The equations to be developed in this section should relate the input parameters (i.e. input

current) to the electrostatic potential. The equation for the electrode on the right hand side is

grounded and thus, the relationship is

0groundedΦ = (4-1)

The relationship between the input current, inI , and the electrostatic potential is developed by first

expressing the input current as a function of the input current density, inJ , with the following

relationship

ininIJA

= (4-2)

, where A is the surface area of the electrode.

We now consider Gauss’s law for displacement fields and Ampere’s law from Maxwell’s

equations

(Gauss’s law) fD ρ∇⋅ = (4-3)

40

(Ampere’s law) fDH Jt

∂∇× = +

∂ (4-4)

, where D is the electric displacement field in C/m2, fρ is the free charge density (not including

bound charges) in C/m3, H is the magnetizing field in A/m, and fJ is the free current density

(not including bound current) in A/ m2.

According to the vector calculus identity that the divergence of a curl of any vector field is

always zero implies that the divergence of Ampere’s Law, equation (4-4), becomes

( )

0

f

f

DH Jt

DJt

∂∇ ⋅ ∇× = ∇⋅ + ∂

∂= ∇ ⋅ +∇ ⋅ =

∂

. (4-5)

Substituting equation (4-3) into equation (4-5) results in

( ) ff DJ Dt t tρ∂∂ ∂

∇ ⋅ = −∇ ⋅ = − ∇ ⋅ = −∂ ∂ ∂

. (4-6)

Consider the Gaussian surface around the electrode as shown in Figure 4.2.

41

Figure 4.2 - Gaussian surface around perfect conductor

Assuming that there is no generation or recombination of charges inside the Gaussian surface (i.e.

no electrochemical reaction or recombination of electrons and holes), equation (4-6) can be

integrated across the Gaussian volume as follows

f fv v

J dv dvt

ρ∂∇ ⋅ = −∂∫ ∫ . (4-7)

Applying the divergence theorem, equation (4-7) becomes

f f fv S v

J dv J dS dvt

ρ∂∇ ⋅ = ⋅ = −∂∫ ∫ ∫ . (4-8)

Since there are no vertical components of the current passing through the top and the bottom side

of the Gaussian surface, then

2 1f fS v

J dS J A J A dvt

ρ∂⋅ = − = −∂∫ ∫ . (4-9)

Because there is no generation or recombination of charges at the interface, the injected current, 1J ,

42

is accumulated at the interface (on the electrode side of the interface) and 2J becomes zero.

1

2 0

inJ J

J

=

=