mathematical modelling of primary alkaline...

Mathematical Modelling of

Primary Alkaline Batteries

Jonathan JohansenBachelor of Applied Science (Honours)Queensland University of Technology

A thesis submitted in partial fulfilment of the requirements for the degree ofDoctor of Philosophy

March 2007

Principal Supervisor: Dr Troy W. FarrellAssociate Supervisor: Professor Ian W. Turner

Queensland University of TechnologySchool of Mathematical Sciences

Faculty of ScienceBrisbane, Queensland, 4001, AUSTRALIA

c© Copyright by Jonathan Johansen 2007

All Rights Reserved

ii

I dedicate this thesis to Tamara.

Keywords

advection, anode, asymptotic analysis, BET surface area, binary electrolyte, boundary

condition, Butler-Volmer equation, cathode, closed circuit voltage, concentration polar-

isation, control volume, current path, discretisation, diffusion, electrochemical reaction,

electrode, electrolytic manganese dioxide, EMD crystals, EMD particles, exchange cur-

rent density, geometric surface area, initial condition, linearisation, macrohomogeneous

porous electrode theory, mathematical model, Nernst equation, ohmic losses, open

circuit voltage, ordinary differential equation, overpotential, partial differential equa-

tion, perturbation techniques, potassium hydroxide, potassium zincate, precipitation

reaction, primary battery, separator paper, simulation, step potential electrochemical

spectroscopy, ternary electrolyte, theoretical capacity, utilisation, zinc, zinc oxide

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Abstract

Three mathematical models, two of primary alkaline battery cathode discharge, and

one of primary alkaline battery discharge, are developed, presented, solved and investi-

gated in this thesis. The primary aim of this work is to improve our understanding of

the complex, interrelated and nonlinear processes that occur within primary alkaline

batteries during discharge.

We use perturbation techniques and Laplace transforms to analyse and simplify an

existing model of primary alkaline battery cathode under galvanostatic discharge. The

process highlights key phenomena, and removes those phenomena that have very little

effect on discharge from the model. We find that electrolyte variation within Elec-

trolytic Manganese Dioxide (EMD) particles is negligible, but proton diffusion within

EMD crystals is important. The simplification process results in a significant reduction

in the number of model equations, and greatly decreases the computational overhead

of the numerical simulation software. In addition, the model results based on this

simplified framework compare well with available experimental data.

The second model of the primary alkaline battery cathode discharge simulates step po-

tential electrochemical spectroscopy discharges, and is used to improve our understand-

ing of the multi-reaction nature of the reduction of EMD. We find that a single-reaction

vii

framework is able to simulate multi-reaction behaviour through the use of a nonlinear

ion-ion interaction term.

The third model simulates the full primary alkaline battery system, and accounts for

the precipitation of zinc oxide within the separator (and other regions), and subsequent

internal short circuit through this phase. It was found that an internal short circuit is

created at the beginning of discharge, and this self-discharge may be exacerbated by

discharging the cell intermittently. We find that using a thicker separator paper is a

very effective way of minimising self-discharge behaviour.

The equations describing the three models are solved numerically in MATLAB R©, using

three pieces of numerical simulation software. They provide a flexible and powerful

set of primary alkaline battery discharge prediction tools, that leverage the simplified

model framework, allowing them to be easily run on a desktop PC.

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Acknowledgements

I would like to thank Troy for his continued support and guidance throughout this

project. He has enabled me to step up to the challenge of completing a doctorate in

the philosophy of applied mathematics. Many hours of involved discussion have given

me invaluable insight into the mind of a very skilled researcher.

I would like to acknowledge the support of Delta EMD Australia Pty. Limited over the

course of this work.

I would like to thank my fiance Tamara for keeping me focused, and for her love during

this time. Tamara helped motivate me when I thought I would never make it, and she

is showing me how much can be done with a life or two. Thank you for your vision.

I would like to thank my family and friends for always being there. I would not have

made it this far without them.

Finally, I would like to thank the ‘person’ behind it all, our Creator. He gives my

life meaning and hope more than anyone else, in a way that makes others pale in

comparison. God’s love, manifested in His son Jesus, is fundamental to my existence.

Thank you.

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Contents

Keywords v

Abstract vii

Acknowledgements ix

1 Introduction 1

1.1 Primary Alkaline Batteries . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Aims, Objectives and Outcomes . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Background on Primary Alkaline Batteries 7

2.1 The Operation of the Primary Alkaline Battery System . . . . . . . . . 7

2.2 The Cathode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 The Cathodic Reaction Mechanism . . . . . . . . . . . . . . . . . 10

2.2.2 The Cathodic Zero Current Potential . . . . . . . . . . . . . . . 15

2.3 The Separator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 The Anode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 The Anodic Reaction Mechanism . . . . . . . . . . . . . . . . . . 19

2.5 The Electrolyte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Modes of Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7 Previous Primary Alkaline Bettery Modelling . . . . . . . . . . . . . . . 22

3 The Simplified Model 33

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.1 The Full Cathode Model . . . . . . . . . . . . . . . . . . . . . . . 34

xi

3.2.2 The Simplified Cathode Model . . . . . . . . . . . . . . . . . . . 42

3.3 The Numerical Solution of the Simplified Equations . . . . . . . . . . . 46

3.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 The Potentiostatic Model 59

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59


4.3 The Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4 Determining the Ion-ion Interaction Term . . . . . . . . . . . . . . . . . 66

4.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5.1 The Initial Exchange Current Density . . . . . . . . . . . . . . . 75

4.5.2 The Diffusion Coefficient of Protons . . . . . . . . . . . . . . . . 77

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 The Precipitation Model 81

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81


5.2.1 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2.2 The Cathode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2.3 The Separator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2.4 The Anode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2.5 Transference Numbers . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3 The Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4.1 The Effect of Changing ZnO Bulk Conductivity . . . . . . . . . . 103

5.4.2 The Effect of Changing Initial KOH Concentration . . . . . . . . 105

5.4.3 The Effect of Changing Separator Thickness . . . . . . . . . . . . 106

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Conclusion 111

6.1 Summary and Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

A Derivation of the Butler-Volmer Like Expression 119

B Laplace Transform Simplification of Crystal Scale Equations 123

C Perturbation Analysis of Electrolyte Conservation Equations 127

C.1 Particle Scale Simplifications . . . . . . . . . . . . . . . . . . . . . . . . 127

C.2 Cathode Scale Considerations . . . . . . . . . . . . . . . . . . . . . . . . 128

List of Symbols 131

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Bibliography 135

xiii

List of Figures

2.1 Cutaway view of a primary alkaline battery . . . . . . . . . . . . . . . . 8

2.2 Schematic diagram of the current path in an alkaline battery . . . . . . 9

3.1 Schematic diagram of the three scales within a cathode . . . . . . . . . 35

3.2 Schematic diagram of the control-volumes in one dimension . . . . . . . 48

3.3 Comparison of simplified model output with experimental data . . . . . 51

3.4 Comparison of experimental data and simplified model results for differ-

ent EMD conductivities . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5 Discharge results for simplified model using either an asymptotic or

Laplace transform crystal scale solution . . . . . . . . . . . . . . . . . . 55

3.6 Mn4+ ion concentration distributions at various times and discharge rates 56

4.1 The potential experienced by a cell during a SPECS discharge . . . . . . 60

4.2 The minimum and maximum power in each potential step of a typical

EMD SPECS discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Comparison of a simulated 5 mV/hr SPECS discharge using a linear

ion-ion interaction term, with experimental data . . . . . . . . . . . . . 68

4.4 Two possible representations of the ion-ion interaction term . . . . . . . 69

4.5 Two 5 mV/hr SPECS simulation results based on different ion-ion in-

teraction terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.6 A polynomial representation of Υ based on “fitting” the Butler-Volmer

equation to experimental 5 mV/hr SPECS discharge data . . . . . . . . 71

4.7 Minimum and maximum power of simulated 5 mV/hr SPECS discharge

using experimentally determined ion-ion interaction term . . . . . . . . 72

4.8 A non-linear form of the ion-ion interaction term used to produce the

simulated SPECS discharge in Figure 4.9 compared to a linear approxi-

mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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4.9 Minimum and maximum power of simulated 5 mV/hr SPECS discharge

using the form of the ion-ion interaction term given in Figure 4.8 . . . . 74

4.10 The effect of i00 on individual current spike shape . . . . . . . . . . . . . 76

4.11 The effect of i00 on the overall SPECS discharges . . . . . . . . . . . . . 77

4.12 The effect of DH+ on individual current spike shape . . . . . . . . . . . 78

4.13 The effect of DH+ on the overall SPECS discharges . . . . . . . . . . . . 79

5.1 Comparison of self-discharge of two AA-cells under 3.3 and 6.6 Ω loads . 101

5.2 ZnO volume fraction within the separator of an AA-cell under a 3.3 Ω

load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.3 External and total current for an intermittent 3.3 Ω discharge simulation 103

5.4 Comparison of model output for different ZnO conductivities . . . . . . 104

5.5 Comparison of precipitation model output for different initial KOH con-

centrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.6 Comparison of precipitation model output for different separator thick-

nesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

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List of Tables

3.1 Dimensionless variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Dimensionless constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Parameter values used in the simplified model simulations . . . . . . . . 43

3.4 Discharge parameters and cell geometry used in the simplified model

simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5 Discharge parameters and cell geometry for AA-cell cathode geometry . 56

4.1 Additional governing equations used in the potentiostatic model . . . . 65

4.2 Additional boundary and initial conditions used in the potentiostatic

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Discharge parameters and cell geometry used in the potentiostatic model

simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.1 Parameter values and cell geometry used to simulate AA-cell discharge . 100

xvii

Statement of Original Authorship

The work contained in this thesis has not been previously submitted for a

degree or diploma at any other higher education institution. To the best of

my knowledge and belief, the thesis contains no material previously published

or written by another person except where due reference is made.

Signature:Jonathan Johansen

Date:

xix

CHAPTER 1

Introduction

1.1 Primary Alkaline Batteries

The primary alkaline battery is a widely used consumer product, which is essential for

powering many portable devices, such as power tools, cameras, radios, torches, toys and

remote controls. With such a wide and pervasive use, the global market surrounding

batteries is worth around £25 billion, and most this is from the sales of alkaline batteries

(Lewis 2005). The primary alkaline battery owes its pervasiveness to several factors.

These include its total energy content, performance and shelf life. In addition, it

contains non-toxic components. These types of batteries are sold commercially in AAA,

AA, A, C and D sizes, depending on the type of portable device and its power demands.

The primary alkaline battery system has been improved since it was invented in 1901

by Thomas Edison. Since first entering the market in the early 1960’s (Scarr & Hunter

1995), battery design has been continuously improved by companies such as Duracell,

Energizer, and Rayovac. Many improvements have been made, resulting in a 60%

increase in the energy output of alkaline batteries since their introduction (Scarr &

Hunter 1995).

2 Chapter 1. Introduction

The alkaline battery is a complex system and its behaviour depends on many phys-

ical, chemical and electrochemical effects that interact in a nonlinear, interconnected

way. The interactions within the primary alkaline battery system can mean that some

outcomes are counter-intuitive. To facilitate battery design, analytical tools such as

mathematical modelling can be used to improve the understanding of the complex

interdependent processes that occur within batteries.

Whilst most mathematical models do not account for all effects or phenomena within

a given system, well targeted models can be used to test and extend explanations of

the key mechanisms that dictate system performance. When such a model correctly

predicts experimental behaviour in several different well validated cases, it may then

be used to investigate other cases where experimental data is unknown. In this way, a

mathematical model can provide theories and insight into the way a system operates.

1.2 Aims, Objectives and Outcomes

The primary aim of this thesis is to use mathematical models of the primary alkaline

battery system to add to the understanding of how the complex, interrelated and

nonlinear processes that define battery discharge, interact. In view of this aim, we

pursue the following three main objectives in this thesis.

Objective 1: To evaluate the importance of the phenomena accounted for in existing

models of cathodic discharge. The phenomena of lower importance will be systemat-

ically eliminated, in an attempt to obtain a simplified model of cathodic discharge.

This process highlights key phenomena that significantly effect discharge. The sim-

plified model will then be solved and validated against experimental data. This will

provide the mathematical framework that the second and third objectives will be based

on.

Objective 2: To improve the treatment of the multi-reaction behaviour of the reduction of

EMD within primary alkaline battery cathode models. The approach of using a single

reaction to describe this process will be evaluated. In particular, we investigate the

possibility of using a single-reaction framework to describe and simulate multi-reaction

1.2 Aims, Objectives and Outcomes 3

phenomena. A model based on this assumption will then be developed, solved and

validated against experimental data.

Objective 3: To investigate the role zinc oxide (ZnO) may play in cell failure. ZnO is

a semiconductor, and there is evidence that precipitation of this material within the

separator paper of a primary alkaline battery may provide a conduction pathway for

internal short circuit, which significantly shortens battery life. A model accounting for

this phenomenon will be developed and solved. Methods of preventing cell failure due

to this mechanism will be investigated using this model.

There are several significant outcomes that should result from the successful completion

of the three objectives stated above. These are as follows.

We will obtain three novel models of primary alkaline battery discharge. The first

will be simpler than its predecessor, whilst still retaining essential cathodic discharge

behaviour. As part of the development of this model, our understanding of the processes

that occur within the primary alkaline battery cathode, and our knowledge of their

importance, will be improved. The second model of primary alkaline battery discharge

will contain an improved and novel description of the multi-reaction nature of the

reduction of EMD. This will also improve our understanding of the effects of the multi-

reaction nature of the reduction of EMD on discharge. The third model will account

for the separator paper and the anode, including internal short circuit through the ZnO

phase in the separator paper. Using this model, we will obtain a wider understanding

of the effects of ZnO precipitation, and subsequent internal short circuit across the

separator paper.

Further to this, a set of major outcomes are the three pieces of numerical simulation

software that provide solutions to the mathematical models corresponding to each of the

three modelling tasks outlined in the above objectives. The simplified model simulation

software will be smaller and more stable than its predecessor, taking less time to arrive

at a solution. The second and third pieces of numerical simulation software will use

the simplified model framework and retain the computational benefits, while providing

extended predictive capabilities. The three pieces of numerical simulation software will

be suitable for use on a desktop PC. This is essential in terms of implementing the

4 Chapter 1. Introduction

models in an industrial environment.

1.3 Thesis Outline

This thesis comprises six distinct chapters, an outline of which is as follows.

Chapter 2 provides the necessary background information to interpret the work pre-

sented herein. The internal structure of a primary alkaline battery is explained and the

individual components are described. The mechanism of discharge is discussed, along

with a short explanation of the several modes in which a battery can be discharged.

Chapter 3 documents the work done in relation to the first objective. The previous

equations of the three-scale model of Farrell, Please, McElwain & Swinkels (2000) are

presented. Using asymptotic techniques, the importance of the phenomena accounted

for within the cathode are analysed. Based on this analysis, the model equations are

simplified. Following the simplification, the numerical solution technique used to solve

the new system of equations is described. The outcomes of the simulations are validated

against experimental data. A journal article (Johansen, Farrell & Please 2006) based

on this research has been published in the Journal of Power Sources.

Chapter 4 documents the work done in relation to the second objective. A model

for simulating the potentiostatic discharge of primary alkaline battery cathodes is pre-

sented, including the framework to include an approximation of the multi-reaction

behaviour of the reduction of EMD. The numerical solution technique employed for the

simulations is outlined, including the process of finding an improved approximation of

the multi-reaction behaviour of the reduction of EMD. The results are validated against

experimental data, and the effect of varying some key parameters is investigated.

Chapter 5 documents the work done in relation to the third objective. In this chapter,

a mathematical model for simulating battery discharge, accounting for failure due to

precipitation of ZnO and internal shorting is developed. Following this, the numer-

ical solution technique used for the simulations is outlined. Using the software that

implements the numerical solution technique, several case studies are investigated and

presented. The effect of several key parameters on battery performance and failure is

1.3 Thesis Outline 5

also investigated.

At the time of submission of this thesis, two further journal articles containing the work

given in Chapters 4 and 5, respectively, are under preparation for submission to the

Journal of Power Sources.

Chapter 6 contains a summary of the results of Chapters 3 to 5, and draws conclusions

from the work. Additionally, the aims, objectives and outcomes are reviewed, and

possiblities for further work are discussed.

CHAPTER 2

Background on Primary Alkaline Batteries

In this chapter we present background information that is helpful in understanding the

modelling work and results that are presented in the following chapters. This chapter is

divided into two halves. The first half focusses on the components of a primary alkaline

battery from a modelling perspective, and how a primary alkaline battery operates. The

second half focusses on how the primary alkaline battery has previously been modelled

in the literature.

2.1 The Operation of the Primary Alkaline Battery Sys-

tem

An alkaline battery is effectively two paired electrochemical cells. They work in se-

ries as an electron pump, absorbing electrons in the positive electrode (cathode), and

producing them at the negative electrode (anode). These electrons travel from the

negative electrode to the positive electrode through an extermal circuit. The energy to

do this comes from the stored chemical energy in the anodic and cathodic materials of

the battery.

8 Chapter 2. Background on Primary Alkaline Batteries

Figure 2.1: A Cutaway view of a primary alkaline battery (Farrell 1998).

A schematic diagram of an alkaline battery is shown in Figure 2.1. The outside of the

battery is the cathodic current collector, which is typically a steel can or container.

This container holds the cathode, separator paper, anode, anodic current collector and

a plastic grommet. The cathode is located just inside the can wall, and is in electrical

contact with it. It is composed of graphite particles and electrochemically prepared

manganese dioxide (EMD) particles that have been mixed together and compressed

into porous annular rings (Williams, Fredlein, Lawrance, Swinkels & Ward 1994). The

graphite is added to improve the electrical conductivity of the EMD (Scarr & Hunter

1995). The separator paper is found between the anode and cathode and is usually made

of a non-woven fibre. This paper acts as an insulator, because its primary function is

to prevent electrical contact between the anode and cathode internally while allowing

liquid electrolyte to pass between them (Scarr & Hunter 1995). The anode is at the

centre of the battery. It is made of powdered zinc in a gel, and is typically 55-70%

zinc by weight (Scarr & Hunter 1995). At the centre of the zinc anode is the anodic

current collector, or nail. The nail is attached to the negative cap at the bottom of

the cell. Throughout the cell, the void space in the pores is filled with potassium

hydroxide electrolyte (KOH). A plastic grommet is included in the cell, because during

discharge, some of the contents of the cell increase in volume, and the plastic grommet

accommodates this. The plastic grommet also insulates the cathodic current collector

from the negative cap to prevent short circuiting.

halla

This figure is not available online. Please consult the hardcopy thesis available from the QUT Library

2.2 The Cathode 9

Figure 2.2: Schematic diagram of the electronic current path in an alkaline battery.

The path of electronic current within an alkaline battery is shown schematically in

Figure 2.2. During discharge, electrons at the cathodic current collector enter the

cathode and travel to the EMD/KOH interface. It is here that the cathodic reaction,

discussed in Section 2.2.1, takes place. This results in the production of hydroxide

ions (OH−) in solution. As this occurs, OH− ions are consumed at the zinc surface in

the anode, via the anodic reaction, which is discussed in Section 2.4.1. This process

produces electrons in the anode, which make their way to the anodic current collector

to complete the circuit.

2.2 The Cathode

As mentioned in Section 2.1, the cathode contains graphite and EMD, and the void

space within and between these components is filled with KOH electrolyte. Naturally

occuring manganese dioxide behaves very poorly in alkaline batteries, and to make it

suitable for use in batteries, it is electrochemically prepared (Scarr & Hunter 1995).

During the electrodeposition of EMD, temperature, electrolyte composition and depo-

sition current density all influence the chemical, electrochemical and physical properties

to some degree (Williams et al. 1994). In addition, EMD deposits in a microporous

crystal structure, which has been comprehensively characterised by Chabre & Pannetier


(1995). Unfortunately, EMD is not a good conductor, and it’s conductivity decreases

as it is reduced. Xi, Li & Chen (1989) found that within the oxidation range from

MnO1.95 to MnO1.6, the conductivity of manganese oxide decreases from 10−2 S.cm−1

to 10−7 S.cm−1. To minimise the ohmic losses due to this decrease in conductivity,

EMD is crushed into small particles, mixed with graphite and compressed into pellets.

This improves the conductivity of the electrical contact between the EMD and the

graphite, however, graphite takes up valuable space that could otherwise be used for

more EMD, thereby increasing the theoretical capacity of the battery, a balance must

be made between total EMD content and solid phase ohmic losses.

A second effect of crushing the EMD is that it now has two different types of porosity.

One is the original porosity of the EMD (now in the form of particles), and the second

is due to the spaces between the graphite and the particles (but not within them). The

porosity of the EMD particles gives rise to a very large electrochemically active surface

area, which facilitates the electrochemical reaction.

2.2.1 The Cathodic Reaction Mechanism

The cathodic reaction mechanism has been the focus of much investigation in the

literature. The process is complex and its characterisation has proved to be a lengthy

process.

The cathodic reaction mechanism in concentrated (7.6M) KOH electrolyte was studied

by Cahoon & Korver (1959). They found that the cell potential decreased continuously

during discharge, and proposed that EMD (MnO2) is reduced to Mn(OH)2 and Mn4O7,

which is followed by a reduction of Mn4O7 to Mn(OH)2 and Mn3O4, and then in a third

stage, Mn3O4 is reduced to Mn(OH)2.

Vetter (1963) developed a set of expressions to predict the OCV for the homogeneous

reduction of non-stoichiometric metal oxides. A homogeneous process means that there

is only one phase, and no new phases are formed by the reactants and products. In

contrast to this, a heterogeneous process involves the formation of an additional solid

phase. Using the findings of Vetter, Bell & Huber (1964) studied the reduction of

manganese dioxide in 9.0M KOH, and, like Cahoon & Korver (1959), proposed a three

2.2 The Cathode 11

stage process. The first stage was a homogeneous reduction of MnO2 to MnO1.7. In

the second stage, the MnO1.7 was heterogeneously reduced to MnO1.47, whilst the third

stage consisted of a heterogeneous reduction of MnO1.47 to MnO. The second and third

stages of the reduction of EMD, as proposed by Bell & Huber, are heterogeneous, which

is characterised by constant potentials over the corresponding parts of the discharge

(potential versus time) curve. In contrast, the homogeneous first stage of reduction is

characterised by a sloping potential, which indicates that the reactant forms a solid

solution with the product (Vetter 1963). In addition, Bell & Huber used X-ray diffrac-

tion to determine that there is no Mn(OH)2 at the beginning of any of the stages of

reduction, as previously suggested by Cahoon & Korver (1959). They also found that

MnOOH is formed in the first stage of discharge.

We note that Cahoon & Korver continuously discharged their EMD through a 30Ω

load, while Bell & Huber discharged their EMD through a 100Ω load for 100 hours per

week. Kozawa & Yeager (1965) noted that this could be the cause of their different

findings and proposed mechanisms. Specifically, Kozawa & Yeager suggested that Bell

& Huber obtain three stages for discharge because at very low discharge rates the

soluble manganese oxide may recrystallize and provide an extra stage of discharge.

Kozawa & Yeager (1965) studied the effect of varying the discharge rate and the KOH

electrolyte concentration on the reduction of EMD. They also studied the effect of

adding triethanolamine, (HOCH2CH2)3N, which increases the solubility of MnOOH in

the electrolyte. They found that the potential decreases continuously during the first

stage of discharge, which is characterised by the reduction of MnO2 to MnO1.5. This

is followed by a second stage with a relatively flat potential, which is characterised by

the reduction of MnO1.5 to MnO. The second stage is observed at low discharge rates,

high KOH concentration, and when (HOCH2CH2)3N is present. Thus, they conclude

that the first stage of the reduction of EMD is a homogeneous process, whereas the

second stage is a heterogeneous process. They propose that the first stage of reduction,

or first-electron reduction (MnO2 to MnOOH) is an electron-proton mechanism given

by the reaction,

MnO2 + H2O + e− MnOOH + OH−. (2.1)


This reaction specifies that an electron (from the cathodic current collector) and a

proton (formed from the decomposition of a H2O molecule) are introduced into the

MnO2 crystal structure to form MnOOH. For the second-electron reduction (MnOOH

to Mn(OH)2), they proposed the reactions

MnOOH Mn(III)(aq), (2.2)

Mn(III) + e− Mn(II), (2.3)

and

Mn(II)(aq) Mn(OH)2. (2.4)

We note that Kozawa & Yeager state that in usual discharges of primary alkaline

batteries, the second-electron reduction does not occur to an appreciable extent, and

thus, we will not discuss the second-electron reduction mechanism in any further detail

here.

Kozawa & Powers (1966) performed a similar study to that of Kozawa & Yeager. They

investigated the effect that different temperatures, using D2O instead of H2O in the

electrolyte solution, and adding zinc ions to the solution, have on the electron-proton

mechanism for the reduction of EMD as given by Reaction (2.1) above. The authors

found that by using D2O instead of H2O, a higher potential drop through the EMD was

formed. They hypothesise that because dueterium is twice as heavy as a single proton,

a larger potential drop would be experienced when it is incorporated into EMD. They

found the potential drop due to this is approximately 60 mV, which supports their

previous electron-proton mechanism. Further, using thermodynamic princpiples, the

authors propose that adding zinc ions would raise the potential of the reduction of

EMD if a new phase was formed (not an electron-proton mechanism). In accordance

with the electron-proton mechanism, they found that the prescence of zinc ions had no

influence on the first stage, but did influence the second stage of discharge, in which a

new phase is formed. In addition, the authors found that increasing the temperature

decreased the potential drop through the EMD. They hypothesise that this is because

protons can move more easily at higher temperatures, and thus, their finding supports

2.2 The Cathode 13

the electron-proton mechanism. Kozawa & Powers also proposed a relationship between

cell potential and the reduction of EMD. This is discussed in more detail in Section 2.2.2.

Since the work of Kozawa and co-workers, several authors have proposed modified

mechanisms of discharge. McBreen (1975) proposed that the first step given by Bell &

Huber (1964) ends at MnO1.5 or even MnO1.3, rather than MnO1.7. Swinkels, Anthony,

Fredericks & Osborn (1984) examined the reduction of many (>100) samples of EMD in

9M KOH. By inspecting the slope of the potential with respect to the amount of charge

passed, the authors found that contrary to previous findings, the reduction from MnO2

to MnO1.5 is composed of three distinct but overlapping reduction processes. They

then linked this to the existence of at least three sites in the crystal structure of EMD

with different equilibrium potentials. Swinkels et al. further investigated the existence

of different energy sites within the EMD by heat treating the EMD, whereby it was

thought that the higher energy configurations in the crystal structure would change

to a lower energy configuration. This change to a lower energy configuration can be

seen in their experimental data through a reduction in the OCV for the heat treated

material and in their plots of the slope of the potential with respect to the amount of

charge passed.

Zhang, Chen & Xi (1989) studied the reduction of EMD by examining the current

response as the cell potential of partially reduced EMD samples is forcibly held at a

slightly lower level. After simulating the diffusion of protons through different geometric

crystal configurations, they suggested the existence of three different insertion sites for

protons in the EMD. They also proposed that the three insertion sites are filled in

different ways throughout discharge.

Chabre (1991) examined the reduction of EMD, using step potential electrochemical

spectroscopy (SPECS), a method similar to the approach of Zhang et al. (1989). SPECS

discharges are explained in more detail in Chapter 4, Section 4.1. The key advantage

of SPECS discharges over galvanostatic and other continuous discharge modes is that

it does not measure overall performance, but measures the electrochemical and physi-

cal response of the cathode at discrete potential levels. The cell is usually allowed to

approach equilibrium between voltage steps, which has the added benefit of avoiding


over-reduction and resolving individual electrochemical processes. In Chabre’s accel-

erated communication, he examined SPECS discharges of EMD at different rates. He

found that there are two potential ranges of distinct electrochemical activity in the

data, and that at higher discharge rates the two processes tend to merge into one. The

author attributed this to the existence of two types of sites for proton insertion into

the EMD, with different equilibrium potentials.

Chabre & Pannetier (1995) used X-ray diffraction together with SPECS discharges to

characterise the reduction process. Through the use of X-ray diffraction, the authors

determined that γ-EMD can be understood as a form of EMD called ramsdellite with

two types of crystallographic defects. The first type of defect is De Wolff disorder

(De Wolff 1959). It corresponds to the extent of the appearance of pyrolusite, which

has the same chemical formula as ramsdellite, but has a different crystal structure. The

second type of defect is microtwinning, which is a measure of how much the crystal units

are rotated or mirror themselves. They found that the method of preparation of EMD

greatly influences the extent of microtwinning. Through characterising EMD samples

based on the extent of De Wolff disorder and microtwinning, the authors found a

dependence between the crystallographic defects and discharge behaviour. In addition,

they proposed a more detailed mechanism for the first electron reduction of EMD. The

first stage of the reduction mechanism is the reduction of sites at the surface of the

crystal structure. When comparing SPECS discharges involving chemically prepared

manganese dioxide (CMD) and EMD, Chabre & Pannetier found that the EMD showed

wider peaks and had more microtwinning, which suggested a link between the number

of surface sites and the amount of microtwinning. In the second stage of the reduction

mechanism, the ramsdellite crystal units of the EMD are reduced to groutite. During

this stage, the crystal structure forms a second phase in a process called Jahn-Teller

distortion. This occurs at different times, depending on how fast the EMD is being

reduced. Only the part of this stage that occurs before the Jahn-Teller distortion is

reversible. Furthermore, in concentrated alkaline electrolytes such as 9 M KOH, Chabre

& Pannetier noted that the second stage becomes more complex because the Mn3+ in

MnOOH has a high solubility in such solutions. Following the second stage, further

reduction occurs, but this is part of the second electron reduction of EMD, and does

2.2 The Cathode 15

not influence the majority of typical discharges.

Although the multi-reaction framework introduced by Chabre & Pannetier is probably

the most accurate in terms of describing the actual reduction mechanism of EMD in an

alkaline battery environment, the incorporation of such a framework in a mathematical

model would require kinetic parameter data which at this point in time is unknown.

As such, and in a manner that is consistent with previous authors (Farrell et al. 2000),

the single-reaction, first-electron, reduction mechanism proposed by Kozawa & Yeager

(1965) and represented by Reaction (2.1) is adopted. This mechanism provides a good

overall representation of the reduction of EMD and requires the least numer of kinetic

parameters to be determined. Howver, as many authors have attested (Atlung &

Jacobsen 1981, Maskell, Shaw & Tye 1982, Ruetschi 1988, Chabre & Pannetier 1995),

the equilibrium potential based on the Nernst equation (Atkins & de Paula 2006)

corresponding to Reaction (2.1) does not match the experimental data. The next

section provides information on the work that has gone into describing this relationship

and rectifying this anomoly.

2.2.2 The Cathodic Zero Current Potential

Even if the mechanism for the reduction of EMD is known, predicting the potential as

this occurs is very difficult. If we consider the electrochemical reaction,

αOx + ne− βRed. (2.5)

where α and β are stoichiometric coefficients, Ox and Red are the oxidant and reduc-

tant, respectively, and n is the number of electrons transfered in the reaction, then the

equilibrium potential of this reaction (or the potential at which zero current flows due

to this reaction) is given by the Nernst equation, namely,

E = E0 +RgasT

nFln

[Ox]α

[Red]β(2.6)

Here, E (V) is the equilibrium potential of the reaction, E0 (V) is the standard po-

tential for the reaction, Rgas (J.K−1.mol−1) is the universal gas constant, T (K) is the


temperature, F (C.mol−1) is Faraday’s constant, and [Ox] and [Red] are the concentra-

tions of the oxidant and reductant, respectively. The problem arising here, in relation

to the reduction of EMD, is that as we have seen in the previous section, it is most likely

that this reduction mechanism does not consist of a single reaction process, and thus, a

single Nernst equation of the form given by Equation (2.6) cannot adequately describe

the equilibrium potential of EMD reduction. Notwithstanding this, Equation (2.6) has

been adopted by several authors as a basis for describing the zero current potential for

the reduction of EMD.

Johnson & Vosburgh (1953) proposed an empirical relationship based on the Nernst

equation, namely,

E = E0 − 0.073 logCMn3+

CMn4+

. (2.7)

Here Ci (mol.cm−3) is the concentration of species i. While qualitatively correct, the

expression does not capture the discharge characteristics specific to the reduction of

EMD. It was only later that Kozawa & Powers (1966) suggested that the equilibrium

potential for the reduction of EMD is given by the Nernst equation corresponding to

Reaction (2.1), namely,

E = E0 − RgasT

Fln

CMn3+

CMn4+

. (2.8)

When Kozawa & Yeager compared this expression to experimental discharge curves,

it was qualitatively correct, but the relationship breaks down after the halfway point

(around oxidation state MnO1.75).

Atlung & Jacobsen (1981) study the reduction of EMD, and conclude that the equilib-

rium potential for the reduction process cannot be accounted for using a homogeneous

reaction mechanism. Using thermodynamics and statistical mechanics, they account

for an electronic term, and for a concentration dependent numbers of proton insertion

sites. They propose the zero current potential relationship,

E = E0 − RgasT

Fln

CMn4+

(

1 + βCMn4+

C0Mn4+

)β

(

C0Mn4+ − (1 + β)CMn4+

)1+β, (2.9)

2.2 The Cathode 17

where β is an emiprical parameter related to the concentration dependence of the

number of proton insertion sites, and C0Mn4+ (mol.cm−3) is the initial concentration of

Mn4+ ions.

Maskell et al. (1982) consider the process, and conclude that the electrons and protons

in the EMD crystal structure are independently mobile, in the same way that the ions of

a highly concentrated aqueous electrolyte may be independently mobile. To account for

this, they include the equivalent concentration of electrons and protons in the Nernst

expression. These may be related to the reduction state, and the expression simplifies

to,

E = E0 − 2RgasT

Fln

CMn3+

CMn4+

. (2.10)

While this relationship is closer to the experimental data, it still breaks down at oxi-

dation states lower than MnO1.75.

Ruetschi (1988) also examined the reduction of EMD, and proposed that there is a

change in the potential energy of the protons in the EMD as it is reduced. He thus

modifies the Nernst equation to account for change in the potential energy of protons

relative to unused EMD. The proposed relationship is

E = E0 +Naϕ

F

(

C0Mn4+ − CMn4+

)

− RgasT

Fln

CMn3+

CMn4+

, (2.11)

where Na is Avogadro’s number, and ϕ (J) is the difference in the energy of a proton

in fresh (MnO2) and reduced (MnOOH) EMD.

A further zero current potential relationship has been proposed by Xi et al. (1989).

They measure the semiconductor properties of EMD, and relate the Seebeck coefficient

to the equilibrium potential of the reduction of EMD. Their relationship is similar in

form to the Nernst equation, namely,

E =

(

EF (ref) − AkT

e+ BT

)

− CT lnCMn3+

CMn4+

, (2.12)

where EF (ref) (J) is the reference Fermi level, k (J.K−1) is Boltzmann’s constant, e (C)

is the charge of an electron, A is a dimensionless constant relating to the way electron

scattering occurs and B and C (V.K−1) are constants empirically relating the Seebeck


coefficient to the degree of reduction. The constant C is given two different values,

which it switches between at the point when CMn3+ = CMn4+ .

Chabre & Pannetier (1995) state that the effects of the various crystal structures and

defects in EMD have not been satisfactorially linked to the reduction of EMD and the

zero current potential. In addition to their comprehensive study of the structure of

several EMD’s, Chabre & Pannetier also propose the use of a modified Nernst equation

as an approximation of the equilibrium potential for the first electron reduction of

EMD, namely,

E = E0 − Υ (CMn4+) − RgasT

Fln

CMn3+

CMn4+

, (2.13)

where Υ (CMn4+) (V) is an unknown concentration dependent term, named the “ion-ion

interaction term”, that accounts for the difference in the equilibrium potential given by

the Nernst equation and that observed experimentally. Chabre & Pannetier do note,

however, that the reduction of EMD is non-reversible after a certain point, and that

this approach assumes that the process is reversible.

Given the difficulty in determining kinetic parameters for multiple reduction mech-

anisms, in this work we will adopt Equation (2.13) for the equilibrium potential of

Reaction (2.1). This approach is consistent with the work of Farrell et al. (2000) and

Farrell & Please (2005). Chabre & Pannetier note that a linear function of reduction

state enables good agreement with experimental data, but up to mid reduction only.

The choice of a more complicated ion-ion interaction function is dealt with in Chapter 4

of this thesis.

2.3 The Separator

The separator derives its name from its function, as its primary use is to electrically

separate the anode and cathode. This forces electrons produced in the anode to travel

through an external circuit before returning to the cathode.

Separators used in the primary alkaline battery system must be electronic insulators,

ionic conductors, and stable in a strong alkaline electrolyte, and strong and flexible

enough to maintain their other functions while being assembled, and during use (Scarr

2.4 The Anode 19

& Hunter 1995).

The separator must be permeable to ions and molecules, otherwise OH− ions would

build up in the cathode and be depleted in the anode, creating large potential losses

and greatly reducing the usable life of the cell. Thus, the separator is made of a non-

conducting and highly porous material that permits the flux of hydroxide ions from

the cathode to the anode and water molecules from the anode to the cathode, allowing

continuous discharge as long as there is active material remaining.

Typical separators have large porosities, and have thicknesses ranging between 0.1 and

0.2 mm.

2.4 The Anode

The anode is constructed from powdered zinc mixed with electrolyte and a gelling

agent (Scarr & Hunter 1995). In addition, several other additives may be present in

small quantities (0-4% by weight, Scarr & Hunter 1995) , including ZnO. Anodic zinc is

created by powdering, or atomizing, zinc in a stream of compressed air. The particles

of zinc produced this way are from 20 to 820 µm in diameter (Scarr & Hunter 1995) and

have an average surface area of approximately 0.02 m2/g. The gelling agent is added at

weight percentages from approximately 0.4 to 2% (Scarr & Hunter 1995). The addition

of a gelling agent makes it easier to process the zinc anode, and holds the zinc particles

in place, which prevents them settling at the bottom of the anode. Details of how we

have modelled the anode are given in Chapter 5 of this thesis.

2.4.1 The Anodic Reaction Mechanism

When considering the oxidation of the (pure) zinc anode in KOH electrolyte, several

reactions are assumed to proceed (Bockris, Nagy & Damjanovic 1972). When the

KOH electrolyte is saturated by the addition of ZnO, Butler (1964) and Boden, Wylie

& Spera (1971) found that the reaction for the dissolution of zinc is

Zn + 4OH− Zn(OH)2−4 + 2e−. (2.14)


Powers & Brieter (1969) found that the exchange current densities for a pure zinc or an

amalgamated zinc anode were similar, even though the crystal structures are different.

From this they concluded that the discharge mechanism is also similar.

In Reaction (2.14), solid zinc is consumed, which leads to a build up of Zn(OH)2−4 ions

in solution, in the form of K2Zn(OH)4. The removal of Zn(OH)2−4 from solution via

the precipitation of ZnO has been described by Dirkse (1971) and Dirkse, Vander Lugt

& Hampson (1971) as,

Zn(OH)2−4 ZnO + H2O + 2OH−. (2.15)

Together, Reactions (2.14) and (2.15) have been used as the anodic reaction mechanism

in several mathematical models of zinc anodes (Sunu 1978, Podlaha & Cheh 1994a,

Podlaha & Cheh 1994b). This dissolution-precipitation mechanism has been adopted

to describe the zinc anode modelled in Chapter 5.

2.5 The Electrolyte

The electrolyte in the primary alkaline battery system is concentrated (7-9M) aqueous

potassium hydroxide (KOH). This type of electrolyte has good conductivity, and fa-

cilitates the normal operation of the cell (Scarr & Hunter 1995). The liquid contains

K+, OH− and Zn(OH)2−4 ions and is effectively a ternary electrolyte. The conductivity

of KOH electrolyte has been measured by See & White (1997), and the conductivity

of KOH electrolyte saturated with K2Zn(OH)4 has been measured by Bennion (1964).

In Chapters 3 and 4 we use the conductivity measured by See & White (1997). In

Chapter 5, we extrapolate the conductivity of the electrolyte from KOH electrolyte to

include the effect of the Zn(OH)2−4 ions using the data of Bennion.

See & White (1997) measured the conductivity of binary KOH electrolyte at concentra-

tions of 15, 20, 25, 30, 35, 40 and 45% KOH by weight (approximately 3 - 11 mol.cm−3),

and for about 24 different temperatures from -15 to 100C. They fitted an empirical

expression to their data. In addition, they found that their data compared well with

Falk & Salkind (1969). However, the conductivity of concentrations lower than 15%

2.6 Modes of Discharge 21

or higher than 45% by weight was not measured, and so the correlation may not be

accurate outside that range.

2.6 Modes of Discharge

Batteries are discharged in a variety of ways, depending on the application the battery

is used for. There are four different modes of discharge that we consider in this work.

The following are examples of where each of the four discharge modes arise, and the

associated relationship between cell voltage and discharge current.

Constant current (galvanostatic) discharge is widely used when testing cells, and it is a

popular discharge mode in battery modelling work because it is simple to specify. The

equation used to specify a galvanostatic discharge is simply,

I = I(t), (2.16)

where I (A) is the current passing through the cell, and I(t) (A) is the desired galvano-

static discharge rate. I(t) is usually specified in terms of current per gram of EMD in

the cathode, which seeks to make it easier to relate the discharge rate with its demand

on the cell.

In step potential electrochemical spectroscopy (SPECS) tests (described in further de-

tail in Chapter 4, Section 4.1), batteries are discharged via a series of constant potential

(potentiostatic) discharges at successively lower voltages, whilst all the time the result-

ing current response is recorded. Ideally, the individual potentiostatic discharges run

long enough for the cell to approach equilibrium before the voltage is “stepped down”.

The equation used to describe potentiostatic discharge is,

φc|Rco− φa|Rai

= Ecell(t), (2.17)

where φc (V) and φa (V) are the potentials of the solid phases in the cathode and

anode, respectively, Rai and Rco (cm) are the radial distances from the center of the

cell to the anodic and cathodic current collectors, respectively, and Ecell(t) (V) is the

desired cell potential.


In “real world” consumer applications (such as the operation of portable power devices),

constant load and power discharges are more common. The equation used to describe

constant load discharge is,

φc|Rco− φa|Rai

I= Rload(t), (2.18)

where Rload(t) (Ω) is the desired load resistance. Furthermore, many high range

portable electronic devices require a source of constant power. The equation used

to describe constant power discharges is,

(

φc|Rco− φa|Rai

)

I = Pload(t), (2.19)

where Pload(t) (W) is the desired power. We note that the constant power discharge is

the most demanding mode of discharge for an alkaline battery since in order to maintain

a constant power at low cell voltages an increase in the current drain is required.

Using the above equations, intermittent discharges can be simulated by switching to

Equation (2.16) and setting the current to zero during rest periods.

2.7 Previous Primary Alkaline Bettery Modelling

In this section we present the modelling techniques and models that have been recorded

in the literature to date, to simulate primary alkaline battery systems.

A framework, known as Macrohomogeneous Porous Electrode Theory, has been estab-

lished by Newman and coworkers (Newman & Tobias 1962, Newman 1967, Newman &

Chapman 1973, Newman & Tiedemann 1975, Newman & Pollard 1979, Newman 1983,

Newman 1991) to simulate and model porous electrodes such as those found in the

primary alkaline battery system. One of the key strengths of this theory is that porous

structures are easily dealt with. All liquid phase quantities are averaged over the total

space, ignoring pore boundaries. Knowledge of the individual pore structure is not

needed, but is characterised by parameters such as total void fraction, tortuosity and

surface area per unit volume, which are substituted into the standard physical equa-

tions such as those describing mass conservation and current conservation. This greatly

2.7 Previous Primary Alkaline Bettery Modelling 23

simplifies model formulation and allows quicker simulations. Great care, however, has

been taken to make sure all phenomena (such as the diffusive flux and advection)

are correctly described. Macrohomogeneous porous electrode theory has been used to

formulate a number of models of primary alkaline battery discharge to date, includ-

ing those of Wruck (1984), Chen & Cheh (1993a), Chen & Cheh (1993b), Podlaha &

Cheh (1994a), Podlaha & Cheh (1994b), Zhang & Cheh (1999a), Farrell et al. (2000),

Kriegsmann & Cheh (2000), and Farrell & Please (2005).

The first alkaline battery cathode model to predict transient discharge behaviour was

that of Wruck (1984). Wruck presented a one-dimensional model for the galvanostatic

discharge of a planar (flat) alkaline battery. He modelled the use of active material

at a non-uniform rate but assumed that the electrolyte concentrations were uniform,

and infinitely dilute. In addition, a linearized Butler-Volmer equation (see for example,

Atkins & de Paula 2006) was used to determine the reaction rate for the first-electron

reduction of EMD.

In a series of papers Cheh and co-workers have significantly contributed to the area

of alkaline battery modelling. This began with the work of Mak, Cheh, Kesley &

Chalilpoyil (1991a). They presented a simple quasi-equilibrium model of the primary

alkaline battery based on the zero current potential expression of Maskell et al. (1982)

(see Equation (2.10)). The model was used to predict cell potential and discharge

current for low rate, constant load, current and power discharges. The results show a

linear relationship between discharge rate and service life (the time taken to reach a

specified cutoff voltage), and this trend was validated against experimental data of the

same form. The authors note that at higher discharge rates, non-equilibrium effects

must be taken into account.

In a second paper, Mak, Cheh, Kesley & Chalilpoyil (1991b) presented a secondary

current model of the primary alkaline battery. They used this model to examine the

extent to which ohmic and kinetic losses contribute to the cell potential. Their model

is applicable at the beginning of discharge, and is not applicable in the presence of

concentration gradients in the solid or solution phases. They used their model to predict

the initial reaction distribution and potential drop across the cathode. The effect of the


curvature of the cell and solid and solution phase conductivities was investigated. Mak

et al. found that larger cells have more non-uniform reaction distributions. In addition,

they found that when the conductivities of the solid and solution are decreased, the

reaction distribution shifts to the internal and external radii of the cathode, and that

increasing the curvature of a cathode decreases it’s potential losses.

Following the work on non-transient primary alkaline battery cathode behaviour, Chen

& Cheh (1993a) presented a one-dimensional model for the galvanostatic discharge of

a cylindrical primary alkaline battery. The model accounted for the anode, separator

and cathode. Importantly, the model also accounted for non-uniform concentration

gradients in the solution phase within the cathode. The electrolyte was assumed to

be concentrated binary KOH electrolyte in H2O, and the cell was modelled as a fixed

volume device, with no swelling or volume change. The cathodic reduction mechanism

was assumed to be that given by Kozawa & Powers (1966) for the first electron reduction

only (see Reaction (2.1)), and was modelled using a full Butler-Volmer equation to

describe the electrode kinetics. The cathode was assumed to consist of small solid

spheres of MnO2. A shrinking core type model was used to simulate the reduction

of the MnO2 spheres in which the layer of MnOOH that forms at the outside of the

spheres, as a result of the reduction process, increases the polarisation of the electrode.

The anode was assumed to be reversible and non-polarisable (totally uniform). In

addition, it was assumed that the anodic reaction mechanism was a mixture of two

reactions, namely,

Zn + 2OH− ZnO + H2O + 2e− (2.20)

and

Zn + 4OH− Zn(OH)2−4 + 2e−. (2.21)

A parameter was used to specify the relative extent of the above reactions with the

discharge current, and found that the best fit to experimental data was when Reac-

tion (2.20) accounted for 75 to 80% of the total discharge current. The closed circuit

potential was calculated with reference to the zero current potential of the cell at equi-

librium, which was given by Expression (2.10). We note, however, that this choice of


expression for the open-circuit potential is not consistent with the form of the Butler-

Volmer expression used in the model to dscribe the cathodic reaction rate. That is

to say, that at equilibrium conditions, the Butler-Volmer expression used by Chen &

Cheh does not yield the open circuit potential given by Expression (2.10).

The results of the model of Chen & Cheh were validated against experimental, gal-

vanostatic discharge data for a D-cell at a rate of 10 mA.g−1. These results show

significant discrepancies between the theoretical and experimental curves, especially at

cell potentials above 1 V.

In further work, Chen & Cheh (1993b) improved upon their previous model by ac-

counting for non-uniform reaction, electrolyte and overpotential distributions within

the anode, and by adopting a more realistic dissolution-precipitation mechanism for

the anodic reaction, previously proposed by Sunu (1978). The separator paper was

assumed to be impermeable to Zn(OH)2−4 ions, and, as such, the concentration of

Zn(OH)2−4 is zero in the separator paper and the cathode. Chen & Cheh found that

the overpotential in the anode is small up until the end of discharge, and that the

polarization of the cell during discharge is mainly due to the cathodic overpotential.

The model results are similar to the mixed-reaction model (Chen & Cheh 1993a) for

the initial stages of discharge, and although the qualitative shape of the simulated dis-

charge curve better represents that observed experimentally, there are still significant

discrepancies between the theoretical and experimental curves.

Podlaha & Cheh (1994a), building upon the work of Chen & Cheh, presented a model

for the high rate galvanostatic discharge of AA primary alkaline batteries. They ac-

counted for non-isothermal effects by modelling changes in the total cell temperature

and concluded that the effect of temperature change on discharge performance, even

at high discharge rates, is insignificant. In addition, the effect of Zn(OH)2−4 on the

cell was investigated by examining three cases. The first case was when Zn(OH)2−4 is

confined to the anode where it can precipitate as ZnO, the second case was when the

Zn(OH)2−4 is not confined to the anode, but can precipitate only in the anode, and the

third case was when the Zn(OH)2−4 is not confined to the anode and can precipitate in

the anode and separator paper. Podlaha & Cheh found that the performance of the


cell was decreased when zincate is unconfined in the cell and the model results from

case three compare favourably to the experimental data presented.

In a second paper, Podlaha & Cheh (1994b) extend their earlier high rate galvanostatic

discharge model to simulate intermittent constant load and power discharges, however

they do not simulate constant potential discharge. The effects of different intermittent

discharge regimes was investigated. They found that short and frequent intermittent

current pulses gave better discharge profiles than long and less frequent intermittent

current pulses with the same total current passed. They found this to be caused by

smaller overpotentials and more uniform cell concentrations. The model results for

intermittent galvanostatic discharge compare reasonably well with the experimental

data presented.

Zhang & Cheh (1999a) solved the model presented previously by Cheh and coworkers

using a differential-algebraic equation solver and in so doing they facilitate a sensitivity

analysis of the model (Zhang & Cheh 1999b). In addition, the relation between the open

circuit potential of the cell and the reduction of the EMD was modified by adopting an

empirical expression based on the experimental work of Kozawa & Powers (1967). An

“effective discharge factor” is introduced, which directly multiplies the reaction rate

within the cathode. This factor is experimentally deterimined, and is calculated solely

on the effective load for the discharge rate, and ranges from a value of 5 to 100 for

different effective loads. With these modifications, the model predicts the experimental

data very well.

The sensitivity analysis of the model is carried out in a second paper by Zhang & Cheh

(1999b), in which the sensitivity of 17 variables, including the closed circuit potential

was calculated with respect to 27 parameters and initial conditions, including reference

exchange current densities, diffusion coefficients of Zn(OH)2−4 and H+, the discharge

rate and initial electrolyte concentration. The parameters found to have the largest

effect on the closed circuit potential were the nail length in the anode and the discharge

current. The EMD particle size and the amount of active material were also found to

have a significant, although smaller effect on the closed circuit potential.

Kriegsmann & Cheh (1999a) use the model of Podlaha & Cheh (1994b) to investigate


the effect of cathode porosity on cell performance. The porosity of the cell is varied in

two different ways. Firstly, by changing the volume of the cathode at the expense of

the anodic volume, and secondly, by decreasing the amount of graphite in the cathode.

We note that as the amount of graphite is decreased, the conductivity of the solid

phase would degrade because the electrons in the cathode would become forced to

travel in the EMD. In addition, as EMD is reduced, its conductivity decreases (Farrell

& Swinkels 1998). These phenomena significantly effect cell performance, especially

later in discharge. However, Kriegsmann & Cheh did not link the conductivity of

the solid phase in the cathode to the graphite content or the manganese oxidation

state. It is for this reason that the authors found that optimal discharge is obtained

by removing all of the graphite from the cathode. They also found that the discharge

of commercial cells could be improved by slightly increasing the porosity of the anode

and cathode. Furthermore, they found that the optimum, initial anodic and cathodic

porosities depend on the amount of graphite in the cathode.

In a separate set of simulations, Kriegsmann & Cheh (1999a) examined the effect

of changing the cathodic solid phase conductivity. They found that lower cathodic

solid phase conductivities gave longer galvanostatic discharges. The authors link this

counter-intuitive result to the current distribution. For low solid phase conductivities

the transfer current distribution is not predominantly at the separator/cathode inter-

face, but is predominantly at the cathode/current collector interface, followed later

in discharge by a more uniform transfer current distribution. This resulted in more

uniform utilisation of EMD and an increase in discharge time.

In an additional article, Kriegsmann & Cheh (1999b) again used the model of Podlaha &

Cheh (1994b) to vary the amount of cathodic and anodic active material and investigate

the effect that this has on cell performance. High rate galvanostatic discharges were

concentrated on here because they do not utilise all of the active material by the time of

cell failure. The active material in both the anode and cathode was varied, however, a

reduction of cathodic active material was found to provide the best improvement in used

cell capacity. They noted that this is to expected since in an earlier article (Kriegsmann

& Cheh 1999a) they found that increased void volume in the cathode improved cell

performance. The authors found that the best cell performance occurred when there


was 17% less EMD in the cathode than the base case, and when the theoretical capacity

of the zinc in the anode matched that given by the EMD. Using this cell configuration

they found that the discharge time for an AA-cell discharged at 1 A to 0.8 V increased

from 0.7 to 0.8 hours.

In a third paper, Kriegsmann & Cheh (1999c) noted that the model of Podlaha &

Cheh (1994b) gives initial closed cell potentials that are below the expected experimen-

tal values. This was linked to increased overpotentials in the cell, which also caused

the model to underpredict discharge time. By examining and simplifying the cathodic

Butler-Volmer equation, a link was made between the overpotential and the interfacial

area of the cathode. They noted that the Podlaha & Cheh (1994b) model of EMD

interfacial area was based solely on geometric calculations of spherical EMD particles,

and did not use any values from the literature. Upon comparing the previous cathodic

interfacial area to literature values, Kriegsmann & Cheh found that the “true” inter-

facial area should be up to three orders of magnitude larger. Thus, they correct the

interfacial area of Podlaha & Cheh by multiplying it by a numerical factor of 50. We

note however, that no modification of the particle geometry was undertaken in order

to account for this increase in active surface area. The authors found that increas-

ing the area gives a longer discharge time, and an initial closed circuit potential that

is closer to experimental values. The longer discharge time allows the Zn(OH)2−4 ion

concentration to go to zero in the middle region of the cathode. In the regions where

there is no Zn(OH)2−4 , Kriegsmann & Cheh switch from solving the mass conservation

equation for ternary electrolyte to that for binary electrolyte. After these changes were

introduced into the model, numerical fluctuations were observed in the transfer current

and cell potential when Zn(OH)2−4 was depleted in the cathode. We will discuss these

numerical fluctuations in more detail in Chapter 5 of this thesis.

Kriegsmann & Cheh (1999d) revised the model of Podlaha & Cheh (1994b), and found

that the expression given for the equilibrium Zn(OH)2−4 ion concentration is not appro-

priate, as it models the equilibrium concentration of zinc dihydroxide, Zn(OH)2, and not

that of Zn(OH)2−4 . They noted that this expression gives unrealistically high Zn(OH)2−4

ion concentrations in the anode during the initial stages of discharge. Thus, Kriegsmann

& Cheh proposed the use of an alternative expression based on the work of Kordesch


(1974), which results in primary alkaline battery simulations with lower Zn(OH)2−4 ion

concentrations throughout discharge. We note, however, that their model does not ac-

count for the dissolution of ZnO after it has precipitated. In addition, they noted that

their model predicts the depletion of Zn(OH)2−4 during high rate discharges, and this

creates a region in the cathode which is a binary electrolyte and that causes numerical

fluctuations in their simulations, as encountered in their previous work (Kriegsmann

& Cheh 1999c). To eliminate the problem, Kriegsmann & Cheh ignore the effect of

Zn(OH)2−4 ion concentration on the overpotential in the cathode compartment of the

battery and treat the electrolyte as a pseudo-binary system. They find that the impact

of doing so, either in the cathode, the cathode and the separator, or in all regions, on

the cell potential is minimal. The resulting simulations show slightly improved cell per-

formance in comparison to the simulations using the previous, and incorrect, Zn(OH)2

equilibrium expression with a ternary electrolyte in the cathode compartment.

Based on their previous four papers, Kriegsmann & Cheh present a binary electrolyte

model (Kriegsmann & Cheh 2000) for the discharge of the primary alkaline battery

system. Justification is presented for the binary electrolyte description, based on the

characteristic times for diffusion and precipitation of Zn(OH)2−4 as ZnO within the cell.

They found that the time for precipitation in the anode is two orders of magnitude

smaller than the time for diffusion in the anode. In the separator they also found that

the representative time for precipitation is smaller than that for diffusion. Because

precipitation is so much faster than diffusion in the anode, Kriegsmann & Cheh assumed

that Zn(OH)2−4 directly precipitates out as ZnO in the anode and does not significantly

affect the cathode or separator paper. Thus, they model the electrolyte as a binary

system in all compartments. The results of the model showed that the numerical

fluctuations evident in the previous work (Kriegsmann & Cheh 1999c, Kriegsmann &

Cheh 1999d) have been removed. The authors demonstrate the wider applicability

of their binary electrolyte model, in comparison with the previous ternary electrolyte

model, by using it in parameter spaces that caused fluctuations in the ternary model.

A second significant body of modelling work has been produced by Farrell and co-

workers. This work has primarily focused on the modelling of primary alkaline battery

cathodes.


In a pair of papers (Farrell & McElwain 1996, Farrell, McElwain & Swinkels 1997),

Farrell and co-workers produced and analysed a model for the secondary current dis-

tribution within primary alkaline batteries. The results of their model simulations are

consistent with those of Mak et al. (1991b) and their major contributions are in the

solution approaches detailed in each of the articles. Farrell & McElwain (1996) give a

detailed analysis, via the application of matched asymptotic techniques, of the bound-

ary layers that characterise the secondary current distribution at the cathode/current

collector and the cathode/separator interfaces when the potential loss due to charge

transfer kinetics is small compared to the ohmic losses in the solid and/or solution

phases. In addition, Farrell et al. (1997) developed a circuit analogue for the secondary

current model that showed how the model outcomes could be equivalently obtained by

considering a distributed system of ohmic resistances.

Farrell, Please, McElwain & Swinkels (2000) presented a one-dimensional model for

the intermittent galvanostatic discharge of a primary alkaline battery cathode that has

three size scales. The model describes the EMD in the porous cathode as particles

that are themselves porous, being made up of many smaller spherical EMD crystals.

The physical shape of the cathode, as well as the graphite phase, are described on the

macroscopic, or cathode, scale. The porous particles of EMD are described on the

microscopic, or particle, scale while the sub-microscopic, or crystal, scale describes the

small, non-porous crystals of EMD. The reduction of EMD was assumed to take place

at the surface of the crystals and the H+ that are subsequently inserted into the EMD

are assumed to diffuse within the crystals, forming a homogeneous phase. The volume

change of the EMD as it is reduced was neglected. All pores, or void space, within the

cathode was assumed to be filled with a concentrated binary KOH electrolyte. The

reduction of EMD was assumed to follow Reaction (2.1), and the zero current poten-

tial (or open circuit potential) was calculated using Equation (2.13) (as suggested by

Chabre & Pannetier 1995) with a linear approximation of the ion-ion interaction term.

An important aspect of this work is that a Butler-Volmer like expression was derived

to describe the electrode kinetics, that, under equilibrium conditions, is consistent with

the modified Nernst equation given by Chabre & Pannetier. Furthermore, this Butler-

Volmer like expression was related to a well defined reference state, namely, the initial,


unreacted state of the cathode, via the inclusion of exchange current density and over-

potential terms that are defined in terms of this well characterised reference state. The

derivation of this Butler-Volmer like expression is important in the present work and

is reviewed in Appendix A of this thesis. Farrell et al. showed that the results of

the model simulations compare well with galvanostatic discharge data for a range of

EMD particle size fractions. The work of Farrell et al. is reviewed in more detail in

Chapter 3, Section 3.2.1.

Following their three-scale modelling paper, Farrell & Please (2005) presented a model

for the galvanostatic discharge of a single, porous, spherical, particle of EMD. The

model equations were simplified using asymptotic techniques and a closed form ana-

lytic solution was obtained for early time discharge behaviour. The late time solution

was obtained numerically, and together the results were used to characterise porous

particle discharge. Farrell & Please compared the results of this model with those of

their previous three-scale model, and it was found that the simplified particle model

accurately describes porous particle discharge for a wide range of relevant conditions.

The authors showed that such discharge is characterised by an initial period of spa-

tially uniform utilisation of the EMD, followed by a spatially non-uniform discharge

period, which signals the rapid onset of particle failure and the ensuing termination

of discharge. Using this, Farrell & Please were able to determine that particle radius,

the applied discharge current and the EMD conductivity are the parameters that are

most influential in determining the active material utilisation for a given porous particle

discharge regime.

Having discussed the operation of the primary alkaline battery system and several of

the associated phenomena, and introduced past research work in the field, we now move

on to the discussion of a simplified model of primary alkaline battery cathode discharge

in Chapter 3.

CHAPTER 3

The Simplified Model

3.1 Introduction

Most investigations and analysis of primary alkaline battery models have involved exam-

ining the effect of a small number of parameters of perceived importance (Kriegsmann

& Cheh 1999a, Kriegsmann & Cheh 1999b, Kriegsmann & Cheh 1999c, Kriegsmann &

Cheh 1999d). Zhang & Cheh (1999b) presented a comprehensive sensitivity analysis of

the primary alkaline battery model of Chen & Cheh (1993b). Their analysis highlighted

the relative importance of several key parameters. In addition, Farrell & Please (2005)

applied perturbation methods in order to simplify and analyse the set of equations

previously developed by Farrell et al. (2000) to model the galvanostatic discharge of

a single porous particle of EMD. Their work led to significant simplifications of the

model equations, and a greater knowledge of the behaviour of particle discharge, how-

ever, no experimental validation of their results was presented because the discharge of

an individual particle with a size typical of that found within commercial cathodes is

difficult.

In this chapter we consider the full cathodic discharge model proposed by Farrell et al.

34 Chapter 3. The Simplified Model

which was shown to compare favourably with available galvanostatic discharge data.

Our aim is to apply Laplace transform and perturbation methods in order to analyse

the key physical, chemical and electrochemical processes that govern the behaviour of

the model and, in so doing, obtain a simplified model system that accounts for these

key processes. This extends the work of Farrell & Please into the cathodic domain

and yields a simplified model having solutions that can be directly validated against

experimental data. In addition, we aim to exploit the simplified nature of the model

to develop stable numerical simulation software that can generate solutions in a timely

manner on standard computer hardware.

We review and present the model equations of Farrell et al. (2000) in dimensionless form

in Section 3.2.1. We then analyse and simplify the model equations in Section 3.2.2

using Laplace transform and perturbation methods. The simplified model equations

are too complex to fully solve analytically, so a numerical approach is adopted and this

is outlined in Section 3.3. The results of the model simulations, and validation of the

model results with experimental data is presented in Section 3.4.

3.2 Model Development

3.2.1 The Full Cathode Model

Farrell et al. (2000) developed a system of model equations and boundary conditions

for primary alkaline battery cathode discharge based on a simplified description of the

cathode. To do so, they made several simplifying assumptions, which are stated as

follows.

The cathode is assumed to be a cylindrical annulus. As described in Section 2.1,

the cathode is located just inside the can wall, and they are assumed to be in good

electrical contact. The inner radius of the cathode (the point normally in contact with

the separator paper) is assumed to be in contact with a source of excess KOH electrolyte

solution, which remains at a constant concentration throughout discharge.

The cathode itself is assumed to be made of particles of EMD surrounded by graphite.

All void space is assumed to be filled with concentrated binary electrolyte composed of

3.2 Model Development 35

Can wall

GraphiteKOH solution

Scale 2: PorousEMD particle

Manganese oxide crystal

Scale 3: EMD crystal

OH-

H+

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Scale 1: Cathode

Typical thickness: 1-5 mmTypical diameter: 5-500 nm

Typical diameter: 400A

Figure 3.1: Schematic diagram of the three scales within a primary alkaline battery cathode.

K+ ions, OH− ions and H2O molecules. The graphite is assumed to be in good electrical

contact with the EMD, and forms a conduction path between all EMD particles and the

can wall. The EMD particles are assumed to be spherical, and as mentioned previously,

are porous. The EMD particles are made up of smaller, spherical EMD crystals, which

are solid. The description of the model equations for the cathode are split up into three

size scales as depicted in Figure 3.1, namely, the cathode, the particle, and the crystal

scales.

On the cathode scale, variation is only considered in the radial direction of the cylin-

drical coordinate system, with no variation in the other directions. On the particle

and crystal scales, variation is only considered in the radial direction of the spherical

coordinate system, with no variation in the other directions. The crystals are assumed

to be very small in comparison to the particles, and the particles are assumed to be

very small in comparison to the cathode.

The electrochemical reduction of the EMD is assumed to occur at the surface of the


Table 3.1: Dimensionless variables

y = yyo

in

= 2πRiH(Ro−Ri)εEMD

Iyoin

r = rro

vp = 2πRiH(Ro−Ri)εEMDF

ro(V H2O−tK+V KOH)I(1−ε∼p)

vp

R = R−RiRo−Ri

v = 2πRiHF

(V H2O−tK+V KOH)I

v

CMn4+ =CMn4+

C0Mn4+

ηp =2πRiH(Ro−Ri)εEMDσ0

EMD∞

r2oI

ηp

COH−p =COH−p

C0OH−

η =2πRiH(Ro−Ri)εEMDσ0

EMD∞

r2oI

η

COH− =COH−

C0OH−

t = I2πRiH(Ro−Ri)εEMDFC0

Mn4+t

i∼ = 2πRiHI i∼ i∼p = 2πRiH(Ro−Ri)εEMD

I(1−ε∼p)roi∼p

DOH−∞ =DOH−

∞

D0OH−

∞

κ∼∞ = κ∼∞

κ0∼∞

EMD crystals. The EMD is assumed to undergo the first electron step as proposed

by Kozawa & Yeager (1965), given by Reaction (2.1). The cell potential is calculated

using Equation (2.13), as suggested by Chabre & Pannetier (1995). In addition, it is

assumed that discharge is isothermal and that the EMD does not change volume during

reduction.

We now present the model equations in dimensionless form. The equations have been

non-dimensionalised using the dimensionless variables given in Table 3.1. The su-

perscript ˆ denotes a dimensionless variable. Note that the non-dimensionalisation is

similar to that used by Farrell & Please (2005) for their particle scale model, but here

it relates to the timescale of cathode discharge, not particle discharge.

For a complete list of the symbols used in this chapter and throughout the thesis, with

descriptions of their meaning, refer to the List of Symbols. However, we note that in

introducing the dimensionless model equations below, we will outline the dimensionless

quantities appearing in these equations.

The Crystal Scale

The distribution of protons withing the solid spherical crystals is described using a

diffusion equation, namely,

α1∂CMn4+

∂t=

1

y2

∂

∂y

(

y2 ∂CMn4+

∂y

)

, (3.1)


where α1 is a dimensionless constant, CMn4+ is the dimensionless Mn4+ concentration,

t is dimensionless time, and y is the dimensionless radial distance from the centre of

the crystals. The crystals are assumed to be symmetric, which gives the boundary

condition,

∂CMn4+

∂y

∣

∣

∣

∣

∣

y=0

= 0. (3.2)

At the outside of the crystal (y = 1) we link the flux of protons due to the electrochem-

ical reduction of EMD to the Mn4+ concentration and impose the condition,

∂CMn4+

∂y

∣

∣

∣

∣

∣

y=1

= α1in, (3.3)

where in

is the dimensionless transfer current crossing the local EMD/KOH interface

in the direction n, which is the unit normal vector to the interface pointing into the

solution. The transfer current has been described using a Butler-Volmer exression,

and the dimensioned equivalent is derived in Appendix A. The dimensionless transfer

current is given by

in

= α2

(

1 − α3CMn4+

1 − α3

)

COH−p exp[

α14ηp + α15

(

1 − CMn4+

)]

−CMn4+

(

1 − α4COH−p

1 − α4

)

exp[

−α14ηp − α15

(

1 − CMn4+

)]

, (3.4)

where α2, α3, α4, α14 and α15 are dimensionless constants, COH−p is the dimensionless

OH− concentration on the particle scale, and ηp is the dimensionless particle scale

overpotential. We note that the overpotential, ηp (V), is defined as the difference

between the potential drop across the EMD/KOH interface and its value at a well

defined reference state, namely,

ηp = φp − φ∼p − E0. (3.5)

Here φp and φ∼p (V) are the potentials in the solid and solution phases on the particle

scale, respectively, and E0 (V) is the equilibrium potential at the reference state, which

is defined here to be the equilibrium condition that prevails prior to any discharge of

the cathode (at t = 0). The term accompanying the overpotential in the Butler-Volmer


expression is the ion-ion interaction term, Υ, as proposed by Chabre & Pannetier

(1995). Farrell et al. (2000) have chosen a linear approximation of this term, namely,

Υ = Υ1

(

1 − V Mn3+ CMn4+ |y=yo

)

. (3.6)

Such terms have been used by several authors (Atlung & Jacobsen 1981, Maskell et al.

1982, Ruetschi 1988, Chabre & Pannetier 1995) in order overcome the fact that the

reduction of EMD is not really a single-step process.

The initial condition for the crystals is,

CMn4+

∣

∣

∣

t=0= 1. (3.7)

The Particle Scale

On the particle scale, Farrell et al. (2000) give an equation describing current conser-

vation, the dimensionless equivalent of which is,

1

r2

∂

∂r

(

r2i∼p

)

= 3in, (3.8)

where r is the dimensionless radial distance from the centre of the particles, and i∼p is

the dimensionless solution phase current density in the particles in the radial direction.

The equation describing the conservation of volume is similar, namely,

1

r2

∂

∂r

(

r2vp

)

= 3in, (3.9)

where vp is the dimensionless volume average velocity of the electrolyte.

The equation governing the concentration of OH− ions is,

α5

∂COH−p

∂t=

1

r2

∂

∂r

(

r2

[

DOH−∞∂COH−p

∂r− α5α6i∼p − α5α7COH−pv

p

])

, (3.10)

where α5, α6 and α7 are dimensionless constants, and DOH−∞ is the dimensionless

diffusion coefficient of OH− ions in bulk electrolyte.


The equation governing the overpotential at each point within the particles in the

cathode is,

∂ηp

∂r= i∼p

(

(

CMn4+

∣

∣

∣

y=0.8

)−XMn4+

+α8

κ∼∞

)

+ α9

(

1 +α10COH−p

1 − α4COH−p

)

∂ ln aKOHp

∂COH−p

∂COH−p

∂r, (3.11)

where α8, α9 and α10 are dimensionless constants, XMn4+ is the exponent of the EMD

conductivity expression, given below, κ∼∞ is the dimensionless liquid phase conductiv-

ity of bulk electrolyte and aKOHp is the dimensionless activity of the KOH electrolyte

on the particle scale. The expression for the activity of KOH electrolyte used by Farrell

et al. (2000) is adopted here. We note that the first term (including XMn4+) multi-

plying the dimensionless particle scale solution phase current describes the solid phase

conductivity of the EMD. The expression used to determine the conductivity of the

EMD is,

σMn(OH)2= σ0

EMD∞εEMDp

(

CMn4+ |y=0.8yo

C0Mn4+

)XMn4+

, (3.12)

where σ0EMD∞ is the theoretical bulk conductivity of unreduced EMD and εEMDp is the

volume fraction of EMD within the particles. This expression is discussed in further

detail in Section 3.4.

The boundary conditions on the particle scale specify symmetry at the centre of the

particle (r = 0), and continuity with the corresponding cathode scale variables at the

outer radius (r = 1). They are

COH−p

∣

∣

∣

r=1= COH− , (3.13)

ηp|r=1 = η, (3.14)

∂COH−p

∂r

∣

∣

∣

∣

∣

r=0

= 0, (3.15)

vp

∣

∣

∣

r=0= 0, (3.16)

∂ηp

∂r

∣

∣

∣

∣

r=0

= 0, (3.17)


and the initial condition is

COH−p

∣

∣

∣

t=0= 1. (3.18)

The Cathode Scale

The equation describing current conservation on the cathodic scale is

1

α17 + R

∂

∂R

((

α17 + R)

i∼)

= 3 i∼p

∣

∣

∣

r=1, (3.19)

where α17 is a dimensionless constant, R is the dimensionless radial distance from the

centre of the cathode, and i∼ is the dimensionless solution phase current density in the

cathode in the radial direction.

The equation for volume conservation on the cathodic scale is,

1

α17 + R

∂

∂R

((

α17 + R)

v

)

= 3 vp

∣

∣

∣

r=1, (3.20)

where v is the dimensionless volume average velocity of the electrolyte on the cathodic

scale.

The equation governing the dimensionless concentration of OH− ions on the cathodic

scale, COH− , is,

α5α18∂COH−

∂t=

1

α17 + R

∂

∂R

(

(

α17 + R)

(

α216DOH−∞

∂COH−

∂r− α5α7α11v

COH−

))

− 3α11

(

DOH−∞∂COH−p

∂r− α5α7v

p COH−p

)∣

∣

∣

∣

∣

r=1

, (3.21)

where α11 and α16 are dimensionless constants, ε∼p is the porosity (void volume frac-

tion) of the particles, and ε∼ is the porosity of the cathode scale that is obtained by

treating the void volume within particles as solid.


The equation governing the dimensionless overpotential on the cathodic scale, η, is,

∂η

∂R=

α12

α216

i∼

(

α13 +1

κ∼∞

)

− α12α13α17

α216

(

α17 + R)

+ α9

(

1 +α10COH−

1 − α4COH−

)

∂ ln aKOH

∂COH−

∂COH−

∂R, (3.22)

where α12 and α13 are dimensionless constants and aKOH is the dimensionless activity

of the KOH electrolyte on the cathodic scale. The overpotential is defined in a manner

analagous to the particle scale overpotential (see Equation (3.5)), however, we note

that the solid phase on the cathodic scale is the graphite. This replaces the variable

conductivity of EMD encountered within particles with the constant conductivity of

graphite (related to α13).

The boundary conditions on the cathodic scale specify a no flux condition at the can

wall (R = 1), and at the inner radius (R = 0), the prescence of a well mixed reservoir.

They are

COH−

∣

∣

∣

R=0= 1, (3.23)

i∼∣

∣

∣

R=0= 1, (3.24)

i∼∣

∣

∣

R=1= 0, (3.25)

v

∣

∣

∣

R=1= 0, (3.26)

and

∂COH−

∂R

∣

∣

∣

∣

∣

R=1

= 0, (3.27)

and the initial condition is

COH−

∣

∣

∣

t=0= 1. (3.28)

Equations (3.1)-(3.28) will be referred to as the full cathode model hereafter.


3.2.2 The Simplified Cathode Model

We now analyse the full cathode model in order to identify those physical, chemical and

electrochemical phenomena that primarily govern discharge within the cathode. Util-

ising appropriate analysis techniques, our aim is to systematically simplify the model

equations to obtain a simpler model that still exhibits the key discharge phenomena.

The dimensionless constants, α1 to α17, appearing in the full cathode model are used

to facilitate the analysis and are given in Table 3.2. The representative ranges shown

for each dimensionless constant are calculated based on the data given in Tables 3.3

and 3.4, and by varying several key parameters over industrially relevant ranges. The

discharge rate has been varied from 5 to 200 mA.g−1 of EMD, the particle radius (ro)

has been varied from 5 to 250 µm, and the cathode thickness (Ro − Ri) from 0.5 to 5

mm.

We note that in this work we will validate our model simulations against the experimen-

tal data of Williams (1995) in which button-cell cathodes, having a planar geometry,

were discharged under galvanostatic conditions. The corresponding discharge param-

eters and cell geometry is specified in Table 3.4. In order to simulate planar cathode

discharge using our model, which assumes cylindrical geometry, the inner and outer

radii of the simulated cathodes are increased, while maintaining the experimentally

observed planar cathode thickness and volume. This is achieved by decreasing the

cathode “height” of the simulated cathodes. In doing this, the curvature of the sim-

ulated electrode is decreased to essentially zero, and the model equations converge to

those of linear geometry.

By considering Equations (3.1) to (3.3) and (3.7), which describe an initial boundary

value problem, and applying the method of Laplace transforms (see for example, Trim

1990b) , we may determine a closed form solution for CMn4+ , namely,

CMn4+

(

y, t)

= 1+3

t∫

0

in

dt∗+2

y

∞∑

m=1

sin (λmy)

sin (λm)

t∫

0

in

(

t − t∗)

exp

(−λ2mt∗

α1

)

dt∗, (3.29)


Table 3.2: Dimensionless constants

Dimensionless constant Typical range

α1 = Iy2o

2πRiH(Ro−Ri)εEMDDH+C0Mn4+F

0.032 ≤ α1 ≤ 1.3

α2 =2πRiH(Ro−Ri)εEMDi00

Iyo0.022 ≤ α2 ≤ 0.87

α3 = V Mn3+C0Mn4+ 1

α4 = V KOHC0OH−

0.16

α5 = r2oI

2πRiH(Ro−Ri)εEMDD0OH−

∞

√ε∼pFC0

Mn4+5.5 × 10−9 ≤ α5 ≤ 0.014

α6 =tK+C0

Mn4+

ε∼pC0OH−

12

α7 =(V H2O−t

K+V KOH)(1−ε∼p)C0Mn4+

ε∼p6.4

α8 =σ0EMD∞

(1−ε∼p)

κ0∼∞

√ε3∼p

9.7 × 104

α9 =4πRiH(Ro−Ri)εEMDRgasTt

K+σ0EMD∞

Fr2oI

3000 ≤ α9 ≤ 7.6 × 109

α10 =C0

OH−V H2O

tK+

0.74

α11 =εEMD

√ε3∼p

(1−ε∼p)√

ε3∼

0.029

α12 =σ0EMD∞

εEMD

κ0∼∞

√ε3∼

2800

α13 =κ0∼∞

√ε3∼

σG8.3 × 10−10

α14 = Fr2oI

4πRiH(Ro−Ri)εEMDRgasTσ0EMD∞

2.9 × 10−11 ≤ α14 ≤ 7.2 × 10−5

α15 =Υ1V Mn3+C0

Mn4+F

2RgasT6.8

α16 = roRo−Ri

2.0 × 10−4 ≤ α16 ≤ 0.5

α17 = RiRo−Ri

2000 ≤ α17 ≤ 2 × 104

α18 =√

ε∼p

ε∼0.6

Table 3.3: Parameter values used in the simplified model simulations

Parameter Value and reference

C0OH−

(mol.cm−3) 0.009 (Williams 1995)

C0Mn4+ (mol.cm−3) 0.0486 (Farrell et al. 2000)

DH+ (cm2.s−1) 1 × 10−15

i00 (A.cm−2) 5.0 × 10−8

Υ1 (V) 0.35σ0

EMD∞ (S.cm−1) 1.5 × 102

XMn4+ 4.328T (K) 298.15

tK+ 0.22 (Falk & Salkind 1969)

V KOH (cm3.mol−1) 17.8 (Sunu & Bennion 1980)

V H2O (cm3.mol−1) 18.07 (Sunu & Bennion 1980)

V Mn3+ (cm3.mol−1) 20.576

V Mn4+ (cm3.mol−1) 20.576yo (cm) 2.6 × 10−6

ε∼p 0.1 (Williams 1995)σG (S.cm−1) 7 × 106 (Aylward & Findlay 1994)


Table 3.4: Discharge parameters and cell geometry used to simulate the experimentaldata of Williams (1995)

Parameter Value

Discharge rate (mA.g−1of EMD) 50Particle radius (µm) 10, 45, 100 and 180Inner radius, Ri (cm) 1000.0Outer radius, Ro (cm) 1000.5

Height, H (cm) 6.603 × 10−4

Total mass of cathode (g) 4.5Mass of EMD in cathode (g) 1.0

Mass of graphite in cathode (g) 2.9

where the values of λm (m = 1, 2, . . . ,∞) are the positive roots of

tan λm − λm = 0. (3.30)

Full details of the solution are given in Appendix B, however, we note that Equa-

tion (3.29) can now be used to replace Equations (3.1) to (3.3) and (3.7). In addition,

we note that in order to obtain the solution (3.29), the transfer current, in, was as-

sumed to be a known function of time. For practical purposes, such as for use in a

numerical simulation, this suffices, as an approximation of the transfer current may be

used, with successive iterations of the solution improving upon the final answer at each

time step.

We now consider the particle scale. In particular, the mass conservation equation (3.10).

Upon examination of Table 3.2, we see that the dimensionless constant α5 is small. This

parameter is representative of the time it takes KOH electrolyte to diffuse a distance

ro, as a fraction of the time it takes to completely discharge the cathode. Noting that

α6 and α7 are both O(1), we assume that α5 → 0 asymptotically and we apply a

regular perturbation analysis to Equation (3.10) and the boundary conditions given by

Equations (3.13) and (3.15). Details of this analysis are given in Appendix C.1. This

analysis shows that to leading order, the spatial variation of KOH concentration within

the particles may be ignored, that is, we may write,

COH−p(R, r, t) = COH−(R, t). (3.31)


Using this result, we may simplify Equation (3.11) to give

∂ηp

∂r= i∼p

(

(

CMn4+

∣

∣

∣

y=0.8

)−XMn4+

+α8

κ∼∞

)

. (3.32)

Furthermore, upon consideration of Equation (3.32) (or even Equation (3.11) and the

Condition (3.15)) in conjunction with Equation (3.17), the symmetry condition on i∼p

becomes evident, namely,

i∼p

∣

∣

∣

r=0= 0. (3.33)

Now, given the similarity of Equations (3.8) and (3.9) and the corresponding Condi-

tions (3.33) and (3.16), we may also write that,

i∼p = vp . (3.34)

Moreover, we may use a similar argument on the cathodic scale because given Equa-

tion (3.34), the source terms of the cathodic current and volume conservation equations

are identical. That is, considering the similarity of Equations (3.19) and (3.20) in lieu

of Equation (3.34) and the similarity of the corresponding Conditions (3.25) and (3.26),

we may write

i∼ = v. (3.35)

A further consequence of the loss of spatial variation within the electrolyte solution

of the porous particles, as given by Equation (3.31), is that the source term within

Equation (3.21) vanishes. This term does, however, provide a vital link within the

model, between the reaction rate on the particle scale and the reaction rate on the

cathode scale. Thus, rather than applying the leading order (i.e., O(

α05

)

) expansion

for COH−p given by Equation (3.31), in this instance a version of Equation (3.10) is

obtained that is accurate to O(α5) (refer to Equation (C-7) in Appendix C.2). Manip-

ulation of this equation yields an expression for the source term in Equation (3.21) that

preserves the link between the particle and the cathode scales (refer to Equation (C-10)


in Appendix C.2). Substituting into Equation (3.21) gives,

α5 (α18 + α11)∂COH−

∂t=

1

α17 + R

∂

∂R

[

(

α17 + R)

(

α216DOH−∞

∂COH−

∂R− α5α7α11i∼pCOH−

)]

− 3α5α6α11 i∼p

∣

∣

∣

r=1. (3.36)

An additional simplification can be performed on the cathodic scale, involving the

overpotential equation (3.22). Upon further examination of Table 3.2, we see that the

dimensionless constant α13 is very small. This parameter represents the ratio of the

effective graphite and electrolyte conductivities on the cathodic scale. By assuming

that α13 → 0 asymptotically, we may simplify Equation (3.22) to obtain

∂η

∂R=

α12

α216κ∼∞

i∼ + α9

(

1 +α10COH−

1 − α4COH−

)

∂ ln aOH−

∂COH−

∂COH−

∂R. (3.37)

Equations (3.4), (3.8), (3.14), (3.17), (3.19), (3.23) to (3.25), (3.27) to (3.30), (3.32), and

(3.34) to (3.37) represent a simplified model of cathode discharge and will be referred

to as the simplified model hereafter.

3.3 The Numerical Solution of the Simplified Equations

The simplified model equations are solved numerically in MATLAB R©. Here we describe

the numerical scheme and the discretisation method employed.

The numerical simulation software starts with the initial distributions of all variables,

and solves the model equations iteratively to determine the distribution of each variable

at every time step. An overview of the algorithm is given below.

1. Set constants and read input data file.

2. Initialise all physical, chemical and electrochemical variables, all main loop’s logic

variables and open output files.

3. Enter main loop:

3.3 The Numerical Solution of the Simplified Equations 47

(a) Determine time step size.

(b) Initialise solver loop’s logic variables.

(c) Enter solver loop.

i. Calculate the coefficient values that correspond to the linearised form

of the model equations.

ii. Solve the matrix system based on the linearised system of equations.

iii. Exit the solver loop if any of the following conditions apply:

• The solution has converged,

• The solution has taken too many iterations without converging,

• A non-physical solution has been predicted. For example, a chemical

concentration that is negative.

iv. Otherwise, update variables and go to 3(c)i.

(d) If the solution converged, output data to files and update time.

(e) Exit the main loop if any of the following conditions apply:

• The maximum time is reached,

• The cell voltage is below the cutoff,

• The solver has failed to converge too many times.

(f) Otherwise, go to 3a.

4. Close output files and exit.

The simplified model equations are able to be solved in steps 3(c)i and 3(c)ii by dis-

cretising a linear version of the equations, formed by applying a combination of lin-

earisation and fixed-point techniques, to create a system of linear equations which may

be solved in matrix form. The model equations are discretised in dimensional form

using a control-volume approach (Patankar 1980), which has the benefit of implicitly

conserving the physical quantities. In this approach the model domain is divided into


Figure 3.2: Schematic diagram of the control-volumes in one dimension. The variables areevaluated at the node points 1 to kmax.

a number of so called control-volumes and the model equations with spatial deriva-

tives are integrated over these representative control-volumes. A schematic diagram

of the control-volumes in one dimension is shown in Figure 3.2. The model equations

containing a time derivative are then integrated over the time step.

The discretisation, in time, of the model equations is achieved through a flexible time

weighting technique, for example,

t+∆t∫

t∗=t

f (t∗) dt∗ ≈ θf (t + ∆t) + (1 − θ) f (t) , (3.38)

where t (s) is the time, ∆t (s) is the time step, f (t) is an arbitrary function of time,

and θ is the time weighting parameter. Note that using Equation (3.38), we can choose

θ = 0 for fully explicit time stepping, θ = 1 for fully implicit time stepping, and θ = 1/2

for Crank-Nicolson time stepping. In our code, fully implicit time weighting (θ = 1)

is used in all simulations. Equation (3.29) is first differentiated with respect to time,

and then discretised as described above. In addition, the sum in Equation (3.29) is

evaluated to 47 terms to ensure accuracy.

As an example of an equation obtained using the control-volume approach, we present

the discretised form of Equation (3.36). It has been integrated over the kth control-

volume, as displayed in Figure 3.2, and integrated over the time step as described

in Equation (3.38). The dimensional form of Equation (3.36) is given in Chapter 4

in Table 4.1. We note that we have used the dimensional form of Equations (3.35)

and (3.37) (also given in Table 4.1) to eliminate the volume average velocity and solution

3.3 The Numerical Solution of the Simplified Equations 49

phase current, respectively. The discretised form is,

(

ε∼ +εEMD

εEMDpε∼p

)

R2k+ 1

2

− R2k− 1

2

2

(

COH− |t+∆tk − COH− |tk

)

=

∆tθ

[

B|mk+ 12

COH− |t+∆tk+1 − COH− |t+∆t

k

Rk+1 − Rk− B|mk− 1

2

COH− |t+∆tk − COH− |t+∆t

k−1

Rk − Rk−1

]

+ ∆t (1 − θ)

[

B|tk+ 12

COH− |tk+1 − COH− |tkRk+1 − Rk

− B|tk− 12

COH− |tk − COH− |tk−1

Rk − Rk−1

]

−(

V H2O − tK+V KOH

)

∆t

FRk+ 1

2×

θ

[

COH− |mk+ 12

κ∼∞|mk+ 12

η|t+∆tk+1 − η|t+∆t

k

Rk+1 − Rk+ COH− |t+∆t

k+ 12

κ∼∞|mk+ 12

η|mk+1 − η|mkRk+1 − Rk

− COH− |mk+ 12

κ∼∞|mk+ 12

η|mk+1 − η|mkRk+1 − Rk

]

+ (1 − θ) COH− |tk+ 12

κ∼∞|tk+ 12

η|tk+1 − η|tkRk+1 − Rk

+

(

V H2O − tK+V KOH

)

∆t

FRk− 1

2×

θ

[

COH− |mk− 12

κ∼∞|mk− 12

η|t+∆tk − η|t+∆t

k−1

Rk − Rk−1+ COH− |t+∆t

k− 12

κ∼∞|mk− 12

η|mk − η|mk−1

Rk − Rk−1

− COH− |mk− 12

κ∼∞|mk− 12

η|mk − η|mk−1

Rk − Rk−1

]

+ (1 − θ) COH− |tk− 12

κ∼∞|tk− 12

η|tk − η|tk−1

Rk − Rk−1

−3εEMD

(

1 − tOH−

)

εEMDproF

R2k+ 1

2

− R2k− 1

2

2

∆t(

θ i∼p|t+∆tk + (1 − θ) i∼p|tk

)

, (3.39)

where the superscript m denotes a trial, or “best guess”, value, subscript k denotes

the value at the kth node, and the subtraction or addition of a half to k denotes the

value at the inner or outer control-volume face, respectively. All variables evaluated

at the inner and outer control-volume faces are approximated by linearly interpolating

between the data at the node points. The parameter B (cm2.s−1) is defined as

B = R

[

DOH−∞ε∼ +2RgasT

(

V H2O − tK+V KOH

)

ε3/2∼

F 2κ∼∞COH−

×(

1 − tOH− +

COH−

CH2O

)

∂ ln aKOH

∂COH−

]

. (3.40)

The time step must be calculated at the beginning of each main loop iteration. This

is done in such a manner so as to maintain stability in the numerical algorithm, and


is governed by several factors, including the maximum allowable time step size, the

previous time step size, whether the solution converged in the previous attempt, and

the proximity to data output times. In addition, during the initial stages of discharge,

the time step is chosen to be small, to accurately capture the rapid changes that occur

when the discharge starts.

In the simplified model equations, certain non-linear terms, namely, the advection

term in Equation (3.36), and all of Equation (3.4) are linearised on the basis of a trial

solution. The advection term is not upwinded because, as observed by Farrell et al.

(2000), the term only accounts for a very small proportion of the electrolyte flux and

does not cause instability in the numerical simulations. The remaining nonlinearities

are treated using a fixed-point iteration approach (Burden & Faires 2001).

After initialising the resulting linear system, using a trial solution, the system is solved.

The new solution is then used as the trial solution to obtain a second approximation.

This process is repeated until the trial solution does not change significantly, using the

criteria that,∥

∥dk∥

∥

2

(‖dk−1‖ − ‖dk‖) ‖xk‖ < tolerance, (3.41)

where

dk = xk+1 − xk. (3.42)

Here, xk represents a vector of the variables’ values on the kth iteration. This criterion is

based on the convergence of a first order process. We note that Expression (3.41) would

give convergence in the undesirable case when∥

∥dk∥

∥ >∥

∥dk−1∥

∥, which corresponds to

the divergence of the iterative process. If this is the case, Expression (3.41) is not used,

and it is assumed that the process is not converged. Upon convergence of a particular

series of iterations, the time is updated to t + ∆t and the linear system of equations is

again used to determine the values of the variables at the new time.

The simplifications carried out in the previous section provide a substantial reduction

in complexity of the model system, which now has only one partial differential equation,

as compared with three in the full model system. The ensuing reduction in the number

of numerical calculations needed to solve the system means that the software that

3.4 Results and Discussions 51

Time (h)

Ece

ll(V

)

0 1 2 3 4 50.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Figure 3.3: Comparison of model output (filled symbols) with experimental data (outlinedsymbols) (Williams 1995). Shown is experimental data for cathodes with radii in the ranges0-22.5 (), 38.5-53 (♦), 75-150 (⊳) and 150-250 (#) µm, and model output for cathodes withradii of 10 (H), 45 (), 100 () and 180 ( ) µm.

implements the numerical solution is easily run on a standard desktop PC. For example,

on a desktop computer with a 2.4 GHz Pentium R© 4 processor and 512 MB of RAM, a

full simulation of a typical discharge is completed in approximately 60 seconds.

3.4 Results and Discussions

In this section we compare the predictions of the numerical simulations against ex-

perimental data. Williams (1995) galvanostatically discharged a series of planar EMD

cathodes in 9 M KOH electrolyte at a rate of 20, 50 and 100 mA.g−1 of EMD. Each cath-

ode consisted of EMD particles that were taken from a specific size fraction, namely:

2ro ≤ 45µm, 77µm ≤ 2ro ≤ 106µm, 150µm ≤ 2ro ≤ 300µm and 300µm ≤ 2ro ≤ 500µm.

A comparison of the output of the simplified model with the experimental data of

Williams is given in Figure 3.3, and Tables 3.3 and 3.4 list the parameter values used

in the model to simulate the experimental data.


From Figure 3.3, it is seen that the model results compare well with the experimental

data. The simplified model accurately captures the polarisation effects seen in the

experimental results with the cathodic discharge time increasing as the EMD particle

size is decreased. This is due to more uniform reaction distributions within the smaller

EMD particles that lead to a greater utilisation of active material within the cathodes

containing these particles.

In simulating the data of Williams, we have taken the value of CMn4+ that appears

in the EMD conductivity function (given by Expression (3.12)) to be that at 80% of

the radius (i.e., 0.8yo) of a given crystal. This corresponds to the value of CMn4+

at y = 0.8 appearing in Equations (3.11) and (3.32). The effect of choosing various

positions within the oxide crystals at which to take the value of CMn4+ in order to

calculate the EMD conductivity is shown in Figure 3.4. The simulations are for the

cathode manufactured by Williams that consists of EMD particles in the size fraction

150 µm ≤ 2ro ≤ 300µm (refer to Table 3.4). The corresponding experimental discharge

result is also given in Figure 3.4 for comparison.

The results in Figure 3.4 indicate that taking the value of CMn4+ at 0.8yo yields a

theoretical discharge that corresponds well with the experimental result. Farrell et al.

(2000) and Farrell & Please (2005) previously took the concentration value appearing

in the conductivity function to be that at the outer radius of the EMD crystals. As

Figure 3.4 demonstrates, however, this approach appears to overestimate the resistance

experienced by electrons moving within the oxide used by Williams and leads to shorter

discharge times in comparison with the experimental data. To obtain more accurate

predictions of the conductivity of non-uniformly reduced EMD, an in-depth study into

the current paths and the connectivity on a crystal scale would be needed.

Upon examination of Table 3.2, the assumption that α1 → 0, seems valid at low

to medium discharge rates. Furthermore, the second assumption, that the transfer

current does not change on the timescale of crystal diffusion, seems valid in the particle

setting, where the total discharge current of the particle may be directly specified (as

constant). However, within an operating primary alkaline battery there are diverse

conditions throughout the cathode, where solid phase conductivity and variations of


Time (h)

Ece

ll(V

)

0 1 2 30.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Figure 3.4: Comparison of experimental data (⊳) and model results (filled symbols) whenEMD conductivity is calculated at 0.6yo (), 0.8yo ( ) and yo (N). The experimental cathodecontains particles of the size fraction 150µm ≤ 2ro ≤ 300µm.

concentrations within the electrolyte contribute to non-uniformity. In this environment,

crystals may experience changes in their individual discharge rates at shorter time scales

than seen in individual particles.

We also note here that, for the case of constant particle discharge current, Equa-

tion (3.29) simplifies to the equivalent of Equation (3.44), and thus we recover the

more specific particle discharge result of Farrell & Please.

In Section 3.2.2, an expression was obtained for the concentration distribution of Mn4+

within an EMD crystal (i.e., Equation (3.29)) by applying the method of Laplace trans-

forms. This expression does not depend on any simplifying assumptions. Nevertheless,

if we are willing to admit assumptions, namely that α1 → 0, and that in

does not change

on the timescale of proton diffusion, then asymptotic methods can be applied to the

crystal-scale proton diffusion problem (defined by Equations (3.1) to (3.3) and (3.7))

in order to obtain approximate expressions for the distribution of Mn4+ within EMD

crystals. The analysis follows closely that reported by Farrell & Please (2005) for the


discharge of porous EMD particles, and we find that the O(1) expression is given by,

CMn4+

(

y, t)

= 1 + 3

t∫

0

in

dt∗, (3.43)

and the O(α1) expression is given by,

CMn4+

(

y, t)

= 1 + 3

t∫

0

in

dt∗ + α1

(

y2

2− 3

10

)

in

− α12i0

n

y

∞∑

n=1

sin (λny)

λ2n sin (λn)

exp

(−λ2nt

α1

)

, (3.44)

where t∗ is a dummy variable.

The discharge results (given in terms of the fraction of the theoretical capacity of the

cathode that is used) of the simplified cathode model at various discharge rates are

presented in Figure 3.5. Either Equation (3.29) or (3.43) is used to model the distri-

bution of Mn4+ within EMD crystals. The cathode configuration for these simulations

is that of a cylindrical AA-cell, details of which are given in Table 3.5. At low dis-

charge rates, the use of either Equation (3.29) or (3.43) within the model yields very

similar discharge curves, however, as the current is increased, significant discrepancies

between the two models are observed. To understand why these discrepancies occur,

distributions of Mn4+ within an EMD crystal were obtained at R = Ri and r = ro as

given by Equations (3.29), (3.43) and (3.44) for discharge rates of 20 (Figure 3.6(a)),

50 (Figure 3.6(b)), and 100 (Figure 3.6(c)) mA.g−1 of EMD. At low discharge rates,

such as that shown in Figure 3.6(a), the distribution of Mn4+ within an EMD crystal

is essentially independent of crystal radius and the simplifying assumptions that con-

stitute the asymptotic solutions, namely, α1 → 0 and that in

does not change on the

timescale of proton diffusion, are well supported (in fact the α1 value corresponding to

Figure 3.6(a) is 0.17). Thus, the results predicted by Equations (3.29), (3.43) and (3.44)

correspond well in this discharge regime. When the discharge rate is increased, as in

Figure 3.6(b) and 3.6(c), the distribution of Mn4+ within an EMD crystal becomes more

non-uniform and the discrepancies between the predictions of Equations (3.29), (3.43)

and (3.44) become significant. Indeed, at a discharge rate of 100 mA.g−1 of EMD, the

3.5 Conclusions 55

Fraction of theoretical capacity used

Ece

ll(V

)

0 0.2 0.4 0.60.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Figure 3.5: Discharge results for simplified model using an asymptotic (filled symbols) (Equa-tion (3.43)) or Laplace (hollow symbols) (Equation (3.29)) crystal scale solution. The simula-tions correspond to 20 (), 50 (), 100 (#) and 200 (♦) mA.g−1 of EMD.

assumption that α1 → 0 can no longer be supported and the α1 value corresponding

to Figure 3.6(c) is 0.86. In this regime, Equations (3.43) and (3.44) become invalid.

3.5 Conclusions

In this chapter we have simplified an existing model of primary alkaline battery cathode

discharge (Farrell et al. 2000) to yield a smaller model that accounts for the important

physical, chemical and electrochemical phenomena. A MATLAB R© program has been

written and used to provide validation and insight into the operation of the primary

alkaline battery system.

In particular, we presented the model equations of Farrell et al. (2000) in dimension-

less form and gave approximate sizes for the dimensionless constants appearing in the

equations. A simplified model was obtained by applying Laplace transform and pertur-

bation methods. In the analysis that ensued, it is shown that the three size scales used

by Farrell et al. to describe the porous EMD cathode can be reduced to two size scales


Table 3.5: Discharge parameters and cell geometry for AA-cell cathode geometry asused in the simulations presented in Figure 3.5

Parameter Value

Discharge rate (mA.g−1of EMD) 20, 50, 100 and 200Particle radius (µm) 100Inner radius, Ri (cm) 0.45Outer radius, Ro (cm) 0.67

Height, H (cm) 4.04Total mass of cathode (g) 10.62

Mass of EMD in cathode (g) 9.24Mass of graphite in cathode (g) 0.8

y (cm)

CM

n4+

(mol

.cm

-3)

0 0.01 0.020

0.01

0.02

0.03

0.04

0.05

t = 5 h

t = 1 h

t = 3 h

t = 7 h

t = 9 h

t = 0 h

(a) Distributions during a 20 mA.g−1 of EMDdischarge.

y (cm)

CM

n4+

(mol

.cm

-3)

0 0.01 0.020

0.01

0.02

0.03

0.04

0.05

t = 1 h

t = 2 h

t = 3 h

t = 0 h

(b) Distributions during a 50 mA.g−1 of EMDdischarge.

y (cm)

CM

n4+

(mol

.cm

-3)

0 0.01 0.020

0.01

0.02

0.03

0.04

0.05

t = 15 min

t = 45 min

t = 0 min

(c) Distributions during a 100 mA.g−1 of EMDdischarge.

Figure 3.6: Mn4+ ion concentration distributions at various times within an EMD crystal atR = Ri, and r = ro, as given by Equation (3.29) (), Equation (3.43) (H) and Equation (3.44)( ).

3.5 Conclusions 57

without the loss of generality. In addition, the analysis demonstrates that the time

taken for electrolyte to diffuse into a porous EMD particle is fast compared with the

cathodic discharge time, and that ohmic losses within the graphite phase of the cathode

can be considered negligible. Furthermore, the simplified model incorporates a closed

form expression for the distribution of Mn4+ within an EMD crystal that is not reliant

on assumptions that may break down at high discharge rates. The simplified model of

primary alkaline battery cathode discharge extends the work of Farrell & Please into

the cathodic domain.

The simplified model equations are too complex to solve analytically, so a numerical

technique is used. Numerical solutions of the simplified model equations have been

generated by writing and implementing a finite-volume code in MATLAB R© that can

easily be run on a standard desktop PC. In addition, the simplified model results

compare favourably with relevant experimental data.

CHAPTER 4

The Potentiostatic Model

4.1 Introduction

The reduction of EMD is a complex process, as evidenced by the number of research

articles that discuss its determination experimentally (see Chapter 2, Sections 2.2.1

and 2.2.2). Chabre & Pannetier (1995) give a comprehensive discription of the reduc-

tion process. They found that reduction consists of a mixture of heterogeneous and

homogeneous processes, some of which are irreversible. In addition, the crystal struc-

ture of the EMD changes during reduction. It is clear that writing down an accurate

mathematical description of the reduction process requires knowledge of the values

of the kinetic parameters. However, many of these values are not easy to measure

experimentally.

Chabre (1991) and Chabre & Pannetier (1995) used Step Potential Electrochemical

Spectroscopy (SPECS) to examine the reduction process. In a SPECS discharge, the

cell is subjected to a series of consecutive potentiostatic discharges, in which the cell

potential is decreased (or stepped) by a fixed amount in each discharge (usually starting

near the OCV), and the current response is recorded. A typical potential versus time

60 Chapter 4. The Potentiostatic Model

Time (h)

Ece

ll(V

)

0 50 100 1500.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Figure 4.1: The potential experienced by a cell during a SPECS discharge. In this SPECSdischarge the potential is stepped by 5 mV each hour.

curve is shown in Figure 4.1. As mentioned in Chapter 2, the advantage of SPECS over

other, continuous, discharge modes such as galvanostatic, constant load, or constant

power is that it emphasizes the electrochemical and physical responses of the cell.

Ideally, the cell is given time to equilibrate at each potential level, thus minimising

transport losses. However, this means that individual SPECS discharges can take

many days to complete, as can be seen from the time axis in Figure 4.1.

The multi-reaction nature of the reduction of EMD is very apparent in the SPECS

results. The minimum and maximum current or power experienced during each po-

tentiostatic discharge, versus potential, are frequently used formats to visualise results.

A typical plot of a SPECS discharge is shown in Figure 4.2. The figure shows sev-

eral clearly visible peaks in the power output (for example, at 1.46, 1.28 and 1.13

V). These peaks are attributed to the steps or reactions that constitute the reduction

process (Chabre 1991, Chabre & Pannetier 1995), and clearly show that it is not a

single-reaction process.

In contrast to the conclusions drawn by many researchers (see Chapter 2, Section 2.2.1)

4.1 Introduction 61

Ecell (V)

Min

and

Max

Pow

er(m

W.g

-1of

EM

D)

0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

2

4

6

8

10

Figure 4.2: The minimum and maximum power in each potential step of a typical EMDSPECS discharge. The minimum and maximum power values are usually found at the endand beginning of each potential step, respectively. Experimental data courtesy of Delta EMDAustralia Pty. Limited.


who have experimentally studied the reduction of EMD, the prevailing approach in

the mathematical modelling literature is to assume that reduction is described by a

single reaction, specifically, Reaction (2.1). Assuming this, a single Butler-Volmer like

expression (for example, see Chapter 3, Equation (3.4)) may be used to describe the

kinetics at the EMD/KOH interface, and a single Nernst like expression is adopted to

describe the zero current potential, or open circuit voltage (OCV).

In taking the above approach, many authors modify the Nernst like expression in order

to produce OCV curves that are closer to those determined experimentally. By doing

this, these authors are really modifying the kinetics of the reaction mechanism, and/or

the reaction mechansim itself. However, this is often done inconsistently, in that, the

Butler-Volmer equation is not modified to reflect these mechanistic changes. Farrell

et al. (2000) do make consistent modifications, based on the work of Chabre & Pan-

netier, however, (as we shall see) their linear approximation of the ion-ion interaction

term fails to account adequately for the true multi-reaction nature of the reduction

process.

In this chapter we aim to improve the treatment of this process in our mathematical

model. However, this is a difficult task because to accurately model the reduction of

EMD, the full reaction mechanism and associated parameter values, including exchange

current densities for each individual reaction at well characterised reference conditions,

should be known. Without the appropriate information, these become “free” param-

eters and there is no guarantee that the values of these parameters obtained by say,

fitting model simulations to experimental data, will be in any way realistic or unique. It

is conceivable that several sets of parameter values exist that give the same behaviour,

making the results and parameter values determined in such a way largely meaningless.

In addition, when there are many unknown parameters, the predictions of the model

may become simply a result of the choice of the values, so that no new information

may be extracted from the results. To avoid such ambiguity, the approach taken here

is to minimise the number of unknown parameters introduced into the model.

The question arises as to whether a modelling framework centred on a single-reaction

mechanism can display multi-reaction behaviour. To investigate this, we have chosen


the zero current potential relationship proposed by Chabre & Pannetier (1995), given

by Equation (2.13) in Chapter 2 of this thesis. We note that this relationship was used

in Chapter 3, albeit with a linear approximation of the ion-ion interaction term. In

this work, however, we seek to find the best form for the ion-ion interaction term to

describe the multi-reaction reduction process. Furthermore, for this purpose, we will

attempt to model the stepped potential discharge of the alkaline battery cathode.


Here we develop the model equations for the potentiostatic discharge of a primary al-

kaline battery in order to simulate SPECS discharges. The model is adapted from the

simplified model developed in Chapter 3. Only the changes to the previous model equa-

tions are detailed here. These are followed by two tables summarising the unchanged

equations (Table 4.1) and boundary conditions (Table 4.2) in dimensional form.

To account for the general form of the ion-ion interaction term, we use the modified

Butler-Volmer like expression (see Chapter 3, Equation (3.4)), namely,

in

= i00

(

CMn3+

C0Mn3+

)(

COH−

C0OH−

)

exp

[

(1 − αc)F

RgasT

(

ηp + Υ − Υ0)

]

−(

CMn4+

C0Mn4+

)(

CH2O

C0H2O

)

exp

[−αcF

RgasT

(

ηp + Υ − Υ0)

]

. (4.1)

The derivation of the general form of the above expression, as given by Farrell et al.

(2000), is reviewed in Appendix A. The process of determining a suitable form for the

ion-ion interaction term, Υ (CMn4+) (V), is detailed in Section 4.5.

In order to simulate potentiostatic discharge, we develop a boundary condition that

replaces Equation (3.24). By specifying the cell potential, Ecell(t) (V), we are fixing

the potential drop across the cathode, from the solid phase at the cathode/current

collector interface (R = Ro) to the solution phase at the cathode/bulk KOH interface

(R = Ri). This is designated by the expression,

φ|R=Ro− φ∼|R=Ri

= Ecell(t), (4.2)


where Ecell(t) is specified. We may rearrange the definition of the overpotential (see

Chapter 3, Equation (3.5)) to give an expression for the liquid phase potential on the

cathodic scale at R = Ri, namely,

φ∼|R=Ri= φ|R=Ri

− η|R=Ri− E0. (4.3)

Substituting this into Equation (4.2), we find that,

φ|R=Ro− φ|R=Ri

+ η|R=Ri+ E0 = Ecell(t). (4.4)

Because we assume that the graphite on the cathodic scale is well connected and con-

tinuous, the potential in the solid phase at the current collector, φ|R=Ro, is the same

as the potential in the solid phase near the bulk KOH interface, φ|R=Ri. Thus,

η|R=Ri= Ecell(t) − E0. (4.5)

The development of this potentiostatic boundary condition allows the simulation of

a SPECS discharge by choosing an Ecell(t) function that “steps” by an appropriate

voltage at regular intervals, such as shown in Figure 4.1.

Table 4.1 contains the governing equations used in this model that have not changed,

from Chapter 3. The boundary conditions associated with these equations are given in

Table 4.2.

4.3 The Numerical Solution

The numerical approach used to solve the potentiostatic model is similar to that used to

solve the simplified galvanostatic model as detailed in Chapter 3, Section 3.3, however,

there are some key differences and these are commented on here.

The major differences in the solution algorithm extend from the change in discharge

mode from galvanostatic to potentiostatic. By changing the discharge variable from

current to potential, and using the new boundary condition (4.5), no solution was

achievable with the previous code, because the fixed point iterative strategy diverged.

4.3 The Numerical Solution 65

Table 4.1: Additional governing equations used in the potentiostatic model. Theseequations are the dimensional equivalents of the simplified model equations developedin Chapter 3.

Governing Equations

Crystal Scale:

CMn4+ = C0Mn4+ +

t∫

t∗=0

3in

Fyodt∗

+∞∑

m=1

2 sin(

λmy

yo

)

Fyo sin(λm)

t∫

t∗=0

in

(t − t∗) e−λ2

mDH+ t∗

y2o dt∗ (3.29)

Particle Scale:

Fvp =

(

V H2O − tK+V KOH

)

i∼p (3.34)1r2

∂∂r

(

r2i∼p

)

= ApεEMDpin (3.8)

∂ηp

∂r = i∼p

(

1εEMDpσ0

EMD∞

(

CMn4+ |y=0.8yo

C0Mn4+

)−XMn4+

+ 1√ε3∼pκ∼∞

)

(3.32)

Cathode Scale:

Fv =(

V H2O − tK+V KOH

)

i∼ (3.35)1R

∂∂R (Ri∼) = 3εEMD

εEMDproi∼p|r=ro

(3.19)

∂η∂R = i∼√

ε3∼

κ∼∞

+2RgasT

F

(

1 − tOH−

+COH−cCH2Oc

)

∂ ln aKOHc∂COH−

∂COH−

∂R (3.37)(

ε∼ + εEMDεEMDp

ε∼p

)

∂COH−

∂t = 1R

∂∂R

R(

DOH−∞ε∼∂COH−

∂R − COH−v

)

−3εEMD(1−tOH−

)εEMDproF i∼p|r=ro

(3.36)

Table 4.2: Additional boundary and initial conditions used in the potentiostatic model.These conditions are the dimensional equivalents of those developed in Chapter 3 forthe simplified model.

Boundary Conditions∂ηp

∂r

∣

∣

∣

r=0= 0 (3.17)

ηp|r=ro= η (3.14)

COH− |R=Ri= C0

OH−(3.23)

∂COH−

∂R

∣

∣

∣

R=Ro

= 0 (3.27)

COH− |t=0 = C0OH−

(3.28)

i∼|R=Ro= 0 (3.25)


This divergence was overcome by reorganising the matrix structure to ensure that key

elements were on the diagonal, and that it was not as ill-conditioned.

A further difference in the solution procedure arises from the introduction of discrete,

or discontinuous steps in the cell potential. This necessitated a change in the adap-

tive time-stepping algorithm in which small time steps are used to maintain stability

immediately following a potential step.

As with that of Chapter 3, the numerical simulation software developed here is imple-

mented in MATLAB R©. The increase in the length of a typical discharge in comparison

to galvanostatic discharges, as well as the smaller time steps used after potential steps,

means that a typical simulation takes longer than the previous model. However, this

simulation time is still several orders of magnitude smaller than an actual experimental

discharge. For example, the simulation of a 5 mV per hour SPECS cathodic discharge

running on a 2.4 GHz Pentium R© 4 processor with 512 MB of RAM takes about 20

minutes.

4.4 Determining the Ion-ion Interaction Term

Here we present discussion and results in relation to determining the functional form of

the ion-ion interaction term, Υ, in Equation (4.1) that best describes the multi-reaction

reduction process of EMD. We then compare the output of our potentiostatic model

with experimental data.

The cell geometry and discharge parameters used in all simulations presented in this

chapter are based on that of the experimental configuration of the button cell cathodes

used by Delta EMD Australia Pty. Limited, and given in Table 4.3. The remaining

parameters are the same as found in Table 3.3 of Chapter 3. However, the value of the

diffusion coefficient of H+ in EMD crystals, DH+ , is chosen to be 1 × 10−16 cm2.s−1,

based on fits to experimental data performed in Section 4.5.2. From Table 4.3 we

also note that the method used to simulate planar button-cell cathode geometry is the

same as that used in Chapter 3. Namely, we increase the inner and outer radii of

the simulated cathodes, while maintaining the experimentally observed thickness and

4.4 Determining the Ion-ion Interaction Term 67

Table 4.3: Discharge parameters and cell geometry used to simulate the experimentaldata

Parameter Value

Potential step size (mV) 5Potential step time (h) 1Particle radius (µm) 25Inner radius, Ri (cm) 1000.0Outer radius, Ro (cm) 1000.0928

Height, H (cm) 2.81 × 10−4

Total mass of cathode (g) 0.5Mass of EMD in cathode (g) 0.3

Mass of graphite in cathode (g) 0.175

volume by decreasing the cathode “height”. This causes the curvature of the electrode

to decrease and the model equations converge to those of linear geometry.

As an initial test of the model, we simulated a SPECS discharge using the same linear

form of the ion-ion interaction term as used in Chapter 3. The comparison of the

model output with experimental data is shown in Figure 4.3. We observe that the

model output compares poorly with the experimental data. The model output does not

display any multi-reaction behaviour, as there is only one peak, while the experimental

data has at least two clearly evident peaks, one at 1.46 V and the other at 1.3 V.

Improving the approximation of the ion-ion interaction term to yield multi-reaction

behaviour is not a straightforward task. However, there are some constraints on the

choice of possible approximations. One constraint is that the domain of Υ (CMn4+) must

be within realistic Mn4+ ion concentrations. In addition, its range should be positive,

because a negative range would increase the zero current potential to voltages above

the predictions of the Nernst expression corresponding to the single reaction given by

Reaction (2.1), which already overpredicts the zero current potential. Furthermore, we

assume that the standard potential, E0 (V), in the modified Nernst equation proposed

by Chabre & Pannetier (1995), namely,


Fln

CMn3+

CMn4+

, (4.6)

takes into account the initial value of Υ when the initial open circuite voltage is mea-

sured. Based on this, the value of Υ(

C0Mn4+

)

is chosen to be zero. This simplifies


Ecell (V)

Min

and

Max

Pow

er(m

W.g

-1of

EM

D)

0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

2

4

6

8

10

Figure 4.3: Comparison of a simulated 5 mV/hr SPECS discharge (#) using a linear ion-ioninteraction term, with experimental data (). The model parameters used are described inTable 4.3.

Equation (4.1) when we choose the reference equilibrium state of the cell to be the

state that exists in the cathode immediately before discharge.

We note that the linear approximation of Υ (CMn4+) used in Chapter 3 and by Farrell

et al. (2000) has yielded realistic galvanostatic discharge behaviour. Thus, as a first

approximation we consider extending the linear form to a higher degree polynomial in

CMn4+ . An example of two different polynomial representations of Υ (CMn4+) is given in

Figure 4.4 and the corresponding 5 mV/hr SPECS results are displayed in Figure 4.5.

The results show very little similarity with the previous experimental data, however,

we do note, by comparison with Figure 4.3, the significant effect that changing the

form of Υ (CMn4+) has on the output of the SPECS simulation. Thus the choice of

the form of this function would seem crucial to successfully simulating multi-reaction

reduction behaviour in a single-reaction framework. However, its form is not obvious,

and guessing it is very unlikely.

To give an estimate of the form of the ion-ion interaction term, we consider using the


CMn4+ (mol.cm-3)

Υ(V

)

0 0.02 0.040

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Figure 4.4: Two possible representations of the ion-ion interaction term. The model resultsbased on these to functions are shown in Figure 4.5. The symbols used in this figure correspondto those used in Figure 4.5

experimental data provided by Delta EMD Australia Pty. Limited. The Butler-Volmer

expression, given by Equation (4.1) is ideal for this. However, to relate the experimental

SPECS data to Υ (CMn4+), we must make several simplifying assumptions, which are

detailed as follows.

Firstly, we must assume that the concentration distributions within the cathode are

close to uniform at the end of each potentiostatic discharge. This assumption may

be poor, especially if the time at which the potential is maintained constant is short.

Secondly, we assume that each potential step occurs instantaneously, so that the con-

centration distributions before and after the potential step are the same. Thirdly, we

assume that ohmic losses in both the solid and solution phases are negligible. This as-

sumption may also be poor, especially later in discharge on the particle scale, because

reduced EMD is not a high resistance. This assumption, however, is necessary because

it allows us to specify that any change in the cell potential is exactly reflected by a

change in the particle scale overpotential at all points in the cathode.


Ecell (V)

Min

and

Max

Pow

er(m

W.g

-1of

EM

D)

0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

2

4

6

8

10

12

14

16

18

Figure 4.5: Comparison of two 5 mV/hr SPECS simulations using the two different ion-ioninteraction terms shown in Figure 4.4. The symbols in this figure correspond to those used inFigure 4.4.

If we admit these assumptions, we may relate the increase in experimentally observed

current at each potential step to the difference between two Butler-Volmer like ex-

pressions, one for the transfer current before, and one after, the potential step. The

concentrations in the Butler-Volmer terms remain the same before and after the po-

tential step because of the first two assumptions, and we may determine the expected

change in the transfer current, ∆in

(A.cm−2), throughout the cathode based on the

experimental data. This yields the following expression, namely,

∆in

i00=

(

CMn3+

C0Mn3+

)(

COH−

C0OH−

)

exp

[

(1 − αc) F

RgasT(ηp − ∆Ecell + Υ)

]

−(

CMn4+

C0Mn4+

)(

CH2O

C0H2O

)

exp

[−αcF

RgasT(ηp − ∆Ecell + Υ)

]

−(

CMn3+

C0Mn3+

)(

COH−

C0OH−

)

exp

[

(1 − αc)F

RgasT(ηp + Υ)

]

+

(

CMn4+

C0Mn4+

)(

CH2O

C0H2O

)

exp

[−αcF

RgasT(ηp + Υ)

]

, (4.7)


CMn4+ (mol.cm-3)

Υ(V

)

0 0.02 0.040

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Figure 4.6: A polynomial representation of Υ based on fitting the Butler-Volmer equation toexperimental 5 mV/hr SPECS discharge data

where ∆Ecell (V) is the size of the potential step, ηp (V) is the average particle scale

overpotential of the cell before the potential step, and each (non-reference) concentra-

tion variable is calculated by volume averaging its distribution over the whole cathode.

Solving Equation (4.7) for Υ at each potential step, we obtain a number of estimates

at different Mn4+ concentrations.

A polynomial approximation for Υ (CMn4+), based on the above process is shown in

Figure 4.6. The maximum and minimum power outputs for the corresponding 5 mV/hr

SPECS simulation are shown in Figure 4.7. The previously introduced experimental

5 mV/hr SPECS results are also shown in the figure. The model output does display

two prominent peaks, one at 1.52 V, and another at 1.3 V. However, overall, the model

output does not correspond well with the experimental data.

It should be noted that due to the very interconnected nature of the phenomena that

govern cathode discharge, the parameter values used in the model simulations impact,

sometimes significantly, on the prediction of Υ, when determined in the above manner.

For example we found that changing the value of the initial exchange current density,

i00, significantly affected the predicted Υ term. This behaviour makes it difficult to


Ecell (V)

Min

and

Max

Pow

er(m

W.g

-1of

EM

D)

0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

2

4

6

8

10

12

Figure 4.7: A comparison of the minimum and maximum power of a simulated 5 mV/hrSPECS discharge (#), obtained using the Υ depicted in Figure 4.6, with the correspondingexperimental data ().

extract useful predictions from the experimental data in the above manner, because

some cell parameters are not measured and there is some level of uncertainty in the

values of these parameters, and using wrong values in the model creates differences

between experimental and simulated cell operation that should not be attributed to Υ.

It is important to note, however, that the above analysis was not futile as it facilitates

two crucial observations in linking the form of Υ to the observed multi-reaction dis-

charge behaviour and the successful simulation of SPECS discharge. The first is that

plateaus in the ion-ion interaction term, for example those observed at Mn4+ concen-

trations of 0.042 and 0.012 mol.cm−3, correspond to peaks in the SPECS discharge at

1.52 and 1.3 V, respectively. We note that plateaus at high Mn4+ concentrations are

reached earlier in reduction and their effects appear in simulated SPECS discharges

at higher voltages than plateaus at low Mn4+ concentrations. The second is that the

width of each plateau corresponds to the size of the predicted peak in the simulated

SPECS discharge.

Given the above observations we attempted to modify the polynomial approximation

4.5 Results and Discussion 73

shown in Figure 4.6 in order to obtain improved SPECS simulations, however in doing

so it became evident that a polynomial cannot adequately represent the information

required. The oscillations found in higher order polynomials limit the amount of infor-

mation that can be represented. Upon investigation, the form of the ion-ion interaction

term that we choose as being better in representing the essential features of Υ that con-

vey accurate multi-reaction behaviour in the SPECS discharge simulations is the sum

of several inverse tan functions, namely,

Υ (CMn4+) =n∑

i=1

hi

π

[

arctan(

si

(

CMn4+ − CMn4+,i

))

− arctan(

si

(

C0Mn4+ − CMn4+,i

))]

,

(4.8)

where hi (V) controls the magnitude of the arctan terms, CMn4+,i (mol.cm−3) denotes

the approximate Mn4+ concentration at which the corresponding plateau occurs, and

si controls the slope of the arctan function and how quickly it flattens off to create

a plateau. The second arctan function ensures that the value of Υ(

C0Mn4+

)

is zero.

Equation (4.8) is able to naturally represent each plateau with a single term in the sum,

and does not cause unwanted numerical oscillations in our model output. In practice, a

satisfactory form for Υ can be obtained using only three terms in the above sum. Such

a form is shown in Figure 4.8. A corresponding 5 mV/hr SPECS discharge using this

ion-ion interaction term is compared to the relevant experimental data in Figure 4.9.

We note that the model output shows a main peak at 1.29 V, with a secondary peak

or shoulder at 1.45 V. These correspond well with the position, width and magnitude

of the peaks in the experimental data.

The values of the initial exchange current density and the diffusion coefficient of H+ in

EMD are also very important for obtaining the agreement seen in Figure 4.9. These

parameters have distinct, yet interconnected, influences on discharge behaviour. The

effects of these two parameters are discussed in the following two sections.

4.5 Results and Discussion

Here we present and discuss the results of the modelling work. In Sections 4.5.1

and 4.5.2 we discuss the effects two key parameters have on the simulation of SPECS


CMn4+ (mol.cm-3)

Υ(V

)

0 0.02 0.040

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Figure 4.8: The non-linear form (♦) of the ion-ion interaction term used to produce thesimulated SPECS discharge in Figure 4.9 compared to the linear approximation () used inChapter 3.

Ecell (V)

Min

and

Max

Pow

er(m

W.g

-1of

EM

D)

0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

2

4

6

8

10

12

Figure 4.9: A comparison of the minimum and maximum power of a simulated 5 mV/hrSPECS discharge (♦) using the form of Υ given in Figure 4.8, with experimental data ().


tests.

4.5.1 The Initial Exchange Current Density

The initial exchange current density, i00 (A.cm−2), is found at the front of the Butler-

Volmer like expression (see Equation (4.1)). It describes the facility of charge transfer

at the EMD/KOH interface (Bard & Faulkner 2001). It directly affects the size of

the response of the current to changes in the potential and chemical concentrations

involved in the reduction of EMD. However, we note that a representative i00 is difficult

to obtain experimentally.

The effect of different i00 values on the current response, over three potential steps in

a 5 mV/hr SPECS simulation, is displayed in Figure 4.10, where the model output is

also compared with the relevant experimental data. We observe that the value of i00

significantly affects the current spike at each step in the potential. For small values

of i00, for example 5×10−9 A.cm−2, the resulting current spikes are small, while for

larger values of i00, for example 5×10−7 A.cm−2, the current response is much more

pronounced. In addition to this, the initial exchange current density also affects the rate

of relaxation. This is less intuitive than its effect on the initial current spike, however, it

may be explained by considering the crystal scale. When i00 is small, protons are inserted

at the surface of the EMD crystals at a slow rate, and are able to be transported away

from the crystal surface faster than they are inserted. This corresponds to a situation

that is kinetically limited, and leads to a very even, or flat, current response. For larger

values of i00, protons are able to be inserted into the EMD crystals faster than they can

diffuse from the surface. This corresponds to a situation where the process is diffusion

limited, and leads to larger current responses that diminish quickly. Based on this,

we observe that the two 5 mV/hr SPECS simulations with i00 values of 5×10−8 and

5×10−7 A.cm−2 are both diffusion limited. Interestingly, the experimental data seems

to match the model predictions for an i00 value of 5×10−7 A.cm−2 for the first half

of each potentiostatic discharge, and seems to match the model predictions for an i00

value of 5×10−9 A.cm−2 for the remainder of each potentiostatic discharge. This would

suggest that there are multiple i00 values in the experimental data. This is consistent

with our understanding of the multi-reaction reduction process, and may be why a


Time (h)

Cur

rent

(A)

49 50 51 520

0.001

0.002

0.003

0.004

0.005

Figure 4.10: A comparison of the current spike shape of several simulated 5 mV/hr SPECStests with i00 values of 5×10−9 ( ), 5×10−8 ( ) and 5×10−7 ( ) A.cm−2, with experimentaldata ().

better fit was not obtained using a single i00 value.

The effect of i00 on overall discharge behaviour for simulated 5 mV/hr SPECS discharges

is shown in Figure 4.11. The fluctuations seen in the data corresponding to the model

output for an i00 value of 5×10−7 A.cm−2 are caused by the difficulty the model has

in numerically capturing the extremely thin current spikes observed at large i00 values.

The primary effect of i00, displayed in Figure 4.11, is that increasing its value increases

the difference between minimum and maximum power. This is consistent with our

observations of the current response displayed in Figure 4.10. However, this effect is

diminished near the end of discharge, below approximately 1.15 V, because the EMD is

almost completely reduced and there is simply not enough there to produce a noticeable

current response.

Revisiting Figure 4.9, and considering in particular the comparison of the experimental

data with the model output using the form of Υ shown in Figure 4.8, we see that the

shoulder peak at 1.45 V in the experimental data has a larger difference between the

minimum and maximum power than it does at the main peak at 1.29 V. In the context


Ecell (V)

Min

and

Max

Pow

er(m

W.g

-1of

EM

D)

0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

5

10

15

20

Figure 4.11: A comparison of the minimum and maximum power of simulated 5 mV/hr SPECSdischarges with i00 values of 5×10−9 (♦), 5×10−8 (#) and 5×10−7 () A.cm−2.

of a multi-reaction reduction process, this suggests that the process occurring at 1.45

V has a larger i00 than the main process. This effect is unable to be reproduced using

our model, but it may be possible with a “variable” i00 which is dependent on CMn4+ .

Using this, i00 could have a higher value at the first peak, and a lower value at the main

peak.

4.5.2 The Diffusion Coefficient of Protons

The diffusion coefficient of protons in EMD crystals, DH+ (cm2.s−1), is found in Equa-

tion (3.29). It is a measure of the ability of inserted protons to move within the EMD

crystals, and more to the point, how fast these protons are able to vacate the reaction

sites at the EMD surface. Figure 4.12 shows the effect DH+ has on three individ-

ual current spikes in a 5 mV/hr SPECS simulation. We observe, as expected, that

changes in DH+ have little to no effect on the initial current spike height. However,

DH+ greatly influences the relaxation response. For small DH+ values, for example

1×10−17 cm2.s−1, the current almost immediately decreases to below half of its initial

value at the potential step. Following this, the current seems to maintain a steady


Time (h)

Cur

rent

(A)

49 50 51 520

0.001

0.002

0.003

Figure 4.12: A comparison of the current spike shape of simulated 5 mV/hr SPECS dischargewith DH+ values of 1×10−17 ( ), 1×10−16 ( ) and 1×10−15 ( ) cm2.s−1, with experimentaldata.

response. This corresponds to a situation where the process is diffusion limited. For

larger DH+ values, for example 1×10−15 cm2.s−1, the current does not experience a

large immediate decrease, but rather a gradual decline. This corresponds to a process

which is kinetically limited. We see that the amount of EMD utilised in a certain

time-frame (which is proportional to the area under the current response) is dependent

upon the value of DH+ . This is especially true when diffusion is the limiting process,

as seen in Figure 4.12, when DH+ takes the values 1×10−16 and 1×10−17 cm2.s−1. We

observe that a small DH+ value of 1×10−17 cm2.s−1 allows less EMD to be utilised than

observed when DH+ has the value 1×10−16 cm2.s−1. This is expected because DH+

directly influences the availability of reaction sites at the EMD surface.

The SPECS simulation shown in Figure 4.12 with a DH+ value of 1×10−16 cm2.s−1

seems to match the experimental data for the beginning of each potentiostatic dis-

charge, but none of the simulations match the experimental data at the end of each

potentiostatic discharge. The inability of the model to accurately predict the current

relaxation curve may be evidence for a variable DH+ . We note that this is consistent

with the multi-reaction reduction process of EMD.


Ecell (V)

Min

and

Max

Pow

er(m

W.g

-1of

EM

D)

0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

1

2

3

4

5

6

7

8

9

10

11

12

Figure 4.13: A comparison of the minimum and maximum power of simulated 5 mV/hr SPECSdischarge with DH+ values of 1×10−17 (♦), 1×10−16 (#) and 1×10−15 () cm2.s−1.

Figure 4.13 shows the effect of DH+ on overall discharge behaviour. We previously

observed in Figure 4.12 that DH+ primarily affects the current relaxation response, and

not the minimum and maximum current. This is seen to be true in the overall discharge

until approximately 1.25 V. Below this voltage, the simulated SPECS discharge with

a DH+ value of 1×10−17 cm2.s−1 has a larger power response. This difference occurs

because the cathodes with larger DH+ values have utilised, or exhausted, most of their

EMD, and cannot produce a sizeable current response.

It is thought that the shape of the minimum power response is evidence of a variable

DH+ (Delta 2005) because the current flowing at the end of each potentiostatic dis-

charge is thought to be related to DH+ . In our model framework, this is true, but the

relationship between the minimum power and DH+ seems to only appear later in the

SPECS simulations, below aprroximately 1.25 V, as noted above. In fact, the variable

Υ has a greater effect on the shape of the minimum power response. Furthermore, our

results clearly show that it is possible that a constant DH+ can produce a curve shape

that is consistent with the relevant experimental data.


4.6 Conclusions

A novel model for the potentiostatic discharge of primary alkaline battery cathodes

has been presented. The model has been solved using a MATLAB R© program that

can be run on a desktop computer. The model has been used to simulate SPECS

discharges on button cell cathodes, and the results have been validated and compared

with experimental data. We note that a potentiostatic model for primary alkaline

battery discharge, let alone one that can accurately predict SPECS discharge, has not

been (to the best of the author’s knowledge) presented and/or solved previously in the

literature.

The modified kinetics single-reaction framework proposed by Chabre & Pannetier

(1995) has been adopted in this model, in order to investigate the multi-reaction be-

haviour of the reduction of EMD as observed in SPECS experiments. We find that the

linear form of the ion-ion interaction term worked well in the previous chapter because

the multi-reaction behaviour of the reduction of EMD is not emphasised under gal-

vanostatic discharge. A single-reaction model framework can simulate multi-reaction

behaviour. Furthermore, we find that the initial exchange current density primarily

affects the height of the current spike produced when the potential is stepped, but also

influences the relaxation of the cell. Moreover, it seems that for the two distinct peaks

in the SPECS tests, the individual reactions have different initial exchange current

densities. We also find that the diffusion coefficient of protons in EMD, as interpreted

by the model framework, does not significantly affect the height of the current spikes.

Instead, its primary influence is to change the shape of the current relaxation curve.

Furthermore, a small value of DH+ decreases the utilisation of EMD, even at the low

discharge rates encountered in SPECS discharges. Moreover, we have found that the

minimum power response may be influenced by other variables, most notably the ion-

ion interaction term, and its shape is not necessarily determined only by a variable

DH+ .

CHAPTER 5

The Precipitation Model

5.1 Introduction

The premature failure of primary alkaline batteries due to the plugging of pores in the

anode has been modelled by Sunu (1978) and Chen & Cheh (1993b), and the plugging

of pores in the anode and separator has been modelled by Podlaha & Cheh (1994a) and

subsequent papers. However, the formation of a zinc oxide (ZnO) phase in the separator

would suggest that an internal short circuit could be created. The conductivity of bulk

ZnO is 0.01 S.cm−1 (Podlaha & Cheh 1994a), and thus it may reasonably be expected

to conduct electrons when connections are made between the anode and cathode. This

phenomena and its effect on primary alkaline battery discharge have not been modelled

previously.

A mechanism for cell failure due to short circuit based on precipitation of ZnO in the

separator is proposed. A model for the discharge of a primary alkaline battery that

accounts for internal short circuits based on the proposed mechanism is presented. The

model includes a description of the cathode similar to that developed in Chapters 3

and 4, and employs a simplified description of the anode. The separator paper is

82 Chapter 5. The Precipitation Model

modelled as a porous insulator, with possible short circuits, depending on whether an

electronic connection between the anode and cathode is made by solid ZnO.

The model is presented in Section 5.2. The numerical solution method used to construct

the numerical simulation software, including a description of the discretisation method,

is outlined in Section 5.3. The results of the model simulations are presented and

discussed in Section 5.4, followed by conclusions in Section 5.5.


The following model describes the discharge of a primary alkaline battery, with a ZnO

internal short circuit failure mechanism. The variables that are determined by the

model are the concentrations of OH− and Zn(OH)2−4 ions, the void volume fraction,

the ZnO volume fraction, the solution phase current distribution within the anode,

separator and cathode, the volume average velocity of the solution within the cathode

and separator, the concentration of Mn4+ ions in the EMD crystals, the volume fraction

of Zn in the anode, and the overpotential and tranfer current due to electrochemical

reactions in the anode and cathode.

Firstly, the assumptions adopted to facilitate the specification of the governing equa-

tions and boundary and initial conditions are presented, and then the derivations of

equations that differ from, or are additional to, those presented in Chapter 4 are pre-

sented. The equations that have not changed since Chapter 4 are also presented for

completeness.

5.2.1 Model Assumptions

Several assumptions are made to facilitate and simplify the description of the cell and

to arrive at the model equations. Many of these assumptions are the same as in the

two models previously presented in this thesis, but are extended from the cathode to

the full cell. These include the assumptions that discharge is isothermal. In addition, it

is assumed that electronuetrality is maintained in the cell, that all void volume within

the cell is filled with liquid electrolyte. The cathode is assumed to be a three-scale


electrode (as previously modelled in Chapters 3 and 4) (see Figure 3.1 in Chapter 3),

where the solid phase is a porous mixture of graphite and EMD particles. The EMD

particles are themselves assumed to be porous agglomerates of spherical, solid, EMD

crystals. The EMD crystals and EMD particles have uniform size and do not change

volume or swell during reduction. The EMD is reduced at the surface of the EMD

crystals according to the electron-proton mechanism given by Reaction (2.1), namely,

MnO2 + H2O + e− MnOOH + OH−. (5.1)

The transfer current is given by the modified Butler-Volmer expression with the non-

linear ion-ion interaction term developed in Chapter 4 (see Figure 4.8). We assume

that there are no solid phase potential losses on the cathodic scale (as in Chapters3

and 4) since we assume that the graphite is a good conductor, and connects all EMD

particles to the cathodic current collector.

Within the anode, the solid phase is assumed to be a uniform porous agglomeration of

solid zinc particles, flooded with liquid electrolyte. The anode is assumed to discharge

uniformly and spatial gradients of the solid and solution phase species are ignored. The

anode is oxidised at the surface of the zinc via a dissolution-precipitation mechanism

given by Reactions (2.14) and (2.15), namely,

Zn + 4OH− Zn(OH)2−4 + 2e−, (5.2)

and

Zn(OH)2−4 ZnO + H2O + 2OH−, (5.3)

as modelled by Sunu (1978) and Podlaha & Cheh (1994a). Furthermore, the zinc is

assumed to be a good conductor, with no solid phase potential losses in the anode.

It is assumed that the full cell has a fixed volume, and the volume changes for the

cathodic and anodic electrochemical reactions are equal and opposite. In addition, it

is assumed that there is no overall volume change due to ZnO precipitation. That is,

ZnO precipitation affects the fraction of solid phase within the cell, however, the total

cell volume is unchanged.


In this work we extend the treatment of the solution phase beyond that given in Chap-

ters 3 and 4 to include a ternary electrolyte consisting of K+, OH− and Zn(OH)2−4

ions, and H2O solvent molecules. The Zn(OH)2−4 ions are assumed to be unconfined

throughout the cell. To avoid the numerical artefacts in the transfer current and cell

potential experienced by Kriegsmann & Cheh (1999c, d) (refer to the review of these

articles in Chapter 2, Section 2.7), we do not assume that the transference numbers

are constant, but rather a function of electrolyte concentration. The link between

the numerical fluctuations and the transference numbers is discussed in Section 5.2.5,

where we also derive specific forms for the variable transference numbers. In addition,

Zn(OH)2−4 may precipitate out of solution as ZnO in all compartments of the cell. We

do note however, that in the cathodic compartment, ZnO precipitation is limited to

the space between porous EMD particles rather than within them. Furthermore, in

addition to precipitation, the dissolution of ZnO (given by the reverse of Reaction 5.3)

is allowed to occur in all compartments of the cell.

The (solid phase of the) separator paper is assumed to be a nonconducting porous

material. The total solid phase conductivity of the separator (including ZnO) is based

on the volume fraction of ZnO. It is proposed that when the ZnO phase reaches a mini-

mum conductivity at all positions within the separator paper, an electronic connection

is made, and current may flow through it between the anode and cathode solid phases,

with the flow of current being directly related to the potential gradient in the ZnO via

Ohm’s law.

With the expansion of the previous cathodic models to include a ternary electrolyte,

several of the modelling equations introduced in the earlier Chapters will need to red-

erived or specified here.

As this model incorporates the anode and separator, as well as the cathode, the nomen-

clature has been changed from that found in Chapters 3 and 4. To distinguish between

variables in the anode, separator and cathode the subscripts a, s and c are used, respec-

tively. To designate the particle size scale in the cathode, subscript, p, is employed.

The radii of the interfaces between the anode and the separator, and the separator and

the cathode are denoted by Ras and Rsc, respectively, and the radii of the anodic and


cathodic current collectors are denoted by Rai and Rco, respectively.

5.2.2 The Cathode

The cathode is modelled using the same equations as developed in the previous two

chapters, but to account for the effects of a ternary electrolyte, changes are be made to

the mass conservation equations, the overpotential equation, and the volume conserva-

tion equation. In addition, we include an equation to govern the changes in porosity

due to precipitation. Furthermore, the boundary conditions are changed to account for

the separator.

We may simplify the ternary electrolyte equations on the cathodic and EMD particle

size scales in the same manner as the binary electrolyte mass conservation equations are

simplfied in Appendix C. In doing so, we find that spatial variations in the electrolyte

on the particle scale are negligible and that the cathode scale OH− and Zn(OH)2−4

concentrations are governed by the equations,

∂

∂t

(

ε∼c +εEMD

εEMDpε∼p

)

COH−c

= ∇ ·

DOH−∞ε∼c∇COH−c − COH−cvc

+tOH−

i∼c

F

+ 2ks

(

Ac +εEMD

εEMDpAp

)

(CKZnc − CKZnc,eq) −3εEMD

εEMDproFi∼p|r=ro

(5.4)

and

∂

∂t

(

ε∼c +εEMD

εEMDpε∼p

)

CKZnc

= ∇ ·

DZn(OH)2−4 ∞ε∼c∇CKZnc − CKZncvc

+tZn(OH)2−4

i∼c

2F

− ks

(

Ac +εEMD

εEMDpAp

)

(CKZnc − CKZnc,eq) , (5.5)

respectively. We note that Equation (5.4) is the ternary analogue of Equation (3.36) in

Table 4.1. The above equations include a source term (2nd term on the RHS) similar

to that proposed by Kriegsmann & Cheh (1999d), that accounts for the precipitation

of ZnO at points in the solution for which the Zn(OH)2−4 concentration exceeds the

equilibrium concentration, CKZnc,eq (mol.cm−3). Unlike Kriegsmann & Cheh however,

we also account for the dissolution of precipitated ZnO when the Zn(OH)2−4 concentra-

tion is lower than the equilibrium concentration. The geometric surface area per total


unit volume available for the precipitation or dissolution of ZnO within the cathode,

Ac (cm−1), is dependent upon the amount of already precipitated ZnO, and is given

by,

Ac =3εEMD

εEMDproδ (ε∼c)

√

ε∼c

ε0∼c

. (5.6)

Here, δ (ε∼c) stops the dissolution of ZnO when it is already fully dissolved. The

boundary conditions on the cathodic scale reflect a zero flux of electrolyte at the cath-

ode/current collector boundary, and the continuity of electrolyte flux and concentration

at the cathode/separator boundary, namely,

∇COH−c|R=Rco= 0, (5.7)

∇CKZnc|R=Rco= 0, (5.8)

DOH−∞ε∼c∇COH−c|R=Rsc= DOH−∞ε∼s∇COH−s|R=Rsc

, (5.9)

DZn(OH)2−4 ∞ε∼c∇CKZnc

∣

∣

∣

R=Rsc

= DZn(OH)2−4 ∞ε∼s∇CKZns

∣

∣

∣

R=Rsc

, (5.10)

COH−c|R=Rsc= COH−s|R=Rsc

, (5.11)

and

CKZnc|R=Rsc= CKZns|R=Rsc

. (5.12)

The initial conditions for the OH− and Zn(OH)2−4 ion concentrations are,

COH−c|t=0 = C0OH−c

, (5.13)

and

CKZnc|t=0 = C0KZnc. (5.14)

The evolution of cathodic porosity (void volume between particles in the cathode) is

given by (Podlaha & Cheh 1994a),

∂ε∼c

∂t= −V ZnOAcks (CKZnc − CKZnc,eq) . (5.15)

This implies that ε∼c is a variable of both R and t. Similarly, the evolution of EMD


particle porosity (void volume within EMD particles in the cathode) is given by,

∂ε∼p

∂t= −V ZnOApks (CKZnc − CKZnc,eq) . (5.16)

The Zn(OH)2−4 and equilibrium concentrations of the precipitation reaction are written

in terms of the cathodic scale because we assume, asymptotically, that there are no

concentration gradients within the EMD particles. Furthermore, the electrolyte con-

centrations within the particles are the same as those immediately outside the particles

on the cathode scale (see Appendix C). This implies that ε∼p is a variable of both R

and t, but not r. The initial cathode and particle porosities are specified, namely,

ε∼c|t=0 = ε0∼c, (5.17)

and

ε∼p|t=0 = ε0∼p. (5.18)

The ternary analogue of the overpotential equation (3.37) in Table 4.1 on the cathodic

scale must now include an additional concentration polarisation term due to the pres-

ence of Zn(OH)2−4 , and is given by (Podlaha & Cheh 1994a),

∇ηc =i∼c

√

ε3∼cκ∼∞

+RgT

F

3

CKZnc

CH2Oc−

tZn(OH)2−4

2

∇ ln aKZnc

+2

(

1 − tOH− +

COH−c

CH2Oc

)

∇ ln aKOHc

]

. (5.19)

The boundary condition on the overpotential is given by,

ηc|R=Rsc= φs|R=Rsc

− φ∼s|R=Rsc− E0

c , (5.20)

where E0c (V) is the equilibrium potential for the cathodic reaction. E0

c is assumed to

be measured with reference to the zinc electrode, and thus, is equivalent to the open

circuit voltage.

The ternary dimensional equivalent to the conservation of volume equation (3.20) is


given by,

∇ · vc =

3εEMD

εEMDproF

tZn(OH)2−4

2V KZn −

(

1 − tOH−

)

V KOH + V H2O

×

i∼p|r=ro. (5.21)

The boundary condition on the volume average velocity specifies zero flux at the ca-

thodic current collector, namely,

vc

∣

∣

∣

R=Rco

= 0. (5.22)

The remainder of the equations are unchanged since Chapter 4 by the assumptions made

in this model, however, they are presented here for clarity. The equation governing the

crystal scale Mn4+ ion concentration is

CMn4+ = C0Mn4+ +

3

Fyo

t∫

t∗=0

in

dt∗+

2

Fyo

∞∑

m=1

sin(

λmyyo

)

sin (λm)

t∫

t∗=0

in|t∗=t−t∗ exp

[−λ2mDH+t∗

y2o

]

dt∗. (5.23)

The transfer current is given by Equation (4.1) in Chapter 4, namely,

in

= i0c0

(

CMn3+

C0Mn3+

)(

COH−

C0OH−

)

exp

[

(1 − αc) F

RgasT(ηp + Υ)

]

−(

CMn4+

C0Mn4+

)(

CH2O

C0H2O

)

exp

[−αcF

RgasT(ηp + Υ)

]

. (5.24)

As mentioned earlier in the model assumptions, we use the non-linear ion-ion interaction

term, Υ (CMn4+) (V), found to best represent the experimental data, given in Figure 4.8

of Chapter 4.

The following cathode equations have not changed since Chapter 4. Current conserva-

tion on the EMD particle scale is given by,

∇ · i∼p = jp = ApεEMDpin. (5.25)


The overpotential on the EMD particle scale is given by,

∇ηp = i∼p

1

εEMDpσ0EMD∞

(

CMn4+ |y=0.8yo

C0Mn4+

)−XMn4+

+1

√

ε3∼pκ∼∞

. (5.26)

The boundary conditions of the particle scale overpotential specify symmetry at the

centre of the particles, and continuity with the cathode overpotential at the outside of

the particles, namely,

∂ηp

∂r

∣

∣

∣

∣

r=0

= 0, (5.27)

and

ηp|r=ro= η, (5.28)

respectively. Current conservation in the cathode is governed by,

∇ · i∼c =3εEMD

εEMDproi∼p|r=ro

, (5.29)

subject to the boundary condition,

i∼c|R=Rco= 0. (5.30)

5.2.3 The Separator

In modelling the separator paper, we include equations for the conservation of current,

the mass conservation of the electrolyte species, the volume average velocity of the

solution, the solid and solution phase potentials and changes in porosity.

The mass conservation equations for OH− and Zn(OH)2−4 ions in the separator are

similar to those given by Podlaha & Cheh (1994a), however, like their cathodic coun-

terparts, they contain the ZnO precipitation source term proposed by Kriegsmann &

Cheh (1999d). They are,

∂

∂t(ε∼sCOH−s) = ∇ · (DOH−∞ε∼s∇COH−s)

+ ∇ ·(

tOH−

Fi∼s − COH−sv

s

)

+ 2Asks (CKZns − CKZns,eq) , (5.31)


and

∂

∂t(ε∼sCKZns) = ∇ ·

(

DZn(OH)2−4 ∞ε∼s∇CKZns

)

+ ∇ ·

tZn(OH)2−4

2Fi∼s − CKZnsv

s

− Asks (CKZns − CKZns,eq) , (5.32)

respectively. The boundary conditions for these equations at the anode/separator in-

terface are derived in the next section (see Equations (5.56) to (5.59)). The initial

conditions for the OH− and Zn(OH)2−4 ion concentrations within the separator paper

are,

COH−s|t=0 = C0OH−s

, (5.33)

and

CKZns|t=0 = C0KZns. (5.34)

Porosity change within the separator is due to the precipitation of ZnO, and is analagous

to the cathodic porosity change equation (5.15), namely,

∂ε∼s

∂t= −V ZnOAsks (CKZns − CKZns,eq) . (5.35)

Here, the geometric surface area, per total unit volume of the separator, available for

the precipitation of ZnO, As, is dependent upon the amount of already precipitated

ZnO, namely,

As = A0sδ (ε∼s)

√

ε∼s

ε0∼s

. (5.36)

The initial separator porosity is specified, namely,

ε∼s|t=0 = ε0∼s. (5.37)

Since we assume that the precipitation of ZnO does not increase or decrease the overall

volume, the divergence of the volume average velocity is given by,

∇ · vs = 0. (5.38)


We specify continuity at the separator/cathode interface, namely,

vs

∣

∣

∣

R=Rsc

= vc

∣

∣

∣

R=Rsc

. (5.39)

As there is no electrochemical reaction within the separator, the divergence of the

solution phase current is zero (Podlaha & Cheh 1994a), namely,

∇ · i∼s = 0. (5.40)

Expressing the above in cylindrical coordinates, we may integrate to find that the total

solution phase current flowing through a certain radius within the separator is given

by

I∼s = 2πHRi∼s = 2πHRasi∼as = 2πHRsci∼sc. (5.41)

The short circuit current that can flow in the separator as a result of ZnO precipitation

in this compartment is governed by Ohm’s law, namely,

∇φs =−is

ε3/2ZnOσZnO∞

. (5.42)

Since, in general, charge is neither created nor destroyed, we can write that,

∇ · (i∼ + i) = 0, (5.43)

where i∼ and i are the solution and solid phase currents, respectively. This relation-

ship applies everywhere within the cell. In the separator (and the anode and cathode),

where the total current flowing in the radial direction is the discharge current, we find

that

i∼s + is =IR

2πHR, (5.44)

where I (A) is the external discharge current. Using Equation (5.44) we may eliminate

the solid phase current from Equation (5.42) to arrive at

∇φs =i∼s

ε3/2ZnOsσZnO∞

− IR

2πHRε3/2ZnOsσZnO∞

. (5.45)


The solution phase potential in the separator is influenced by ohmic losses and concen-

tration gradients in solution. The governing equation is given by,

∇φ∼s =−i∼s

κ− RgT

F

3

CKZns

CH2Os−

tZn(OH)2−4

2

∇ ln aKZns

+2

(

1 − tOH− +

COH−s

CH2Os

)

∇ ln aKOHs

]

. (5.46)

We apply continuity conditions to the potentials in both the solid and solution phases

at the anode/separator interface. Since we assume that the anode is uniform and has

no Ohmic losses, the solid and solution phase potentials in the anode are uniform, and

we may write the continuity conditions as,

φs|R=Ras= φa, (5.47)

and

φ∼s|R=Ras= φ∼a, (5.48)

for the solid and solution phase potentials in the separator, respectively. Similarly, we

apply continuity conditions to the potentials in the solid and solution phases at the

separator/cathode interface. Since we assume that there are no solid phase losses in

the cathode, we write,

φs|R=Rsc= φc, (5.49)

and

φ∼s|R=Rsc= φ∼c|R=Rsc

, (5.50)

for the solid and solution phase potentials in the separator, respectively. We may

simplify Equation (5.47) by choosing the value of the potential applied to the anodic

current collector to be zero, as it determined by our choice of an arbitrary reference

potential, namely,

φa = φs|R=Ras= 0. (5.51)

Given this reference potential, the potential of the cathodic current collector is the total

cell potential, Ecell.


The discharge mode provides an additional condition. If galvanostatic discharge is

desired, the condition is,

I = I(t), (5.52)

where I(t) (A) is the externally applied current. If potentiostatic discharge is desired,

the condition is,

φc = φs|R=Rsc= Ecell(t). (5.53)

where Ecell(t) (V) is the applied cell potential. We note that this is the second boundary

condition for the solid phase potential in the separator, which is governed by the first

order ordinary differential equation (5.45). This does not overdetermine the equation

system since I is able to change to satisfy the two boundary conditions. If constant

load discharge is desired, the condition is,

φs|R=Rsc− φs|R=Ras

I= Rload(t), (5.54)

where Rload(t) (Ω) is the applied load resistance. Finally, if constant power discharge

is desired, the condition is,

(

φs|R=Rsc− φs|R=Ras

)

I = Pload(t), (5.55)

where Pload(t) (W) is the required power. During the relaxation periods in intermittent

discharges, the condition (5.52) is used, and the desired current is set to zero.

5.2.4 The Anode

The anode is assumed to be of uniform composition throughout discharge, and is mod-

elled using a specially derived set of boundary conditions, similar to the approach taken

by Chen & Cheh (1993a). The boundary conditions are derived by assuming that there

is no spatial variation within the anode, and writing the equations as if averaged over

the whole anodic volume. This is done because our aim is not to accurately model the

anode, but to model the aforementioned phenomena in the separator without signifi-

cantly increasing the computational overhead of the numerical solution.


In modelling the anode, we include equations for the transfer current, the conservation

of current, the mass conservation of the electrolyte species and changes in porosity.

The bulk mass conservation equations for OH− and Zn(OH)2−4 ions in the anode are

given by,

∂

∂t(ε∼aVaCOH−a) = 2πRasH (DOH−∞ε∼s∇COH−s)|R=Ras

+

2πRasH

(

tOH−

Fi∼s − COH−sv

s

)∣

∣

∣

∣

∣

R=Ras

−

2Va

Fja + 2VaAaks (CKZna − CKZna,eq) (5.56)

and

∂

∂t(ε∼aVaCKZna) = 2πRasH

(

DZn(OH)2−4 ∞ε∼s∇CKZns

)∣

∣

∣

R=Ras

+

2πRasH

tZn(OH)2−4

2Fi∼s − CKZnsv

s

∣

∣

∣

∣

∣

∣

R=Ras

+

Va

2Fja − VaAaks (CKZna − CKZna,eq) , (5.57)

respectively, where again we note the inclusion of the ZnO precipitation term. The con-

centrations in the anode are linked to the separator by specifying continuity conditions

at the anode/separator interface, namely,

COH−s|R=Ras= COH−a, (5.58)

and

CKZns|R=Ras= CKZna. (5.59)

The above two continuity conditions also specify the initial concentrations of OH− and

Zn(OH)2−4 ions, through the initial conditions (5.33) and (5.34). Porosity change within

the anode is due to the precipitation of ZnO (as in Equations (5.15) and (5.35)) and the

dissolution of Zn via the electrochemical reaction (5.2) according to Equation (5.62),


and is governed by (Podlaha & Cheh 1994a),

∂ε∼a

∂t=

ja

2FV Zn − V ZnOAaks (CKZna − CKZna,eq) . (5.60)

The initial anodic porosity is specified, namely,

ε∼a|t=0 = ε0∼a. (5.61)

To determine the anodic transfer current density, ja (A.cm−3), corresponding to the

electrochemical reaction (5.2), we use the Butler-Volmer expression given by Podlaha

& Cheh (1994a), namely,

ja = Aai0a0

(

CeffKZn

C0KZn

)0.06(

CeffOH−

C0OH−

)2.59

exp

(

2 (1 − αa) F

RgasTηa

)

−

(

CeffKZn

C0KZn

)0.94(

CeffOH−

C0OH−

)−0.92

exp

(−2αaF

RgasTηa

)

, (5.62)

where the superscript “eff” denotes effective concentration at the surface of the zinc.

The surface concentrations are different from the bulk concentrations because it is

assumed that the electrolyte must diffuse through a ZnO layer. The diffusion of elec-

trolyte is assumed to proceed at a quasi-steady rate, and is specified by Sunu (1978)

and Podlaha & Cheh (1994a). The interfacial area per total unit volume in the anode,

Aa (cm−1), is

Aa = A0a

(

1 − ε∼a

1 − ε0∼a

)2/3

. (5.63)

The overpotential in the anode, ηa, is defined to be the difference between the potential

drop at the solid/solution interface and the equilibrium potential of the anodic reaction

at the initial state, namely,

ηa = φa − φ∼a − E0a . (5.64)

As noted earlier, E0a is zero because we assume that it is measured with respect to the

zinc electrode. We also note that ηa may be related to the solid and solution phase

potentials in the separator at R = Ras, by Equations (5.47) and (5.48).


The total current which enters the solution phase due to the electrochemical reac-

tion (5.2), is related to the transfer current density by,

I∼s = Vaja. (5.65)

where the current entering the solution phase in the anode must account for all of the

current flowing in the solution phase in the separator, I∼s, and Va (cm3) is the volume

of anode. We note that any short circuit current will increase I∼s above the external,

or applied, current, I.

5.2.5 Transference Numbers

The numerical fluctuations found in the ternary electrolyte model of Kriegsmann &

Cheh (1999c, d) were eliminated by these authors by reverting to a binary electrolyte

model (Kriegsmann & Cheh 2000). However, modelling the electrolyte as a ternary

system without numerical fluctuations should be possible. As noted by Kriegsmann

& Cheh (1999c), the fluctuations are related to the depletion of Zn(OH)2−4 within

the cathode. By examining the terms of the ternary electrolyte mass conservation

equation used in the cathode region of our model, given by Equation (5.5), we see

that the causes for changes in the Zn(OH)2−4 ion concentration are due to diffusion,

advection, precipitation and charge migration. At a given point in the cathode, if the

Zn(OH)2−4 is depleted or nearly depleted, diffusive flux will tend to bring Zn(OH)2−4

from nearby regions of higher Zn(OH)2−4 concentration. Furthermore, advective flux of

electrolyte is minimal in the primary alkaline battery system (Farrell & Please 2005),

and thus, we expect only a minimal decrease in the Zn(OH)2−4 concentration. At

low Zn(OH)2−4 concentrations and high OH− concentrations, any available ZnO will

be dissolved to increase the Zn(OH)2−4 concentration (according to Reaction (5.3)).

Migration, however, will tend to transport Zn(OH)2−4 ions from the cathode at all

times. If the transference numbers are constant, then when Zn(OH)2−4 is depleted and

the governing equations are switched from ternary to binary electrolyte, a discontinuity

in the transference numbers will be experienced. Accordingly, the fluctuations should

not occur if there is a continuous transition between ternary and binary transference


numbers. We therefore seek to derive forms for variable transference numbers that

make the appropriate ternary to binary transition.

The tranference number is the fraction of current carried by a particular ionic species

as a fraction of total current carried by the solution. For infinitely dilute electrolytes,

Newman (1991) gives the definition of the transference number of an ion i as

ti =κi∞κ∼∞

, (5.66)

where

κi∞ = z2i Ciui. (5.67)

Here κi∞ (S.cm−1) is the bulk conductivity of the ion i, κ∼∞ (S.cm−1) is the bulk

conductivity of the solution, zi is the charge of the ion i, and ui (S.cm2.mol−1) is the

mobility of the ion i. We note that Equation (5.66) applies in any concentration of

electrolyte. The solution conductivity, κ∼∞, is the sum of the conductivities of all ions

in solution, which is the reason that all the tranference numbers of a particular system

add to one.

All models reviewed in this thesis, and the models presented in Chapters 3 and 4 of this

thesis, assume that the transference numbers are constant. For a binary electrolyte,

this may be considered consistent because the ratio of the number of K+ ions to OH−

ions is constant. However, in the ternary electrolyte, if one of the ionic species, such

as Zn(OH)2−4 , becomes infinitely dilute, its transference number may be described by

Equations (5.66) and (5.67). It is then a linear function of its concentration relative

to the solution conductivity, and approaches zero. This way, if Zn(OH)2−4 is nearly

depleted, and then becomes depleted, the transference number is not forced to change

discontinuously. Thus, we propose a more consistent variable tranference number defi-

nition using Equations (5.66) and (5.67).

Data for concentrated ternary KOH/K2Zn(OH)4 electrolyte is not readily available

(Kriegsmann & Cheh 1999d). However, Bennion (1964) measured the conductivity of

KOH electrolyte saturated with ZnO. Here, we assume that the Zn(OH)2−4 is infinitely


dilute, and apply Equation (5.67). We assume that the mobility is constant, and deter-

mine an approximation of the mobility of Zn(OH)2−4 ions in 9 M KOH by comparing

the two conductivity functions of See & White (1997) and Bennion (1964).

The determination of the transference numbers requires knowledge of their respective

ionic conductivities (see Equation (5.66)). We may calculate approximations of the

ionic conductivities of OH− and K+ ions if we assume that they have a similar ionic

conductivity in binary or ternary electrolyte, and thus, we use the data of See & White

for binary electrolyte, namely,

κOH−∞ = t⊙OH−

κKOH∞ (5.68)

and

κK+∞ = t⊙K+ κKOH∞. (5.69)

Here, κKOH∞ (S.cm−1) is the conductivity of KOH electrolyte as measured by See

& White, and t⊙OH−

and t⊙K+ are the constant transference numbers of OH− and K+

in KOH electrolyte (they are given the values 0.78 and 0.22, respectively, as used in

Chapters 3 and 4). We obtain a reference value for the ionic conductivity of Zn(OH)2−4

by calculating the difference between the conductivity of the ternary electrolyte and

the conductivy of the binary electrolyte at 9 M KOH. We relate this to the mobility of

Zn(OH)2−4 , uZn(OH)2−4, using Equation (5.67) (the saturation concentration of Zn(OH)2−4

in 9 M KOH is 1.0 M). The mobility of Zn(OH)2−4 is found to be approximately 10.1

S.cm2.mol−1. The expression for the transference number of Zn(OH)2−4 then becomes,

tZn(OH)2−4

=4uZn(OH)2−4

CKZn

κ∼∞(5.70)

where the liquid phase conductivity, κ∼∞, is the sum of the ionic conductivities, namely,

κ∼∞ = t⊙OH−

κKOH∞ + t⊙K+ κKOH∞ + 4uZn(OH)2−4

CKZn. (5.71)

Importantly, as the concentration of Zn(OH)2−4 ions approaches zero in the cathode (or

indeed, in any region), using Expression (5.70) will allow a smooth transition between

5.3 The Numerical Solution 99

ternary and binary electrolyte.

Equations (5.4)-(5.39), (5.41) and (5.45) to (5.71) represent a model of primary alkaline

battery discharge including precipitation of ZnO and internal short circuits, and will

be referred to as the precipitation model hereafter.

5.3 The Numerical Solution

The numerical simulation used to solve the precipitation model is similar to that used

to solve the simplified and potentiostatic models in Chapters 3 and 4, respectively. The

additional equations used to model the separator and anode are discretised in a similar

way as those for the cathode.

The model equations are discretised using a combination of linearisation and fixed-

point techniques, which creates a linear system of equations which we solve in matrix

form. However, Equations (5.15), (5.16) and (5.35), which govern the porosity changes

in the separator and cathode, are not included in the matrix, but are used to update

the solution inbetween iterations.

The time step is chosen in a similar way as previously done, however, the amount of

cathode active material remaining is incorporated into the adaptive time stepping al-

gorithm. When the active material decreases below a minimum amount, the maximum

time step able to be chosen is decreased, because this signals the rapid decrease in cell

potential (Farrell & Please 2005).

The changing conductivity of the solid phase in the separator necessitates a change

in the numerical solution to account for this. When the solid phase conductivity is

below a specified minimum, it is assumed that there is no electrical connection, and the

internal short circuit current is set to zero. Only when an electronic connection is made

across the separator, are the discretised equations governing solid phase potential in the

separator included in the matrix (which contains the linearised system of equations).


Table 5.1: Parameter values and cell geometry used to simulate AA-cell discharge.

Parameter Value

Load resistance (Ω) 3.3Particle radius (µm) 25Particle porosity, ε∼p 0.1066

Effective crystal radius, yo (cm) 210 × 10−8

Diffusion coefficient of protons, DH+ (cm2.s−1) 1 × 10−16

EMD conductivity coefficient, σ0EMD∞ (S.cm−1) 150

EMD conductivity exponent, XMn4+ 4.328ZnO conductivity, σZnO∞ (S.cm−1) 0.01 (Sunu 1978)Anode/separator radius, Ras (cm) 0.43

Separator/cathode radius, Rsc (cm) 0.45Outer cathode radius, Rco (cm) 0.67

Height, H (cm) 4.04Total mass of cathode (g) 10.62

Mass of EMD in cathode (g) 9.24Mass of graphite in cathode (g) 0.8Initial separator porosity, ε0

∼s 0.8Initial anode porosity, ε0

∼a 0.74initial zinc volume fraction 0.251

Initial OH− ion concentration, C0OH−

(mol.cm−3) 0.009

Initial Zn(OH)2−4 ion concentration, C0KZn (mol.cm−3) 8.7641 × 10−4

Diffusion coefficient of Zn(OH)2−4 , DZn(OH)2−4 ∞ (cm2.s−1) 6.9 × 10−6 (Sunu 1978)

Initial zero current potential (V) 1.65Initial cathodic exchange current density, i0c0 (A.cm−2) 5 × 10−8

Initial anodic exchange current density, i0a0 (A.cm−2) 0.06 (Sunu 1978)Initial anodic interfacial area, A0

a (cm−1) 50 (Sunu 1978)

5.4 Results and Discussions

Here we present and discuss the results of the modelling work. In Sections 5.4.1 to 5.4.3

we discuss the effects of three key parameters on the simulation of primary alkaline

battery discharges. The parameters investigated are the bulk ZnO conductivity, the

initial KOH concentration, and the separator thickness. All simulations presented here

use the parameter values given in Table 5.1, which describe AA-cell geometry, unless

otherwise noted.

Figure 5.1 shows the external discharge current and the total discharge current of two

AA-cells discharged through 3.3 and 6.6 Ω loads. The external discharge current is

assumed to be the externally applied current which flows through the load resistance,

while the total discharge current is assumed to be the sum of the external current

and the internal self-discharge current. We note that the difference between the total


Time (h)

Cur

rent

(A)

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure 5.1: Comparison of self-discharge of two AA-cells under 3.3 () and 6.6 (#) Ω loads.Shown is the external cell discharge current (hollow symbols, ) and the total current (filledsymbols, ) (including short circuit current). The configuration of the cell is given in Table 5.1

and external current (as seen in the figure) essentially represents the self-discharge (or

short circuit) current. The amount of capacity lost to self-discharge does not seem

to be greatly influenced by the discharge rate. However, in both cases self-discharge

occurs at the beginning (of discharge). This is confirmed by examining the volume

fraction of the ZnO within the separator. Figure 5.2 shows the volume fraction of ZnO

within the separator for several times during the 3.3 Ω discharge. The formation of a

continuous ZnO phase within the separator occurs at very early times, and is dissolved

during the operation of the cell. As the ZnO phase grows in size, it allows an increasing

self-discharge current. However, after the first hour of discharge, the OH− ions from the

cathode begin to dissolve the ZnO in the separator at the separator/cathode boundary

(R = 0.45 cm), eventually severing the electrical connection, halting self-discharge and

not allowing it to reoccur later.

Interestingly, the information shown in Figures 5.1 and 5.2 suggest that the reduction

in overall cell capacity, shortening battery life, is a result of self-discharge that occurs


Radius (cm)

ε ZnO

0.435 0.44 0.445 0.450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure 5.2: ZnO volume fraction within the separator of an AA-cell under a 3.3 Ω load. TheZnO volume fraction is shown for the times 0 (#), 15 (), 30 (♦), 45 (), 60 (), 120 (⊲) and180 (⊳) minutes.

during the initial moments of cell discharge, rather than as a result of a short circuit

event which occurs just prior to cell failure.

During intermittent discharges, self-discharge is seen to occur repeatedly. Figure 5.3

shows the external and total discharge current of an AA-cell undergoing a simulated

3.3 Ω constant load intermittent discharge. The load is applied for 4 minutes at the

beginning of each hour, for 8 hours at the beginning of each 24 hour period. Only the

first 8 hour period of this discharge is shown in Figure 5.3. It is seen that self-discharge

starts at the beginning of the first hour and lasts until after the second discharge has

started. Further self-discharge is observed, although to a lesser extent, at the beginning

of each of the following 6 hours. These small repeated losses also occur during the hourly

discharges of the next four 24 hour periods. Overall, more capacity is lost through self-

discharge in the simulated intermittent discharge than in the simulated continuous 3.3

Ω discharge.

Note that in the last 6 hours shown in Figure 5.3, the self-discharge current increases


Time (h)

Cur

rent

(A)

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

Figure 5.3: External () and total () current for an intermittent 3.3 Ω discharge of an AA-cell. The load is applied for 4 minutes at the beginning of the first 8 hours of each 24 hourperiod. Only the first 8 hours of this discharge are shown.

while the load is applied, and decreases (almost immediately) after the load is discon-

nected. We note that during these rest periods, the OH− ions that dissolved the ZnO

at R = Rsc start to equilibrate, and this lets some ZnO precipate during each following

discharge pulse. Because the OH− ions do not have long enough to fully equilibrate,

less ZnO is able to precipitate, and it is dissolved in a shorter time. This results in the

small short circuits observed in the last six hours.

From this we conclude that self-discharge occurs when discharge is initiated and the

cell has almost spatially uniform electrolyte concentrations.

5.4.1 The Effect of Changing ZnO Bulk Conductivity

Here we present discussion and results in relation to the bulk ZnO conductivity, σZnO∞

(S.cm−1). It is found in Equation (5.45), which governs the solid phase potential

within the separator. It has been found that two types of ZnO may be formed during

the discharge of a primary alkaline battery (Powers & Brieter 1969). Because the


Time (h)

Cur

rent

(A)

0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 5.4: Comparison of model output for ZnO conductivities of 0.001 (), 0.005 (), 0.01(#) and 0.02 () S.cm−1. Shown is the external (hollow symbols) and the total (filled symbols)discharge current.

morphology of precipitated ZnO can be varied (Szpak & Gabriel 1979, Horn & Shao-

Horn 2003), the connectivity of the ZnO phase itself may be lower than expected, or

calculated, from the volume fraction, εZnO.

Figure 5.4 shows the effect of different ZnO conductivity values on discharge behaviour.

In Figure 5.4, the conductivity of ZnO has been given the values 10%, 50%, 100% and

200% of it’s bulk conductivity (0.01 S.cm−1). We see that larger ZnO conductivity in-

creases the amount of capacity lost to self-discharge in an almost linear fashion. It is not

expected to increase in an exactly linear fashion with respect to ZnO conductivity be-

cause extended discharge elevates the OH− ion concentrations at the separator/cathode

interface, which dissolves the ZnO connection in the separator. Thus, when the ZnO

conductivity increased, the self-discharge current is also increased, however the ZnO

connection is then dissolved at an earlier time.


5.4.2 The Effect of Changing Initial KOH Concentration

In this section we present discussion and results in relation to the initial KOH con-

centration and compare the model output of several discharges with different KOH

concentrations. In the simulations presented in this section, the initial Zn(OH)2−4 ion

concentration, C0KZn has been set at the saturation concentration, CKZn,eq (used in the

precipitation source terms that appear in the mass conservation equations for OH− and

Zn(OH)2−4 ions, and also the porosity equations in the cathode, separator and anode)

based on the initial KOH concentration. By doing this, the cell is in equilibrium before

discharge proceeds.

A definite relationship between the initial KOH concentration and the self-discharge

current is observed. Figure 5.5 shows the external and total discharge current for initial

KOH concentrations of 0.005, 0.007, 0.009 and 0.011 mol.cm−3 (the corresponding equi-

librium Zn(OH)2−4 ion concentrations are 0.000418, 0.000694, 0.001038 and 0.001450

mol.cm−3, respectively). The observed trend is that as the initial KOH concentration

is increased, the total self-discharge current decreases. This is expected, because the

precipitation reaction (5.3), shows that it will be more sensitive to high OH− con-

centrations than to high Zn(OH)2−4 concentrations. This means that at higher OH−

concentrations, less ZnO will precipitate within the separator.

We should note that the trend observed in Figure 5.5 may be dependent upon the

model of the anode. Since the anode is assumed to be uniform, it is as if the anode is

well mixed, and transport within the anode is fast. A non-uniform anode will change

the concentrations of OH− and Zn(OH)2−4 ions entering the separator from the anode,

influencing, and perhaps changing the trend observed in Figure 5.5. Kriegsmann &

Cheh (2000), while justifying their binary electrolyte model, note that in the anode,

precipitation is several orders of magnitude faster than diffusion. This means that

most of the Zn(OH)2−4 may precipitate out as ZnO before it gets the chance to leave

the anode. In addition, this directly contrasts with our model, and implies that the

amount of Zn(OH)2−4 leaving the anode will tend to be decreased. When the OH− ion

concentration is decreased, this may further decrease the amount of Zn(OH)2−4 entering

the separator, decreasing the total self-discharge current.


Time (h)

Cur

rent

(A)

0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure 5.5: Comparison of model output for initial KOH concentrations of 5 (), 7 (#), 9 ()and 11 () M. Shown is external (hollow symbols) and total (filled symbols) discharge current.

5.4.3 The Effect of Changing Separator Thickness

Here we present discussion and results in relation to the separator thickness, and com-

pare the model output of several discharges with different separator thicknesses.

The discharge current of three 3.3 Ω constant resistance discharges with separators of

three different thicknesses, namely, 0.1, 0.15 and 0.2 mm, is shown in Figure 5.6. We

observe that decreasing the separator thickness greatly increases the total self-discharge

current. The thickness of the separator is expected to influence the self-discharge of

a cell, because the thickness directly influences the length of the conduction path.

However, the relationship between the separator thickness and the self-discharge current

is seen to be very nonlinear.

At the beginning of discharge, ZnO is observed to precipitate throughout the separator

in all cases (the precipitation behaviour in the separator is shown explicitly for a 0.15

mm thick separator in Figure 5.2). In addition, all of the simulations had less ZnO at

the separator/cathode interface than anywhere else in the separator (for example, see

5.5 Conclusions 107

Time (h)

Cur

rent

(A)

0 1 2 3 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Figure 5.6: Comparison of model output for separator thicknesses of 0.1 (), 0.15 () and 0.2(#) mm. Shown is external (hollow symbols) and total (filled symbols) discharge current.

Figure 5.2). Thus, it is the thickness of the ZnO at the separator/cathode interface

that crucially determines the presence of self-discharge. To precipitate in the separa-

tor, Zn(OH)2−4 ions must diffuse from the anode, so decreasing the thickness of the

separator makes it easier for Zn(OH)2−4 to reach the separator/cathode interface before

precipitating out as ZnO. Thus, thin separators will have elevated levels of ZnO at the

separator/cathode interface, facilitating increased self-discharge.

5.5 Conclusions

There are several outcomes from this work. These include a novel model for the dis-

charge of a primary alkaline battery, a piece of numerical simulation software that solves

the model, and from the model and the solutions obtained, improved knowledge of the

possible failure mechanism of ZnO induced internal short circuit.

The novel model describes the discharge and relaxation of a primary alkaline battery,

including the effects of ZnO precipitation and internal short circuit. The cathode is


modelled as a 2-scale electrode, using a set of equations and boundary conditions that

are similar to those presented in Chapters 3 and 4. The anode is modelled using

bulk-average equations, which are used as boundary conditions for the separator. The

separator is modelled as an inert porous material. Care was taken to ensure that the

boundary conditions that link the anode, separator and cathode are consistent. This

is the first time the simplified cathode model has been solved in the context of the full

cell. The electrolyte is modelled as a ternary system of K+, OH− and Zn(OH)2−4 ions in

H2O. The precipitation of Zn(OH)2−4 out of solution as ZnO, according to Reaction (5.3)

as proposed by Kriegsmann & Cheh (1999d), however, we allow for the precipitation

and dissolution of ZnO. Furthermore, the depletion of Zn(OH)2−4 in the cathode is

avoided by using variable transference numbers. This combination of model equations,

phenomena and variables has not been presented and/or solved before.

When Zn(OH)2−4 precipitates out of solution as ZnO in the separator, its solid phase

may conduct electricity. To model this phenomenon it has been necessary to solve for

the solid and solution phase potentials separately. This internal short circuit has not

been modelled previously.

The numerical simulation software provides solutions to the model, and includes the

failure mechanism for internal short circuit through the ZnO in the separator. It

is written in MATLAB R©, and is implemented using control-volume discretisations of

the model equations. An adaptive time-stepping algorithm is incorporated into the

software, improving the stability of the model predictions. The model uses two matrix

structures, the second of which is used during self-discharge. Great care has been taken

in implementing the switching between the two matrix structures, and this has resulted

in a piece of numerical simulation software that is stable in many discharge parameter

spaces. Discharge may be chosen as intermittent or continuous, and there is a choice

of galvanostatic, potentiostatic, constant load or constant power discharge modes. The

numerical simulation software can be run on a typical desktop computer, and one hour

of simulation on a computer is equivalent to running a discharge for approximately 20

hours.

Using the simulation results, we have arrived at several conclusions, based on the

5.5 Conclusions 109

mechanism of ZnO precipitation. The most interesting and counter-intuitive result

is that self-discharge occurs at the beginning of discharge. We also found that the

discharge rate does not necessarily change the extent of self-discharge, however, the

overall effect of self-discharge is greater in intermittent discharge modes. This is linked

to the spatial uniformity of the electrolyte species. Prolonged discharge causes the OH−

from the cathode to dissolve the ZnO in the separator that is nearest the cathode, and

prevents self-discharge from reoccuring. However, the equilibration of the electrolyte

during the rest periods in intermittent discharges allows repeated self-discharge. Thus,

cell failure due to self-discharge through ZnO is not due to a runaway process, but due

to lost capacity at the beginning of discharge.

While this mechanism allows novel investigation into the effects of Zn(OH)2−4 precipita-

tion on discharge, the results may still be compared with that of Cheh and co-authors.

In the model of Chen & Cheh (1993b), Zn(OH)2−4 precipitates in the anode near the

separator, and forms a film. This results in greater anodic polarisation at the end of

discharge. Their finding is contrary to our result, but we note that their finding is based

on a non-uniform anode, and that our finding is based on a non-uniform separator. In

the model of Podlaha & Cheh (1994a), they find that adding more KOH to a cell ex-

tends cell discharge. Our findings are similar, even though they are due to different

mechanisms. Their finding is based on OH− being depleted in the anode near the end

of discharge, while our findings are based on self-discharge through the separator.

The conductivity of ZnO, the initial KOH concentration, and the separator thickness

all significantly effect the amount of capacity lost to self-discharge. The conductivity

of ZnO is almost linearly related to the amount of capacity lost to self-discharge. In-

creasing the initial KOH concentration is found to decrease the short circuit current,

but this trend may be influenced by the model of the anode. The separator thickness is

the most influential on self-discharge, where changing the separator thickness from 0.2

to 0.1 mm disproportionately increases the amount of capacity lost to self-discharge.

This is thought to be related to the transport of Zn(OH)2−4 within the separator.

CHAPTER 6

Conclusion

6.1 Summary and Outcomes

The primary aim of this thesis was to use mathematical models of the primary alkaline

battery system to add to the understanding of how the complex, inter-related and

nonlinear processes that define battery discharge, interact. In accomplishing this, we

completed three main objectives. They are outlined as follows.

In Chapter 3 we simplified the three-scale model of primary alkaline battery cathode

discharge (Farrell et al. 2000) to yield a smaller model that accounts for the important

physical, chemical and electrochemical phenomena. This was achieved by applying

Laplace transform and perturbation methods. It was shown that the three size scales

used by Farrell et al. to describe the porous EMD cathode can be reduced to two size

scales without the loss of generality. In addition, we demonstrated that the time taken

for electrolyte to diffuse into a porous EMD particle is fast when compared to cathodic

discharge time, and that ohmic losses within the graphite phase of the cathode can be

considered to be negligible. This analysis decreased the number of partial and ordinary

differential equations within the model from 9 over three scales, to 5 over two scales.

112 Chapter 6. Conclusion

This work adds to the literature by extending the analysis of Farrell & Please on the

discharge of EMD particles into the cathodic domain.

The simplified model equations were discretised using a finite-volume technique. Nu-

merical simulation software was written in MATLAB R© to solve the model using the

discretised equations, also incorporating an adaptive time stepping algorithm. The

simplified model results compare favourably with relevant experimental data. This

provides a model framework that facilitates the investigation of additional phenomena

and mechanisms and their effects on cathodic discharge. In this regard, the simplified

model is distinct from others because it is based on the complex three-scale model of

Farrell et al. (2000), yet its numerical solution has a relatively small computational

overhead and may be run on a standard desktop computer.

In Chapter 4 we developed a model, based on the simplified model, to simulate potentio-

static discharge. This model was also extended to investigave whether a single-reaction

framework could simulate the multi-reaction behaviour of the reduction of EMD in the

context of primary alkaline battery cathode discharge. The single-reaction framework

proposed by Chabre & Pannetier (1995), as used in the simplified model, was extended

to use a nonlinear ion-ion interaction term (as a function of reduction state through

CMn4+). Using this model, we simulated SPECS discharges, noting that they empha-

size the effects of the multi-reaction reduction of EMD, as observed in the experimental

data. Furthermore, we demonstrated that a linear approximation of the ion-ion inter-

action term, as used by Farrell et al. (2000), and in Chapter 3, which was adequate

for the prediction of galvanostatic discharges, was not adequate for the prediction of

SPECS discharges.

We found that determining an appropriate nonlinear form for Υ is not trivial. However,

by interpreting the experimental data in the context of the model, two key observations

were made. The first was that plateaus in Υ (CMn4+) correspond to peaks in the power

output of the cell, and the second was that the width of the plateaus correspond to the

size of the peaks. Following these crucial observations, a satisfactory nonlinear form

for Υ was obtained. The simulated 5 mV/hr SPECS discharge displayed a main peak

at 1.29 V, and a shoulder peak at 1.45 V, as seen in the experimental data. In addition

6.1 Summary and Outcomes 113

to this, two key parameters were found to significantly affect SPECS discharges, the

initial exchange current density, i00, and the diffusion coefficient of protons in EMD,

DH+ .

The initial exchange current density, i00, was found to greatly influence the size of the

current response when the potential is stepped. It was also found to effect the relaxation

of the cell. Evidence was found that the EMD reduction process occurring at 1.45 V has

a higher i00 than the main reduction process. DH+ was found to greatly influence the

relaxation of the cell, and did not significantly influence the size of the current response

when the potential was stepped. A small DH+ was found to decrease the utilisation

of the EMD. Furthermore, we found that the minimum power response in a SPECS

discharge may be influenced by variables other than DH+ , most notably the ion-ion

interaction term, and its shape is not necessarily determined by a variable DH+ .

The potentiostatic model was discretised and solved using numerical simulation soft-

ware in MATLAB R© in a manner similar to that used to solve the simplified model.

The differences between the simplified model and the potentiostatic model mean that a

typical SPECS discharge has a larger computational overhead, but the simulation time

is still orders of magnitude less than it takes to perform SPECS experiments.

In Chapter 5 we developed a model of primary alkaline battery discharge. It de-

scribes the cathode (as modelled in Chapter 4), the separator, and a simplified, or

spatially uniform, anode. The electrolyte is modelled as a ternary system of K+, OH−,

and Zn(OH)2−4 ions and H2O solvent molecules. It accounts for the precipitation of

Zn(OH)2−4 out of solution as ZnO, in all regions of the cell. It was used to investigate

the failure of the cell due to internal short circuit through ZnO that has precipitated in

the separator. In addition, the discharge mode may be chosen as galvanostatic, poten-

tiostatic, constant load or constant power. We note that a model for primary alkaline

battery discharge that accounts for internal short circuit through the separator has not

been presented or solved in the literature.

The model is solved in a similar manner to that employed for the simplified and poten-

tiostatic models. The numerical simulation software is written in MATLAB R©. Simu-

lations may be run on a desktop computer, and a typical discharge, for example, a 3.3


Ω constant load continuous discharge takes of the order of 5 minutes.

Using the results of the numerical simulation software, it was found, counter-intuitively,

that self-discharge through the ZnO phase occurs at the beginning of continuous dis-

charges. This was found to be true for intermittent discharges, as self-discharge oc-

curred at the beginning of each discharge pulse. However, self-discharge was not sig-

nificantly affected by the rate of discharge in continuous modes. It was concluded

self-discharge is stopped because OH− ions from the cathode dissolve the ZnO connec-

tion.

The effects of three key parameters were also investigated, namely, the bulk conduc-

tivity of ZnO, the initial KOH concentration and the separator thickness. The amount

of self-discharge a cell experienced was found to be almost linearly related to the con-

ductivity of the ZnO, as expected. The investigation into effect of different concentra-

tions of KOH electrolyte was done with the electrolyte saturated with Zn(OH)2−4 . No

trend was expected, but it was found that increasing the KOH concentration decreased

self-discharge. However, this outcome may be influenced by the model of the anode.

Changing the separator thickness was found to have the most impact on self-discharge.

Decreasing the separator thickness from 0.2 to 0.1 mm caused extended self-discharge,

severely shortening the discharge time. Conversely, increasing the separator thickness

would be the most effective way of eliminating self-discharge.

6.2 Further Work

During the course of this study, several issues have been identified as important in the

modelling of primary alkaline batteries. These issues, including some questions they

raise and why they are important are detailed in this section.

The graphite content of commercial cathodes is a lot lower than is usually found in

experimental cathodes, such as used in ADA tests (Williams 1995). This may violate

the assumption that graphite forms a well connected and continuous phase in the cath-

ode. This assumption has been adopted in the three models presented in this thesis,

and by Farrell et al. (2000). Without this assumption, the conductivity of the solid

phase on the cathodic scale will depend on the conductivity of the EMD particles and

6.2 Further Work 115

may change during reduction. Farrell & Swinkels (1998) measured the conductivity of

various partically reduced EMD samples mixed with a variety of graphite powders, and

found that as the graphite content is lowered, the conductivity makes a transition from

behaving like graphite, to behaving like EMD. However, this work was performed on

a dry mix, and the EMD had been given time for proton equilibration. During a dis-

charge, the EMD is non-uniformly reduced, and may have a very different conductivity.

In addition, as the graphite content becomes lower, EMD may clump together to form

bigger agglomerations. All these factors should influence simulated cell discharge in

some way, but an in depth study is needed to accurately model these phenomena.

Furthermore, cathodes contain a range of EMD particle sizes, having a random distri-

bution. The distribution of particle sizes may be changed by modifying the crushing

technique, and Williams (1995) screened EMD particles to obtain only those within

certain size fractions. In addition, larger agglomerations may be present when the

graphite content is lowered. Small EMD particles are known to perform better than

large ones under high drain, but the effect of distributions (perhaps bimodal) of particle

sizes on the model predictions is unknown.

EMD changes volume as it is reduced because the incorporation of protons into the

EMD crystal structure causes them to swell. While this increase in volume may be

accommodated to some degree by the grommet in commercial cells (see Chapter 2,

Figure 2.1), the swelling may also decrease the porosity on the particle and/or cathode

scale. In addition, the swelling will be non-uniform, and hard to predict. Swelling

has been assumed not to occur in the models presented in this thesis. The proper

treatment of this phenomenon would require consideration of the mechanical stress

and strain within the particle and cathode scales, leading to a very complex set of

additional model equations on both size scales. However, a simpler approach, using

a parameter to specify the change in particle scale porosity relative to the change in

cathode scale porosity, when validated against experimental data, may give qualitative

trends as to the effect of swelling.

An extension of the precipitation model to incorporate non-uniformity within the anode

would also improve its applicability. This has been done before (Sunu 1978, Chen &


Cheh 1993b), however, the effect of this extension on the precipitation of ZnO within

the separator and subsequent self-discharge could be investigated. In addition, the

differences and similarities of the two cathode models of Podlaha & Cheh (1994a) and

Farrell et al. (2000) could be compared. Of further interest is the behaviour of the

model with a non-uniform anode during SPECS discharges.

The precipitation of Zn(OH)2−4 is known to produce solid ZnO of two morphologies,

namely, type I, and type II (Powers & Brieter 1969). Powers & Brieter note that Type I

ZnO is loose, flocculent and white, while type II ZnO is more compact and may appear

light gray to black. The type of ZnO that precipitates was found to depend on the

presence of electrolyte advection and also the potential at which oxidation of the zinc

occurred. These two types of ZnO have different properties, and may effect discharge

differently. The effect of these phenomena on the model predictions are unknown.

All of the models considered in this thesis have been applied to one-dimensional cells.

An obvious extension of this work is to account for a second dimension. This would

allow the modelling of cathodes with different and/or irregular geometry, and with

additional current collectors. It would be applicable to cylindrical cathodes in contact

with current collectors on the side and bottom, or planar cathodes in contact with

current collectors on multiple sides. In an analagous manner, a two-dimensional model

of EMD particles with graphite only in contact with some of the surface, would help

in predicting the effect of lower graphite content on particle utilisation.

There are several experiments which would facilitate the investigation and validation

of the above future work. These include experiments to measure key parameters, and

experiments investigating key effects. A key parameter to be measured is the porosity

distribution of the EMD particles and cathode at different stages of discharge. This

would provide direct validation of any models of EMD swelling on the particle and cath-

ode size scales. Other key parameters that could be measured include the conductivity

of the two different types of ZnO. In measuring these conductivities, the mechanisms

and/or electrolyte conditions that give rise to type I and II ZnO precipitation could be

investigated. In addition to the measurement of key parameters, discharge results of

experiments in which key effects are investigated would provide invaluable validation of

6.3 Conclusions 117

the above future work. These include discharges of cells with very low graphite content,

and of cathodes with EMD particles from two size fractions. Furthermore, discharge

results of cylindrical cells with current collectors at the top and bottom could be used

to validate a two-dimensional model of cell discharge.

6.3 Conclusions

Three mathematical models, two of primary alkaline battery cathode discharge, and

one of primary alkaline battery discharge, have been developed and solved in this work.

Perturbation techniques and Laplace transforms have been used to provide a simplified

model framework, and this process highlighted several key phenomena. This resulted

in a significant reduction in the number of model equations, and greatly decreased

the computational overhead of the numerical simulation software. In addition, the

model results based on this simplified framework compared well with the available

experimental data. The second model (of the cathode) simulated SPECS discharge, and

was used to improve our understanding of the multi-reaction nature of the reduction

of EMD. It was found that the single-reaction framework was able to simulate multi-

reaction behaviour through the use of a nonlinear ion-ion interaction term. The third

model accounted for the precipitation of ZnO within the separator (and other regions),

and subsequent self-discharge through this phase. It was found that self-discharge

occurs at the beginning of discharge, and may be exacerbated by discharging the cell

intermittently. The effects of several key parameters on discharge behaviour and on

self-discharge behaviour was investigated. The three pieces of numerical simulation

software used to generate solutions of the models provide a flexible and powerful set of

primary alkaline battery discharge prediction tools, that leverage the simplified model

framework, allowing them to be easily run on a desktop PC.

APPENDIX A

Derivation of the Butler-Volmer Like Expression

Here we review the derivation of the Butler-Volmer like expression presented by Farrell

et al. (2000) in detail. Derivations of the basic Butler-Volmer expression may be found

in a number of texts (Bard & Faulkner 2001, Newman 2004, Atkins & de Paula 2006).

The expression derived here is used to describe the kinetics of the electrochemical

reactions in the anode and cathode in the models presented in this thesis. A key

feature of the derived Butler-Volmer like expression is that it must be consistent with

the zero current potential relationship given by Chabre & Pannetier (1995), namely,


Fln

CMn3+

CMn4+

, (A-1)

However, the derivation is given in terms that are as general as possible. To this aim,

consider the general electrochemical reaction,

sR(1)R(1) + sR(2)R(2) + . . . + sR(nR)R(nR) + nee−

sP1P(1) + sP2P(2) + . . . + sP(nP)P(nP), (A-2)

120 Appendix A. Derivation of the Butler-Volmer Like Expression

where nR and nR are the number of reactant and product species in the reaction, ne is

the number of electrons transferred, si denotes the stoichiometric coefficient of species

i.

The form of the reaction rates are crucial in determining the ability to generate a zero

current potential relationship like Equation (A-1). The forwards and backwards rates

may be expressed as proportional to the relevant concentrations to their stoichiometric

coefficients, and the reaction rates may be expressed in Arrhenius form (Newman 2004).

For the purposes of arriving at a zero current potential relationship similar to Equa-

tion (A-1), we insert a term (Υ, V) into the exponential. The reaction rates then

become,

inf = nFk0

f

nR∏

i=1

CsR(i)

R(i)exp

[−αnF

RgasT(φ − φ∼ + Υ)

]

(A-3)

and

inb = nFk0

b

nP∏

i=1

CsP(i)

P(i)exp

[

(1 − α) nF

RgasT(φ − φ∼ + Υ)

]

. (A-4)

The transfer current expressed in terms of the above forwards and backwards transfer

currents is,

in

= inb − i

nf . (A-5)

At equilibrium (when inb = i

nf , and φ − φ∼ = E), we solve for E and obtain a zero

current relationship, namely,

E =RgasT

nFln

k0f

k0b

− RgasT

nFln

nP∏

i=1C

sP(i)

P(i)

nR∏

i=1C

sR(i)

R(i)

− Υ. (A-6)

Upon comparison with Equation (A-1), it is evident that the form adopted for the

reaction rates is consistent. The first term is a constant and contributes to the formal

potential (E0).

To arrive at the final form of the Butler-Volmer expression, we wish to rewrite Equa-

tion (A-5) without using the hard to measure k0f and k0

b . We replace the rate constants

with parameters (constant) based on a reference equilibrium state, denoted by su-

perscript ⊙. The equal and opposite current flowing in the forward and backward

121

directions when the process is at equilibrium, known as the exchange current density,

i0 (A.cm−2), is given (at the reference state), based on Equations (A-3) and (A-4),

namely,

i⊙0 = nFk0f

nR∏

i=1

(

C⊙R(i)

)sR(i)

exp

[−αnF

RgasT

(

E⊙ + Υ⊙)]

(A-7)

and

i⊙0 = nFk0b

nP∏

i=1

(

C⊙P(i)

)sP(i)

exp

[

(1 − α) nF

RgasT

(

E⊙ + Υ⊙)]

. (A-8)

We use these equations to eliminate k0f and k0

b from Equations (A-3) and (A-4), respec-

tively, and substitute them into Equation (A-5) to arrive at,

in

= i⊙0

nP∏

i=1

(

CP(i)

C⊙P(i)

)sP(i)

exp

[

(1 − α) nF

RgasT

(

φ − φ∼ − E⊙ + Υ − Υ⊙)]

−nR∏

i=1

(

CR(i)

C⊙R(i)

)sR(i)

exp

[−αnF

RgasT

(

φ − φ∼ − E⊙ + Υ − Υ⊙)]

. (A-9)

The term overpotential (η) is used to collect some of the terms in the above exponentials.

It is defined as,

η = φ − φ∼ − E⊙. (A-10)

The overpotential may be thought of as the departure of the potential drop across the

solid/solution interface from the equilibrium potential drop at a well known reference

state.

122 Appendix A. Derivation of the Butler-Volmer Like Expression

APPENDIX B

Laplace Transform Simplification of Crystal

Scale Equations

Here we detail the Laplace transform solution method applied to Equations (3.1)-(3.7)

from Section 3.2.1. We take the Laplace transform of Equations (3.1)-(3.3) and we

obtain the following equations independent of time, namely,

1

α1y2

∂

∂y

y2∂L

CMn4+

∂y

= pL

CMn4+

− 1, (B-1)

∂L

CMn4+

∂y= 0, (B-2)

and∂L

CMn4+

∂y= α1L

in

, (B-3)

124 Appendix B. Laplace Transform Simplification of Crystal Scale

Equations

where p is the transformation variable, and Lf is the Laplace transform of f . We

apply the following transformation to simplify the system, namely,

L

CMn4+

=1

p+

C (y, p)

y, (B-4)

and arrive at the following set of equations,

∂2C (y, p)

∂y2= α1pC (y, p) , (B-5)

C (0, p) = 0, (B-6)

and

∂C (1, p)

∂y− C (1, p) = α1L

in

. (B-7)

After obtaining the solution to Equations (B-5)-(B-7), and substituting it into Equa-

tion (B-4), we find that,

L

CMn4+

=1

p+

α1L

in

sinh(√

α1py)

y(√

α1p cosh(√

α1p)

− sinh(√

α1p)) . (B-8)

By applying the Convolution Theorem of Laplace transforms to Equation (B-8), we

obtain,

CMn4+ = 1 +

t∫

0

in

(

t − t∗)

L−1 g (p) dt∗, (B-9)

where t∗ is a dummy variable for integration, L−1 g (p) is the inverse Laplace trans-

form of the function g (p), which is given by,

g (p) =α1 sinh

(√α1py

)

y(√

α1p cosh(√

α1p)

− sinh(√

α1p)) . (B-10)

To obtain the inverse transform found within Equation (B-9), we apply an inversion

integral. In this case we apply an extension of Cauchy’s integral formula (Trim 1990),

to give,

L−1 g (p) =1

2πilim

β→∞

γ+iβ∫

γ−iβ

exp(

pt)

g (p) dp, (B-11)

125

which is integrated in the complex plane along the line ℜ (z) = γ, where z is a complex

variable and γ is a real constant. By applying the Residue Theorem (Trim 1990), we

can express the integral in the complex plane as the sum of the residues of the integrand,

at its singularities. The singularities are denoted by pm, where m = 0, 1, 2, . . . ,∞. The

singularity locations for this integrand are p0 = 0 and pm = −λ2m/α1 (m = 1, 2, . . . ,∞),

where the λm values are the positive roots of Equation 3.30. The value of the residue

at p0 is 3, while the other residues are,

2 sin (λmy)

y sin (λm)exp

(−λ2mt

α1

)

; (m = 1, 2, . . . ,∞) . (B-12)

Using these residues, we may evaluate the integral in Equation (B-9), and obtain the

solution of the initial boundary value problem given by Equations (3.1)-(3.3) and (3.7),

an expression for the concentration of Mn4+ ions within a crystal, namely,

CMn4+

(

y, t)

= 1+3

t∫

0

in

dt∗+2

y

∞∑

m=1

sin (λmy)

sin (λm)

t∫

0

in

(

t − t∗)

exp

(−λ2mt∗

α1

)

dt∗, (B-13)

126 Appendix B. Laplace Transform Simplification of Crystal Scale

Equations

APPENDIX C

Perturbation Analysis of Electrolyte

Conservation Equations

Here we perform the simplifications of the electrolyte conservation equations, Equa-

tions (3.10) and (3.21) subject to the relevant boundary conditions. Explain particle

simplification, and ramifications on cathode scale.

C.1 Particle Scale Simplifications

Here we review the perturbation analysis of the particle scale electrolyte conservation

equation, Equation (3.10), subject to the boundary and initial conditions given by

Equations (3.13), (3.15) and (3.18) from Chapter 3, in the limit α5 → 0. We express

COH−p as an asymptotic expansion in powers of α5, namely,

COH−p ∼ COH−,0 + α5COH−,1 + α25COH−,2 + . . . , (C-1)

128 Appendix C. Perturbation Analysis of Electrolyte Conservation

Equations

where COH−,0, COH−,1, COH−,2, . . . are independent of α5. By substituting this expan-

sion into Equation (3.10) we obtain the following set of O(1) equations, namely,

1

r2

∂

∂r

(

DOH−∞r2∂COH−,0

∂r

)

= 0, (C-2)

COH−,0

∣

∣

∣

r=1= ˆCOH− , (C-3)

∂COH−,0

∂r

∣

∣

∣

∣

∣

r=0

= 0 (C-4)

and

COH−,0

∣

∣

∣

t=0= 1. (C-5)

Solving the above set of equations gives,

COH−,0(R, r, t) = COH−(R, t). (C-6)

Thus, to O(1), there is no electrolyte variation within any porous EMD particle, and the

concentration within each particle is equal to the electrolyte concentration immediately

outside the particle.

C.2 Cathode Scale Considerations

Here we review the perturbation analysis of Equation (3.21) in lieu of the particle scale

simplifications made in the previous section. By adopting Equation (3.31), the source

term in Equation (3.21) vanishes, and the flux of OH− ions across the particle boundary

is unknown. To obtain an expression to replace the present source term, we examine the

O(α5) equation obtained by substituting Equation (C-1) into Equation (3.10), namely,

∂COH−,0

∂t=

1

r2

∂

∂r

(

r2

[

DOH−∞∂COH−,1

∂r− α6i∼p − α7COH−,0v

p

])

. (C-7)

By multiplying by r2, integrating with respect to r and evaluating at r = 1, we obtain,

1

3

∂COH−,0

∂t=

(

DOH−∞∂COH−,1

∂r− α6i∼p − α7COH−,0v

p

)∣

∣

∣

∣

∣

r=1

. (C-8)

C.2 Cathode Scale Considerations 129

Note that Equation (C-1) and (C-6) imply that,

α5

∂COH−,1

∂r=

∂COH−p

∂r+ O

(

α25

)

. (C-9)

Using the above, and rearranging Equation (C-8), we obtain the source term given in

Equation (3.21),

(

DOH−∞∂COH−

∂r− α5α7COH− v

p

)∣

∣

∣

∣

∣

r=1

= α5

(

1

3

∂COH−

∂t+ α6 i∼p

∣

∣

∣

r=1

)

. (C-10)

List of Symbols

Roman Symbols

A Area of solid phase per total unit volume (cm−1)

ai Activity of species i (mol.cm−3)

Ci Concentration of ion or species i (mol.cm−3)

Di Diffusion coefficient of species i (cm2.s−1)

E Equilibrium or zero current potential of an electrochemical reaction (V)

e Charge of an electron (1.60217646 × 10−19 C)

E0 Formal or standard potential of an electrochemical reaction (V)

Ecell Cell potential (V)

EF Fermi level (J)

F Faraday’s constant (96485.309 C.mol−1)

g Function of the laplace transform variable

H Height of the cell (cm)

I External current drain of cell (A)

I∼s Liquid phase current flowing through separator (A)

i Current density (A.cm−2)

i0 Exchange current (A.cm−2)

in

Transfer current at a liquid/solid interface (A.cm−2)

j Transfer current density of an electrochemical reaction (A.cm−3)

k Boltzmann’s constant (1.3806503 × 10−23 J.K−1)

(continued on next page)

132 List of Symbols

(continued from previous page)

k0b Backwards electrochemical rate constant (cm.s−1)

k0f Forwards electrochemical rate constant (cm.s−1)

ks Precipitation reaction rate (cm.s−1)

L Laplace transform function

Na Avogadro’s number (6.0221415 × 1023)

Pload Power drawn by external load (W)

p Laplace transform variable

r Radial position in a spherical EMD particle (cm)

R Radial position in a cylindrical cell (cm)

Rgas The universal gas constant (8.31451 J.K−1.mol−1))

Rload Load resistance used to discharge cell (S−1)

T The temperature (K)

t Time (s)

ti Transference number of species i referenced to the volume average

velocity

ui Mobility of species i (S.cm2.mol−1)

Va Volume of the anode (cm3)

V i Partial molar volume of species i (cm3.mol−1)

v Volume average velocity (cm.s−1)

XMn4+ Exponent in expression for the conductivity of EMD

y Radial position in a spherical EMD crystal (cm)

zi Ionic charge of species i

Greek Symbols

αi Dimensionless constants (1 ≥ i ≥ 18) used in Chapter 3

αi Transfer coefficient (i = a, c) of an electrochemical reaction

εi Volume fraction occupied by phase i

η Overpotential (V)

φ Potential level (V)

κ Solution phase conductivity (S.cm−1)

λm Positive roots of tanλm − λm = 0

σ0EMD∞ Conductivity of unreduced EMD (S.cm−1)

σi Conductivity of solid substance i (S.cm−1)

Υ Ion-ion interaction term (V)

Υ1 Approximate linear coefficient of the Ion-ion interaction term (V)

List of Symbols 133

Subscripts, Superscripts and Miscellaneous

⊙ Superscript denotes reference value of variable

ˆ Superscript denotes dimensionless variable

∼ Subscript denotes liquid phase

Subscript denotes solid phase

0 Superscript denotes initial value of variable

∞ Subscript denotes bulk value

a Subscript denotes the anode

c Subscript denotes the cathode

G Subscript denotes graphite phase

H+ Subscript denotes proton species

i Subscript denotes value at the inner radius

KZn Subscript denotes potassium zincate (K2Zn(OH)4) species

o Subscript denotes value at the outer radius

p Subscript denotes the particle scale

s Subscript denotes the separator paper

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