Haverkort JW — Electrolysers Fuel Cells And Batteries — Analytical Modelling

Title Open Textbook: Electrolysers, Fuel Cells and Batteries: analytical modelling Author: J.W. Haverkort Publisher: TU Delft OPEN Publishing Year of publication: 2024 ISBN (softback/paperback): 978-94-6366-853-8 ISBN (E-book): 978-94-6366-855-2 DOI: https://doi.org/10.59490/tb.93 Attribution cover image: made using COMSOL Multiphysics® by W.L. v.d. Does

Preface
Prerequisites
This book was written to accompany the lectures of the M.Sc. course ME45203:
Electrolysers, Fuel Cells and Batteries at Delft University of Technology. Its main aim is to create an understanding of the mathematical description of the relationship between current and voltage in electrochemical devices. A solid background in physical transport phenomena is assumed, while some training in electrochemistry, batteries or fuel cells is highly beneficial. When this is lacking, reading and re-reading the first chapter before continuing with the other chapters may be advisable.
Reader guidelines Important equations will be boxed





𝑗⊥ = 𝑗∗ e𝜂/𝑏a − e−𝜂/𝑏c .

(1)

We will indicate logical steps that are important but perhaps not immediately obvious, and may require one or a few lines of derivation, with the blue “exercise figure”:
. The student is encouraged to work out these steps independently. Footnotes, appendices, and sections denoted with * are optional additional reading material and are not part of the curriculum of ME45203.
Content The first three chapters of the book explore the fundamental concepts related to electrochemistry, transport, and porous electrodes, respectively. The final four chapters are named after particular applications: batteries, fuel cells, electrolysers, and redox flow batteries, respectively. However, from the perspective of analytical modelling, there are more similarities than differences between these devices, so these chapters focus on those aspects that are distinctively associated with these applications. For batteries, treated in chapter 4, a distinctive feature is that their porous electrodes are involved in the reaction as products or reactants. Chapter 5 on fuel cells focuses on the management of water produced in the reaction. This includes how to model the 5

6
multiphase flow in diffusion layers or gas diffusion electrodes and inside a flooded catalyst layer. While these aspects are perhaps most often associated with fuel cells, they can be equally important for e.g. proton exchange membrane electrolysers or CO2 electrolysers. In chapter 6 on electrolysers we examine the effects of gas bubbles.
Contrary to the previous chapters, the direction normal to the current is modelled here. Finally, in chapter 7 on redox flow batteries, both directions parallel and normal to the current are considered in a quasi-two-dimensional model that considers the flow of reactants through a flow field.
Acknowledgements This textbook reports on knowledge developed over decades and centuries. Because it would be impossible to exhaustively credit all authors of works that have been relevant in creating this book, only a very limited number of references are included.
I would like to thank the many people who contributed to this work by providing their lecture notes, proofreading, and correcting many of the mistakes and things that were unclear. In particular, I want to mention the keen observations of Hadi Rajeai, Aviral Rajora, Sohan Phadke, Joe Blake, Wouter van der Does, Nico Valle Marchante, Jelmer Postma, Nikhilesh Kodur Venkatesh, Remco Hartkamp, Emile Craye, and last but not least Gilles Deiters. I am also grateful for the professional support received from Saskia Roselaar in editing, great help from Isis Verhaag with the figures, and Jacqueline Michielen-van de Riet for help in the publishing process. I would be most grateful if readers could let me know of mistakes or suggestions by sending an e-mail to J.W.Haverkort@tudelft.nl.
Delft, April 2024

J.W. (Willem) Haverkort

Contents
1

2

Electrochemistry
1.1 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Current and resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 An electrochemical cell . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Redox reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 Example: alkaline water electrolysis * . . . . . . . . . . . . . . . . .
1.3.4 Reduction potentials * . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Faraday’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Activation energy . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Current-overpotential relations . . . . . . . . . . . . . . . . . . .
1.5 Simplified cell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19
19
20
22
22
23
25
26
27
27
28
31
34
37
39

Appendices *
1.A Multiple reaction steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.B Alternative Butler-Volmer formulations . . . . . . . . . . . . . . . . . . . .
1.B.1 Changing reference concentrations . . . . . . . . . . . . . . . . . . .
1.B.2 Changing overpotentials . . . . . . . . . . . . . . . . . . . . . . . .
1.C The Nernst equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41
41
42
42
42
44

Transport
2.1 Conservation equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Charge transport . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Concentration overpotential . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Transport of charged species . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Boltzmann distribution . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Einstein relation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Nernst-Planck flux . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Multicomponent transport equations . . . . . . . . . . . . . . .
2.4 Binary electrolytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 General binary electrolytes . . . . . . . . . . . . . . . . . . . . .

45
45
46
47
49
49
51
52
52
54
54

7

8
2.4.2 Monovalent binary electrolytes . . . . . . . . . . . . . . . . . . .
2.4.3 Zero-flux ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Supporting electrolyte . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transient diffusion, constant current, Sand’s analysis . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54
56
58
59
62
63

Appendices *
2.A Examples of conservation equations . . . . . . . . . . . . . . . . . . . . . .
2.A.1 Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.A.2 Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.A.3 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.B Stefan velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.C Mobility of a macroscopic particle . . . . . . . . . . . . . . . . . . . . . . .
2.D Simple conductance model . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.E Boltzmann atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.F Balance sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.F.1 Example: hydrogen-bromine redox flow battery . . . . . . . . . . . .
2.F.2 Example: alkaline water electrolysis . . . . . . . . . . . . . . . . . .
2.G General binary electrolytes . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.G.1 General equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.G.2 Nernstian boundary layer with a cation with zero flux . . . . . . . .
2.G.3 Ternary electrolyte . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.H Transient diffusion: Sand’s solution . . . . . . . . . . . . . . . . . . . . . .

65
65
65
65
66
66
67
69
69
70
70
71
72
72
74
75 76

Porous electrodes
3.1 Porous structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Tortuosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Volumetric surface area . . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Superficial, interstitial, and local velocity . . . . . . . . . . . . .
3.1.5 Effective diffusivity in porous materials . . . . . . . . . . . . . .
3.1.6 Bruggeman approximation . . . . . . . . . . . . . . . . . . . . .
3.2 Macro-homogeneous porous electrode approach . . . . . . . . . . . . .
3.2.1 Macro-homogeneous approach . . . . . . . . . . . . . . . . . . .
3.2.2 General porous electrode equations . . . . . . . . . . . . . . . .
3.2.3 Ohm’s law in porous electrodes . . . . . . . . . . . . . . . . . . .
3.3 Simplified 1D porous electrode equations . . . . . . . . . . . . . . . . .
3.3.1 The area multiplier 𝑎𝐿 . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Activation overpotential . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Electrode effectiveness factor . . . . . . . . . . . . . . . . . . . .
3.3.4 Summary and characteristic dimensionless numbers. . . . . . .
3.4 Transport losses in porous electrodes . . . . . . . . . . . . . . . . . . . .
3.4.1 No transport limitations, E = 1 . . . . . . . . . . . . . . . . . . .
3.4.2 Diffusion limitations . . . . . . . . . . . . . . . . . . . . . . . . .

77
77
77
78
78
79
81
82
84
84
84
86
87
88
89
91 92 93 93 94

2.5
2.6
2.7
2.8

3

CONTENTS

3.5

9

3.4.3 Ohmic limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.4.4 Combined ohmic and diffusion limitations - approximation . . 98 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Appendices *
101
3.A Pore versus effective medium . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.A.1 Effectiveness factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.A.2 Thiele modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.A.3 Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.B Dimensionless porous electrode equations . . . . . . . . . . . . . . . . . . . 105 3.C Combined ohmic and diffusion limitations - exact . . . . . . . . . . . . . . . 106 4

Batteries
109
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.1.1 Types of batteries * . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.1.2 Battery terminology . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.2 Moving reaction zone model . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.2.2 The terminal voltage . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.2.3 Optimal battery electrode thickness . . . . . . . . . . . . . . . . 115 4.3 Single particle models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.3.1 Solid diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.3.2 The single particle model . . . . . . . . . . . . . . . . . . . . . . 117 4.3.3 Diffusion in a spherical particle . . . . . . . . . . . . . . . . . . . 118 4.3.4 State of charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.3.5 Polarisation relation . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.3.6 Pseudo-2D Model * . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.4 Binary electrolyte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Appendices * 129 4.A Parabolic polynomial approximation . . . . . . . . . . . . . . . . . . . . . . 129 4.B Dimensionless binary electrolyte porous electrode model . . . . . . . . . . . 129

5

Fuel cells
131
5.1 Types of fuel cells * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.2 Water management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.2.1 Catalyst layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.2.2 Flow channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.2.3 Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.3 Multiphase gas diffusion layer model . . . . . . . . . . . . . . . . . . . . 136 5.3.1 Capillary pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.3.2 Capillary pressure-saturation curve . . . . . . . . . . . . . . . . 140 5.3.3 Darcy’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

10

5.4

5.5

5.3.4 Relative permeability . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.3.5 An equation for 𝑠 . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.3.6 A solution for 𝑠 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.3.7 Effective oxygen diffusivity . . . . . . . . . . . . . . . . . . . . . 144 Flooded agglomerate model . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.4.1 Ohmic and diffusion limitations . . . . . . . . . . . . . . . . . . 145 5.4.2 Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.4.3 Overpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.4.4 Example calculation . . . . . . . . . . . . . . . . . . . . . . . . . 149 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Appendices *
153
5.A Generalised Thiele modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.A.1 General particle shape . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.A.2 Effectiveness factor approximation . . . . . . . . . . . . . . . . . . . 154 5.A.3 Dimensionless flooded agglomerate model . . . . . . . . . . . . . . . 155 6

Electrolysers
157
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.2 A current distribution model . . . . . . . . . . . . . . . . . . . . . . . . 158 6.2.1 Cell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.2.2 The bubble rise velocity . . . . . . . . . . . . . . . . . . . . . . . 161 6.3 One-dimensional model . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.4 Vertical gas fraction and current distributions . . . . . . . . . . . . . . . 163 6.4.1 Constant current density . . . . . . . . . . . . . . . . . . . . . . . 163 6.4.2 Variable current density . . . . . . . . . . . . . . . . . . . . . . . 164 6.5 Natural liquid circulation . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Appendices * 173 6.A Bubble release diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.B Dimensionless current distribution . . . . . . . . . . . . . . . . . . . . . . . 174 6.C An expression for the liquid recirculation velocity . . . . . . . . . . . . . . . 175

7

Redox Flow Batteries
177
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 7.2 Types of redox flow batteries * . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.3 Flow-through electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.4 Optimal electrolyte channel width . . . . . . . . . . . . . . . . . . . . . 182 7.5 Flow-by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.5.1 Quasi-2D model . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.5.2 Channel problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7.5.3 MEA problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.5.4 Cell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

CONTENTS

11

Appendices *
195
7.A Dimensionless transport equations . . . . . . . . . . . . . . . . . . . . . . . 195 7.B Dimensionless cell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 8

Additional exercises

197

9

Formula sheet

211

12

Nomenclature
Dimensionless variables
𝑐¯

Concentration 𝑐/𝑐0

B

Bruggeman’s coefficient in 𝜏2 ≈ 𝜖 −B

E
𝜙¯

Electrode effectiveness factor

r

Reaction order

𝑠

Liquid water saturation

S

Transfer coefficient, inverse mass transfer resistance

SoC

State of charge

t𝑖

transference/transport number of ion 𝑖

𝑋

Conversion (𝑐 in − 𝑐 out )/𝑐 in

z

Ion charge number

Electrolyte potential 𝐹𝜙/R𝑇

Constants
𝑒

Elementary charge 1.602176634 · 10−19

𝐹

Faraday constant 96485.3329

𝑘B

Boltzmann constant 1.38064852 · 10−21

[J/K]

𝑁A

Avagadro constant 6.02214076 · 1023

[-]

R

Gas constant 8.31446

[C/mol]

[J/molg /K]

Dimensionless numbers
Gz

Graetz number ⟨𝑤⟩𝑙 2 /ℎ𝐷

M

Thiele modulus 1/E → 0

Pe

Péclet number ⟨𝑤⟩𝑙/𝐷

Re

Reynolds number 𝑤𝑙/𝜈

[C]

13

14
Sh

Sherwood number 𝑁 𝑙/𝐷Δ𝑐

Greek variables
𝛼

Charge transfer coefficient

[-]

𝛿

Boundary layer thickness

[m]

𝜖

Porosity

𝜂

Activation overpotential Φ − 𝜙 − Φ − 𝜙 eq

𝛾

Surface tension

[N/m]

𝜅

Effective electrolyte conductivity

[S/m]

𝜆

Pore size distribution parameter

[-]

𝜇

Dynamic viscosity

[Pa s]

𝜈

Kinematic viscosity

[m2 /s]

Φ

Electrostatic potential in electrode

[V]

𝜙

Electrostatic potential in electrolyte

[V]

𝜌

Density

[kg/m3 ]

𝜌𝑞

Charge density

[C/m3 ]

𝜎

Effective electrode conductivity

[S/m]

𝜏

Tortuosity

[-]




𝜀

Gas fraction

𝜑e

Voltage efficiency

[V]

[-]
[m3g /m3 ]
[-]

Lower case roman
𝑎

Volumetric surface area

𝑏

Tafel slope 𝑅𝑇/𝛼𝐹

𝑐

Concentration

[m2s /m3 ]
[V]
[mol m−3 ]

∫𝑙

𝑐m

1
Cup mixing concentration ⟨𝑤⟩𝑙

ℎ

Flow channel height

𝑗

Magnitude of total current density

[A/m2 ]

𝑗∗

Superficial exchange current density

[A/m2s ]

𝑘

Areal reaction rate constant

𝑛

Electrons per reactant/product molecule

𝑝

Pressure

[Pa]

𝑞

Electric charge

[C]

𝑟

Pore radius [m]

0

𝑤𝑐𝑑𝑥

[mol m−3 ]
[m]

[mol1−r s−1 m3r−2 ]
[mole− /mol]

NOMENCLATURE

15

𝑡

Time

[s]

𝑥

Coordinate in the direction of the main current

[m]

𝑦

Coordinate normal to 𝑥 and 𝑧

[m]

𝑧

Coordinate parallel to flow, normal to main current

[m]

z

Charge number 𝑞/𝑒

[-]

Mathematical operators
cosh 𝑥 Hyperbolic cosine 21 (e𝑥 + e−𝑥 ) ln 𝑥

Natural logarithm eln 𝑥 = 𝑥

sinh 𝑥 Hyperbolic sine 12 (e𝑥 − e−𝑥 )
tanh 𝑥 Hyperbolic tangent sinh 𝑥/cosh 𝑥 Notation ′

A spatial derivative 𝑑/𝑑𝑥 (𝑑/𝑑 𝑥¯ dimensionless)

¯

Overbars denote dimensionless quantities

Δ

Difference right minus left, or after minus before

𝒏ˆ

Unit vector, normal to a surface into the fluid

⟨.⟩

Spatial average, 𝑥1

∫𝑥
0

.𝑑𝑥 or 1𝑧

∫𝑧
0

.𝑑𝑧 in one dimension

𝜕
+𝒖·∇
𝐷/𝐷𝑡 Material derivative 𝜕𝑡

opt

Optimised for energy efficiency

Subscripts
⊥

Normal to a surface, into the fluid

∥

Parallel to particle/parcel trajectory

𝑖

Species index

𝑥, 𝑦, 𝑧 Vector component in the 𝑥- 𝑦- or 𝑧-direction +

Positively charged ions, cations

-

Negatively charged ions, anions

0,𝐿

At 𝑥 = 0 or 𝑥 = 𝐿

a

Anode

c

Cathode

cap

Capillary

d

Discharge/discharged

diff

Diffusion (superscript)

eq

Equilibrium

16
fl

Fluid

g

Gas

in,out At the inlet 𝑧 = 0 or outlet 𝑧 = ℎ, respectively l

Liquid

lim

Limiting

lin

Linear

m

Molecular / material

mig

Migration (superscript)

O

Oxidation

opt

Optimal with respect to energy efficiency

R

Reduction

ref

Reference

s

Solid/Surface

tn

Thermoneutral

tot

Total

Upper case roman
[m2 ]

𝐴

Projected electrode area

𝐴𝑅

Areal resistance

𝐶𝑝

Heat capacity, at constant pressure

𝐷

Effective diffusion coefficient

𝐸

Electrode potential

[V]

Ea

Activation energy

[J]

𝐺

Gibbs free energy

[J/mol]

𝐼

Current

𝐽∗

Total superficial exchange current density 𝑎𝐿𝑗∗

[A/m2 ]

𝐽𝜅

Characteristic current density 𝑏𝜅/𝐿

[A/m2 ]

𝑙

Flow channel thickness

[m]

𝐿

Electrode thickness

[m]

M

Mobility, 𝑢/𝐹 in equilibrium

𝑀

Molar mass

[kg mol−1 ]

𝑁

Molar flux

[mol m−2 s−1 ]

𝑅

Radius of a particle or bubble

𝑆

Source term

[Ωm2 ]
[J kg−1 K−1 ]
[m2 s−1 ]

[A]

[s/kg]

[m]
[something/m3 /s]

NOMENCLATURE

17

𝑇

Temperature

[K]

U

Potential energy

[J]

𝑉

Voltage

[V]

V

Volume

[m3 ]

Vm

Molar volume 𝑀/𝜌

[m3 /mol]

Vectors
[m2 ]

𝑨

Area, normal to surface

𝑬

Electric field

𝑭

Force

[N]

𝑰

Current

[A]

𝒊

Electronic current density

[A m−2 ]

𝒋

Ionic current density

[A m−2 ]

𝑵

Molar flux

𝑼

Superficial velocity with components 𝑈 , 𝑉 , 𝑊

[m/s]

𝒖

Interstitial velocity with components 𝑢, 𝑣, 𝑤

[m/s]

[V/m]

[mol m−2 s−1 ]

Other conventions
1. The cathode is the electrode at which the reduction takes place and oxidation occurs at the anode.
2. Cell potentials are that of the cathode minus that of the anode 𝐸cell = 𝐸c − 𝐸a , so positive for Galvanic cells and negative for electrolytic cells.
3. We will draw the cathode on the right and anode on the left.
4. Surface local fluxes 𝑁⊥ and 𝑗⊥ are always directed from the solid surface into the electrolyte, so 𝑗⊥ is positive on the anode and negative on the cathode.
5. Since anodic currents are positive, the overpotential will also be positive on the anode, and negative on the cathode.
6. Tafel slopes 𝑏 will be used on an e-folding basis, rather than per decade.
7. The number of electrons 𝑛 > 0 transferred in the reaction O + 𝑛e− →R; per number of reactant or product molecules as indicated.

18

Chapter 1

Electrochemistry
This introductory chapter constitutes a refresher of the basic electrochemical terminology and notation necessary for the subsequent chapters. In this chapter, the basic foundations are laid in terms of electrostatics, equilibrium thermodynamics, and electrochemical kinetics culminating in a simplified cell model.

1.1

Electrostatics

An electric charge 𝑞 [C] in an electric field 𝑬 [V/m] experiences an electrostatic force 𝑭 = 𝑞𝑬.

(1.1)

Since this is a conservative force, the electric field can be expressed in terms of an electrostatic potential 𝜙 as 𝑬 = −∇𝜙.

(1.2)

Combining these two equations we get 𝑭 = −𝑞∇𝜙, where we see that a positive charge moves down along a potential gradient, while a negative charge moves up in potential. The electric potential energy of a charge is the negative of the work done by an electric field in moving the charge within such a field

∫
U =−

𝑭 · 𝑑𝒙 = 𝑞Δ𝜙,

(1.3)

where Δ𝜙 is 𝜙 at the end minus the start of the path that the charged particle traverses.
Positively charged particles thus increase their potential energy when the potential increases, while negatively charged particles increase their potential energy when the potential decreases. Note that in all these equations only differences in potential, termed voltages, are physically relevant.

19

20

1.2

Current and resistance

When thinking of conduction and current we usually think about the movement of electrons. While this is most relevant inside metals, electrochemistry is also concerned with the flow of charged molecules, or ions. Media that conduct ions are called electrolytes. In general, we can define current as the flow of charge, regardless of whether the charge carriers are electrons, ions, or other charged particle. The movement of ion charge in the form of a current 𝐼 [C/s=A] through an area 𝐴 constitutes a current density 𝑖 [A/m2 ] 𝐼 .
(1.4) 𝐴 Since the flow of ions is not constrained to any preferential direction, in general, the ionic current density 𝒊 is a vector field that can have a different direction and magnitude at each location.
𝑖=

Often, there is an approximate proportionality between the current density 𝒊 and the electric field, expressed through1 𝒊 = 𝜅𝑬.

(1.5)

In the next chapter, we will get into why this is often the case and also when it does not hold. The constant of proportionality is the conductivity 𝜅 [S/m = Ω−1 m−1 ], which is the inverse of the resistivity. These material properties depend strongly on the charge carrier and physical parameters, such as temperature, and can differ widely between materials. Metals, for example, typically conduct electrons very well. Aqueous salt solutions, on the other hand, are typically insulating for electrons but conduct ions well. We will refer to such ionic conductivity with the symbol 𝜅, while reserving the symbol 𝜎 for the conductivity of a material towards electrons.
When speaking of conductivity, it is therefore important to specify with respect to what particle. We will also use a different symbol 𝒋 for the electronic current density so that Ohm’s law for electrons reads 𝒋 = 𝜎𝑬. Since the ionic current is converted to electronic at the electrode, charge conservation gives 𝑗 = 𝐼/𝐴 as well. We will typically use the symbol 𝑗 for the current density unless explicitly referring to the current density magnitude inside the electrolyte.
Referring to Figure 1.1, we consider the case in which there is a difference in potential ∆𝜙 ≡ 𝜙 (𝐿) − 𝜙 (0) < 0 between the back and front of a block of material of thickness 𝐿. The average electric field strength in the 𝑥-direction is obtained by integrating Eq. (1.2) to give 𝐸 𝑥 = −∆𝜙/𝐿, which after inserting in Eq. (1.5) gives the positive current density in the 𝑥-direction, 𝑖𝑥 = − 1This is the continuum formulation of Ohm’s law.

𝜅∆𝜙
𝐿

(1.6)

CHAPTER 1. ELECTROCHEMISTRY

𝜅
𝑖 = 𝐴𝐼

21

𝑉 = |Δ𝜙|

𝐴

0

𝐿
𝑥

Figure 1.1: A schematic to illustrate Ohm’s law 𝑉 = 𝐼𝑅 and Pouillet’s law 𝑅 = 𝐿/𝜅𝐴 in a planar domain with ion conductivity 𝜅.
Multiplying with the area 𝐴 and taking the absolute value with 𝐼 = |𝑖 𝑥 |𝐴 and 𝑉 = |Δ𝜙| we obtain 𝑉 𝐿 , where 𝑅 = .
(1.7) 𝑅 𝐴𝜅 The first of these equations is Ohm’s law, while the second is referred to as Pouillet’s law. The latter relates resistance to geometry and the material property 𝜅 in a logical way. The larger the distance ions have to travel or the lower the electrolyte conductivity, the larger the resistance. The area appears only because we introduced the current. Dividing Eq. (1.7) by the area 𝐴, we obtain an expression for the current density magnitude:
𝐼=

𝑖=

𝑉
,
𝐴𝑅

where 𝐴𝑅 =

𝐿
𝜅

(1.8)

Here the area-specific resistance (ASR) 𝐴𝑅 [Ωm2 ] may be viewed as a separate new symbol or as the product of area and resistance.
Since the rate at which a desirable product is made in an electrochemical device scales with the current 𝐼, and the cost of electrodes, membranes, etc. scale with area 𝐴 the ratio 𝑗 = 𝐼/𝐴, the current density, is a good performance metric. Electrochemical processes are usually only commercially attractive above a certain current density. A high current density allows for a large amount of product to be made per unit area of the cell components used.
On the other hand, the potential drop 𝑉 has to be maintained by a power source, for which the power consumption is given by 𝑃 = 𝐼𝑉 [W].
Often, a balance has to be found between maximising the current density 𝑖 and minimising the required potential drop 𝑉. From Eq. (1.8), minimising the areaspecific resistance is therefore crucial. This means that the distance 𝐿 that the charge has to travel has to be as small as possible, through a medium with conductivity as large as possible. Therefore, electrodes have to be positioned as close together as

22
Electrolyte
conducts ions

Membrane / Diaphragm

Anode
oxidation
(of reductant)
Electrons out

Cathode
reduction
(of oxidant)
Electrons in

𝒊, 𝑬

𝑰, 𝑬

𝑉cell

Electrode
electrons

e−

Figure 1.2: A schematic of a general electrochemical cell. Often, but not always, the anode is depicted on the left and the cathode on the right. Electrons move through a metal from the anode, where the oxidation reaction takes place, to the cathode, where reduction occurs.
possible. For this reason, electrodes are often directly placed next to a gas-separating membrane, in a zero-gap configuration or membrane-electrode assembly (MEA).

1.3

An electrochemical cell

A typical electrochemical cell, schematically shown in Figure 1.2, consists of at least the following functional parts:
1. An electric connection (𝜅 = 0, 𝜎 ≠ 0) between the electrodes via a power source or load 2. An electrolyte, conducting ions but not electrons (𝜅 ≠ 0, 𝜎 = 0) 3. Two electrodes, an anode and a cathode. The function of the electrodes is twofold: they should conduct electrons and be catalytically active for the desired reaction.2 Only in the case of batteries does the electrode material itself actually participate in the reaction. In this case, the reaction stalls when the electrodes are used up.

1.3.1

Redox reactions

The reactions taking place at electrodes are redox (reduction-oxidation) reactions in which electrons are transferred 2Sometimes these functions of an electrode are split, in which case there is a catalyst support and a catalyst layer, for example in the form of spray-coated catalyst nanoparticles.

CHAPTER 1. ELECTROCHEMISTRY

23

1. Reduction takes place at the cathode3, where an electron (e− ) is transferred to an oxidant (O) 2. Oxidation takes place at the anode, where an electron is transferred from a reductant (R)4 Such a reaction can be schematically depicted as O + 𝑛e− ↔ R.

(1.9)

We will consider the stoichiometric coefficient 𝑛 to be a positive integer and will use it somewhat flexibly to denote the total number of electrons transferred in a redox reaction, or, depending on the context, the number of electrons per reactant or product molecule, respectively.
The reactants may be solids, as in the case of batteries. Alternatively, reactants may be dissolved in a liquid electrolyte, or the liquid itself may be the reactant, for example, in a water electrolyser or in various redox flow batteries. Finally, gases may also be brought in contact with the electrodes as in the case of fuel cells.
Figure 1.2 shows the anode on the left and the cathode on the right, a convention we will adhere to in this book but is not always followed in the literature. Electrons generated in an oxidation reaction at the anode move through the external circuit to the cathode where the reduction reaction takes place. To avoid negative charge accumulation at the cathode, negative ions (anions) move back through the electrolyte to the anode. Alternatively, positive ions (cations) move from anode to cathode. Either way, an electric field is set up and directed from the anode to the cathode to transport this charge.

1.3.2

Thermodynamics

Electrochemists often use the symbol 𝐸 to denote the potential of an electrode.5 We hope that this will cause no confusion with the magnitude of the electric field.
Neglecting ohmic losses in the external circuitry and the electrodes for now, the cell voltage 𝑉cell reads 𝑉cell = 𝐸c − 𝐸a ,

(1.10)

3To avoid renaming an electrode depending on whether a battery is charging or discharging, for a battery this naming convention only holds during discharging. During charging the oxidation reaction takes place at the cathode.
4Other terms for an oxidant are oxidiser/oxidising agent/electron acceptor. Other terms for a reductant are reducer/reducing agent/electron donor. Reduction refers to a reduction in charge, so the addition of electrons. Some chemists may insist it is rather the oxidation state or oxidation number that is reduced.
This hypothetical charge is usually +1 for hydrogen, -2 for oxygen. The sum of oxidation states of a molecule equals its charge, in units of electron charge, which is called the charge number or valence.
5This notation derives from the electromotive force (EMF), a term used to describe the transformation from a different type of energy (like chemical bonding energy) to electrical energy. Electrode potentials are actually voltages since they are measured relative to some reference electrode; see the optional section 1.3.4 for more information. Sometimes the terminology cell potential is used, for which we will consistently use cell voltage.

24
where the subscripts 𝑎 stand for anode and 𝑐 for cathode, respectively. If we switch off the current, the oxidation and reduction reactions proceed at an equal rate at the equilibrium cell voltage 𝑉eq = 𝑉c,eq − 𝑉a,eq .6 From equilibrium thermodynamics, it follows that the amount of useful work that can be done at constant temperature and pressure is given by (minus) the Gibbs free energy [J] of the reaction. Per mole of product [molp ] we will denote this as −∆𝐺 [J/molp ]. Upon slightly opening the circuit, an infinitesimal amount of current will run that does electrical work, given by Eq. (1.3) as 𝑞𝑉eq [J].
For one mole of product, we convert a charge 𝑞 = 𝑛𝐹, with 𝑛 [mole− /molp ] here representing the number of moles of electrons per mole of product. Here, 𝐹 ≈ 96485 C/mole− is Faraday’s constant, which gives the charge of a mole of electrons.7 Combining this gives ∆𝐺 = −𝑛𝐹𝑉eq .

(1.11)

1. Spontaneous reactions have ∆𝐺 < 0, so such galvanic or voltaic cells have 𝑉eq > 0.8 2. Non-spontaneous reactions have ∆𝐺 > 0, so such electrolytic cells have 𝑉eq < 0.
Besides useful work, at a finite current, there are also various types of losses. For example, the collisions of electrons and ions result in resistive heating. In defining energy efficiency, we will compare the total energy consumption by the cell to the amount of useful work.
For a galvanic cell the useful work that can be done per mole of product is −∆𝐺 = 𝑛𝐹𝑉eq , while dissipation lowers the cell voltage to 0 < 𝑉cell < 𝑉eq , leaving 𝑛𝐹𝑉cell for useful work. Hence we define the energy efficiency or voltage efficiency as 𝜑e =

𝑉cell
.
𝑉eq

(1.12)

For an electrolytic cell, dissipation makes that a more negative cell voltage 𝑉cell < 𝑉eq < 0 is needed to drive a current. Hence, the voltage efficiency, in this case, reads 𝜑e =

𝑉eq
𝑉cell

.

(1.13)

6This open-circuit voltage (OCV) may be different when side-reactions like corrosion determine the cell voltage. Here we consider that an equilibrium of the desired reactions prevails.
7With 𝑒 = 1.602(..) · 10−19 C the elementary charge and 𝑁A Avogadro’s number, or the number of particles in a mole, we have 𝐹 = 𝑒 𝑁A .
8This is a consequence of the second law of thermodynamics that states that the entropy of the universe increases. We have Δ𝐺 = Δ𝐻 − 𝑇ΔS where ΔS > 0 is the entropy increase in the reaction and Δ𝐻/𝑇 > 0 is entropy decrease of the environment due to heat and work. The Gibbs free energy, therefore, gives the total entropy decrease, which is negative for a spontaneous process. For a non-spontaneous process, it can be positive, but this requires an entropy increase elsewhere in the system.

CHAPTER 1. ELECTROCHEMISTRY

25

1.3.3 Example: alkaline water electrolysis *
As an example, we consider in this section the reactions taking place in an alkaline water electrolyser producing hydrogen and oxygen. The word alkaline here refers to a pH>7. Note also that this process is named water electrolysis, for the reactant, rather than for the products hydrogen and oxygen. The overall reaction 2H2 O→2H2 +O2 , proceeds through two redox reactions. The reduction reaction at the cathode, the hydrogen evolution reaction (HER), reads 2H2 O + 2e− →H2 +2OH− .

(1.14)

The pre-factors for each term in this reaction are called stoichiometric coefficients.
The hydrogen gas forms bubbles that leave the electrolyte. Hydroxide ions (OH-) take over the transfer of current from electrodes and move to the anode, where they take part in the oxidation reaction, the oxygen evolution reaction (OER):
4OH− →4e− +O2 +2H2 O.

(1.15)

At the anode, the current is continued by transport of electrons in the electrodes.
A power source closes the circuit and gives the electrons enough Gibbs free energy to participate in the reactions. Note that water is consumed at the cathode but formed at the anode. However, of course, more water molecules are consumed at the cathode than are produced at the anode so that, overall, water is consumed. The equilibrium voltage of the water electrolysis reactions 𝑉eq = −

∆𝐺
237 kJ/mol
≈
≈ 1.23 V.
𝑛𝐹
2 · 9.65 · 104 C/mol

(1.16)

When 𝑉cell ≈ 𝑉eq the process will be endothermic, causing local cooling, which is resupplied by heat drawn from the environment. This additional heat is taken into account in the enthalpy of reaction ∆𝐻 = ∆𝐺 + 𝑇∆S. Here ∆S is the entropy of reaction, associated with the higher entropy of the produced gases compared to those of the consumed liquid. The cell voltage at which this additional heat is produced by internal dissipation 𝑉tn = −

286 kJ/mol
∆𝐻
≈
≈ 1.48 V,
𝑛𝐹
2 · 9.65 · 104 C/mol

(1.17)

is called the thermoneutral voltage. At higher voltages, the process becomes exothermic, as more heat and entropy are produced than is required for the reaction. To create gaseous hydrogen and oxygen bubbles requires a further small heat/enthalpy of vaporisation.
The equilibrium voltage of Eq. (1.16) at standard conditions can be found in tables of standard electrode potentials, like Table. 1.1 in the next section. The energies of reaction ∆𝐺 and ∆𝐻 are referred to as the lower and higher heating values, respectively. In the combustion of hydrogen, this difference arises due to the additional energy required to vaporize the product water.

26
Often, instead of ∆𝐺 as was used in Eqs. (1.12) and (1.13), ∆𝐻 is used to define the energy efficiency of water electrolysis. In this way of bookkeeping, only the electrical energy is used for the minimum required energy, and the required heat is assumed to be freely available. This makes for a maximum efficiency ∆𝐻 ∆𝐺 ≈ 118%, that can be reached for the non-spontaneous reaction, by using heat from the environment.
For the spontaneous reactions of hydrogen and oxygen in a fuel cell, the reverse of ∆𝐺 Eqs. (1.14) and (1.15), on the other hand, a maximum efficiency of ∆𝐻 ≈ 85% can be reached. The generated heat is then discarded as not being useful enough to contribute to energy efficiency. This reasoning has some logic, although so does having 100% as a maximum efficiency.9 Note that, however we define it, at low enough current density the reactions can, in theory, be performed back and forth with 100% efficiency in an electrochemical cell.
This is in stark contrast with the energy efficiency of producing work from the heat released in burning hydrogen, which is limited by the Carnot efficiency. Therefore, at ambient or moderate temperatures, the energy efficiency of electrochemical cells easily beat their mechanical equivalents.

1.3.4

Reduction potentials *

An electrode potential 𝐸 is defined as the cell voltage 𝐸right − 𝐸left,SHE measured when the left electrode is a reference electrode, often a standard hydrogen electrode (SHE).
This is a theoretical, idealised, electrode but can be approximated by bubbling pure hydrogen past a platinum electrode in an acidic solution at ambient conditions. The term “standard" here refers to the standard conditions of 25◦ C, 1 atm, 1 M of all reactants, or actually, to account for non-ideal behaviour, an activity of 1. Electrode potentials at standard conditions are typically dressed with a superscript 0 (or sometimes ⊖) so that a standard hydrogen electrode has 𝐸 0 = 0 by definition. For other reactions, the standard electrode potentials can be looked up in tables.
These potentials are usually given as reduction potentials; for reduction reactions examples are shown in Table 1.1 Rows of Table 1.1 can be combined to give new reactions. For example, the final line follows by subtracting twice the second line from the one before it. Also, the potential follows from subtracting -0.83 from 0.4 to give 1.23 V. However, this is fortuitous, as, in general, we have to realise that Gibbs’ free energy can be added in this way and not the potential. By Eq. (1.11), the factor 𝑛 must be corrected when adding potentials.
9The EU in its “harmonized protocols for testing of low temperature water electrolysis” advises a 𝑉tn definition also propagated in Ref. [16], which amounts to 𝜑e = 𝑉 +𝑉 −𝑉eq . Note that in this definition too, the maximum cannot exceed 100 %.

cell

tn

CHAPTER 1. ELECTROCHEMISTRY
Reduction reaction Li+ + e− → Li 2H2 O + 2e− → H2 + 2OH− 2CO2 + 2H+ + 2e− → CO + H2 O 2H+ + 2e− → H2 O2 + 2H2 O + 4e− → 4OH− O2 + 4H+ + 4e− → 2H2 O

27

𝐸0 [V]
-3.04
-0.83
-0.52
0
0.4
1.229

Table 1.1: An example list of reduction reactions and their reduction potentials. All values are relative to SHE.

1.4

Kinetics

We conclude this chapter in section 1.5 with a simple cell model relating the cell voltage 𝐸cell to the current density 𝑗 and various geometrical and material parameters of the cell. Two important elements, the equilibrium voltage 𝑉eq and the ohmic losses 𝐼𝑅 have already been described. In this section, we introduce a final important ingredient, the activation overpotential.

1.4.1

Faraday’s law

Consider a redox reaction taking place at an electrode. Such a surface reaction is called a heterogeneous reaction because of the two or more phases involved. We will use a normal-to symbol ⊥ to denote vector components of the molar flux normal to this surface in the direction away from the surface into the fluid:
𝑁⊥ = 𝒏ˆ · 𝑵

(1.18)

Here 𝑵 is the molar flux [mol/m2 /s] and 𝒏ˆ is a unit normal vector directed into the electrolyte. Since reactants move towards the electrode surface in the −𝒏-direction, ˆ 𝑁⊥ [mol/m2 /s] is negative for reactants. Product fluxes are always positive because products move away from the electrode in the direction of 𝒏.
ˆ In the case of the redox reaction of Eq. (1.9), for each mole of products made or reactants consumed, 𝑛 mole of electrons are used. This proportionality between charge and product flux is referred to as Faraday’s law and can be expressed as 𝑗⊥ = 𝑛𝐹𝑁⊥,O ,

(1.19)

This equation relates the molar flux 𝑁⊥,O of oxidants to the charge flux 𝑗⊥ = 𝒏ˆ · 𝒋, or current density. A positive local current density 𝑗⊥ is directed from the electrode into the electrolyte. In a reduction reaction, oxidants move towards the electrode so that 𝑁⊥,O < 0 and 𝑗⊥ < 0, so cathodic currents are negative. Conversely, in an oxidation reaction oxidants move away from the electrode so that 𝑁⊥,O > 0 and 𝑗⊥ > 0, so anodic currents are positive.

28
anode

cathode
𝒏ˆ

0 < 𝑗⊥a

𝒏ˆ

𝒊

𝒋

𝒋

𝑵

𝑵

𝑗⊥c < 0

Figure 1.3: The local electronic current density 𝒋 ⊥ , product flux 𝑵 , ionic current density 𝒊, and wall-normal unit vector 𝒏.

1.4.2

Activation energy

To have a chemical reaction proceed at a significant rate, an activation energy Ea [J/mol] usually has to be supplied. This additional energy that molecules should have to overcome the reaction energy barrier is schematically depicted in Figure 1.4.
This schematic shows a spontaneous reaction, for which the Gibbs free energy of reaction ∆𝐺 < 0. The energy of the molecules in their final state is below that of the energy of their initial state.
This activation energy is associated with overcoming repulsive interactions between the reactants before the attractive binding energies can take over. The activation energy can be lowered using a suitable catalytic surface. The right catalyst binds the reactant not too strongly but neither too weakly and may also align the reactants for the reaction in order to proceed more easily. This results in a lower activation energy barrier, as is illustrated with the dashed blue curve in Figure 1.4.
Consider a reaction at the surface of an electrode, immersed in a liquid electrolyte.
For many reactions, the product flux 𝑁⊥ [mol/m2 /s] can approximately be written as 𝑁⊥ = 𝑘𝑐 r , (1.20) where 𝑐 [mol/m3 ] is the reactant concentration, 𝑘 [mol1−r s−1 m3r−2 ] the reaction rate constant and r is the reaction order. For first-order reactions r = 1 and the reaction rate is proportional to the reactant concentration. The rate constant often approximately satisfies the semi-empirical Arrhenius equation Ea

𝑘 ∝ e− R𝑇 .

(1.21)

Ea

Here, R𝑇 is the ratio between the activation energy and the characteristic kinetic energy R𝑇 for a mole of reactants.10 The higher the temperature, the larger the kinetic energy of the reactants, and the faster the reaction proceeds. This is the reason that chemical reactors typically operate at high temperatures. They often also operate at a high pressure to increase the concentration 𝑐 of gaseous or dissolved reactants, further increasing the reaction rate. Finally, often a catalyst helps to lower Ea . These are the main means available to accelerate chemical reactions.
10The characteristic kinetic energy per particle is 𝑘 B 𝑇, with 𝑘B the Boltzmann constant. The gas constant can be written as R = 𝑁A 𝑘 B , with 𝑁A Avogadro’s constant.

CHAPTER 1. ELECTROCHEMISTRY

29

potential energy

Ea
Δ𝐺
reaction coordinate
Figure 1.4: The activation energy Ea "barrier" for a spontaneous reaction (∆𝐺 < 0) to proceed from left to right along the reaction coordinate, which may be seen as a re-scaled time coordinate. For a chemical reaction, the height of this barrier can be lowered using a better catalyst (dashed curve).
Electrochemical reactions, however, have an additional “knob" that can be used to speed up a reaction, which allows them to proceed rapidly even at low temperatures 𝑇 ≪ Ea /R. This has to do with the potential energy of Eq. (1.3). By lowering the potential of the cathode 𝐸c , the potential energy of electrons is increased. The most energetic electrons, at the Fermi energy, can then transition to a lower energy state by participating in a reduction reaction.
Similarly, by increasing the anode potential 𝐸a relative to the potential 𝜙 of the adjacent electrolyte,11 The potential energy of electrons in the anode is lowered. This facilitates electrons from oxidation reactions near the surface to enter the anode.
As argued by a geometrical argument in the caption of Figure 1.5 an increase 𝐸 in electrode potential relative to that of the electrolyte lowers the activation energy for
an oxidation reaction by adding −𝛼 O 𝐹 𝐸 − 𝜙 .12 Here, the charge-transfer coefficient 𝛼O for the oxidation reaction is typically around ½, and adds up to unity when combined with that for the reverse reduction reaction:
𝛼O + 𝛼 R = 1.
(1.22) We consider a first-order, single-electron transfer oxidation reaction so that 𝑛 = r = 1. As argued above, an increase in the
anode potential 𝐸 will decrease the activation energy Ea by adding −𝛼O 𝐹 𝐸 − 𝜙 . Inserting into Eqs. (1.19), (1.20) and 11We take here the electrostatic potential in the electrolyte, measured using an electrode made of the same material as that used to measure the electrode potential. The measurement position should be very close to the electrode to exclude any ohmic drops or concentration differences over the distance to the electrode. So, it should be well within any diffusion layer thickness, but we consider it outside the electric double layer (EDL). At high electrolyte concentrations, this layer can be as thin as a few molecules, much smaller than the continuum scales considered in this book.
12If you have never seen this derivation before, the present exposition may be too short to appreciate the intricacies involved. In that case, you are recommended to study Fig. 1.5 and its caption in detail and/or consult electrochemistry textbooks like [3]. However, the details here will not be crucial to appreciate and apply the final results, so you may also skim over some technicalities.

30

−𝛼 O 𝐹 𝐸 − 𝜙

Electron
energy

−𝐹 𝐸 − 𝜙

𝛼R 𝐹 𝐸 − 𝜙

−𝐹 𝐸 − 𝜙













Into electrolyte

Figure 1.5: Left: a schematic depiction of the energy landscape for a general single-electron transfer redox reaction 𝑎A + e− → B. The 𝑥-axis is a reaction coordinate, which may be seen as a sequence of reaction steps like adsorption, reorientation of molecules, the breakup of bonds, etc. Heuristically, it may also be seen as a re-scaled spatial coordinate from the anode into the electrolyte. The energy barrier of Figure 1.4 is decreased by increasing the potential 𝐸 of the metal electrode relative to the electrolyte 𝜙. This would lower the potential energy of electrons, accelerating oxidation reactions. Right: the activation barrier is approximated by two lines under an angle.

Lowering the left line while keeping the right one fixed
increases the peak by 𝛼R 𝐹 𝐸 − 𝜙 as seen from the left but decreases it by −𝛼 O 𝐹 𝐸 − 𝜙 as seen from the right.
This decrease makes it easier for electrons to move to the left into
the electrode, accelerating the oxidation reaction. Since these changes add up to −𝐹 𝐸 − 𝜙 we have 𝛼 O + 𝛼 R = 1. The exact values depend on the angle between the two lines, but 𝛼 O = 𝛼R = 1/2 for a symmetric peak.

CHAPTER 1. ELECTROCHEMISTRY

31

Eq. (1.21) gives for the positive anodic current 𝑗⊥ = 𝑘 O 𝐹𝑐e

𝛼 O 𝐹 (𝐸−𝜙 )
R𝑇

,

(1.23)

where the oxidation rate constant 𝑘O ∝ e−Ea,O /R𝑇 . Simultaneously, reduction reactions will also take place at the anode, but at a much lower rate than the oxidation reactions. An increase
in anode potential 𝐸 will increase the activation energy Ea by adding 𝛼R 𝐹 𝐸 − 𝜙 . Inserting this into Eqs. (1.19), (1.20) and Eq. (1.21) adds a negative reduction current so that 𝑗⊥ = 𝑘O 𝐹𝑐R e

𝛼 O 𝐹 (𝐸−𝜙 )
R𝑇

− 𝑘R 𝐹𝑐O e−

𝛼 R 𝐹 (𝐸−𝜙 )
R𝑇

.

(1.24)

At the anode, the first term will be referred to as the “forward” reaction and will exceed the second “reverse reaction” term so that 𝑗⊥a > 0. At the cathode a similar expression will hold, although the kinetic parameters 𝑘O , 𝑘R , 𝛼 O , and 𝛼 R will generally be different when different cathode materials are used. At the cathode, the second term in Eq. (1.23) will dominate giving 𝑗⊥c < 0. Note that for smooth planar electrodes, by charge conservation 𝑗⊥a = −𝑗⊥c = 𝑗.

1.4.3

Current-overpotential relations

Nernst equation
Under equilibrium conditions 𝑗⊥ = 0 and Eq. (1.24) gives the single electron transfer Nernst-equation describing the equilibrium electrode potential ′

𝐸eq = 𝐸0 +
′

where 𝐸0 = 𝜙eq + R𝑇
𝐹 ln



𝑘R
𝑘O



𝑐O,eq
R𝑇
ln
,
𝑛𝐹
𝑐 R,eq





(1.25)

is the formal potential and 𝑛 = 1, since we consider a

single-electron transfer reaction. In 1.A, we consider the more general case of several reaction steps, which gives again Eq. (1.25), but now with 𝑛, the total number of electrons transferred in the reaction and a different formal potential. If the reaction is extremely fast, we speak of a reversible or Nernstian process for which this equilibrium equation can be used.
Butler-Volmer equation We define the activation overpotential or surface overpotential 𝜂 ≡ 𝐸 − 𝜙 − 𝐸 − 𝜙 eq







(1.26)

as the difference between the potential 𝐸 in the electrode and the potential 𝜙 in the electrolyte, relative to that same difference in equilibrium. Therefore, under equilibrium conditions at zero current, the activation overpotential vanishes by definition.

32
You may wonder where in the electrode and in the electrolyte we take these potentials. At the high electrolyte concentrations used in most applications, the change in electrostatic potential from 𝐸 inside the metal to 𝜙 in the electrolyte takes place over a distance of the order of a few molecules: the electric double layer (EDL).13 Using Eq. (1.26) in Eq. (1.24), we obtain the Butler-Volmer equation oxidation

𝑗⊥ = 𝑗∗

reduction

z }| { z
𝑐R

𝑐 R,eq

e

𝛼 O 𝐹𝜂
R𝑇

−

𝑐O

}|

𝑐 O,eq

Here the exchange current density 𝑗∗ ≡ 𝑛𝐹 𝑘O 𝑐R,eq

{

𝛼 𝐹𝜂

R
− R𝑇

e


𝛼O

.

𝑘 R 𝑐 O,eq

(1.27)


𝛼R

.

The Butler-Volmer equation, Eq. (1.27), cannot in general be inverted analytically to give an exact expression for the overpotential 𝜂 as a function of current density 𝑗⊥ .
Tafel equation A noteworthy case in which an explicit expression for the overpotential can be obtained  is when  one of the exponentials dominates. When 𝑗⊥ ≫ 𝑗∗ or 𝜂 ≫ R𝑇 𝐹 ln

𝑐O 𝑐R,eq
𝑐O,eq 𝑐R

, the oxidation reaction dominates over the reduction reaction and

we obtain the Tafel equation
𝑗⊥ = 𝑗∗

𝑐R
𝑐 R,eq

e𝜂a /𝑏a .

(1.28)

Solving for 𝜂 gives the concentration-dependent Tafel equation 𝑗⊥ 𝑐 R,eq 𝜂a = 𝑏 a ln .
𝑗∗ 𝑐 R





(1.29)

Here the anodic Tafel slope 𝑏 a ≡ R𝑇/𝛼 O 𝐹 is around 0.05 V at ambient temperature when 𝛼O = 1/2. This means that as long as 𝑐 R ≈ 𝑐R,eq we can increase the current by an order of magnitude with an anodic activation overpotential of only 𝜂a = 𝑏 a ln (10) ≈ 120 mV14.
13This thin region near any surface in an electrolyte consists of a layer of ions that are chemically adsorbed to the surface and a layer of ions of opposing charge that are attracted to this surface charge.
Hence the name electric double layer. This second layer screens the surface charge over the Debye length and is at relevant electrolyte concentrations ≲ 1 nm thick. It is also sometimes called the diffuse layer.
This is not to be confused with the diffusion layer. This boundary layer where diffusion dominates over advection, typically 10 − 100 µm, is many orders of magnitude thicker.
14Tafel slopes are often reported in this way, per decade rather than per e. If this is the case, the value mentioned in the literature has to be divided by ln (10) ≈ 2.3 to obtain Tafel slopes as defined here.

CHAPTER 1. ELECTROCHEMISTRY

33
𝑑𝜂
𝑑 ln 𝑗⊥ = 𝑏 a

𝜂

𝑗⊥

− ln 𝑗∗
𝜂

ln 𝑗∗

ln 𝑗⊥

𝑑𝜂
𝑑 ln −𝑗⊥ = 𝑏 c

Figure 1.6: A plot of current density 𝑗⊥ versus activation overpotential 𝜂 as  described by the  concentration-independent Butler-Volmer equation 𝑗⊥ = 𝑗∗ e𝜂/𝑏O − e−𝜂/𝑏R on a linear scale

(left) and a lin-log plot with the logarithm of the current density on the 𝑥-axis (right). From and 𝑏 R = 𝛼R𝑇 can be obtained as the slope of this latter Tafel plot, the Tafel slopes 𝑏 O = 𝛼R𝑇 R𝐹 O𝐹 the linear sections. The intercept with the horizontal axis gives the exchange current density 𝑗∗ .
Concentration overpotential Using ln 𝑎𝑏 = ln 𝑎 + ln 𝑏 we can write Eq. (1.29) as surface overpotential

𝜂a =

concentration overpotential

z }| {
 
𝑗⊥
𝑏 a ln
𝑗∗

z
+

𝑏 a ln

}|


𝑐 R,eq
𝑐R

{


.

(1.30)

The surface overpotential is independent of concentration and describes purely kinetic activation losses.15 The second part, the concentration overpotential, increases as the reactant concentration 𝑐 R decreases.
Similarly, we obtain for the reverse reduction reaction, in terms of the cathodic Tafel slope 𝑏 c ≡ R𝑇/𝛼 R 𝐹:
−𝑗⊥ 𝑐 O,eq 𝜂c = −𝑏c ln .
𝑗∗ 𝑐 O





(1.31)

At the cathode, both 𝑗⊥ and 𝜂c are negative quantities.

Concentration-independent symmetric kinetics
When there are no appreciable concentration gradients throughout the cell so that 𝑐O 𝑐R 𝑐R,eq = 𝑐 O,eq = 1, we obtain in the concentration-independent Butler-Volmer equation 15Sometimes another convention is used in which part of the concentration effect is attributed to the surface overpotential, see the end of 1.B.2.

34





𝑗⊥ = 𝑗∗ e𝜂/𝑏a − e−𝜂/𝑏c .

(1.32)

Figure 1.6 illustrates the shape of this curve and how to use it to derive 𝑏a , 𝑏 c , and 𝑗∗ .
In case of equal charge transfer coefficients 𝛼 O = 𝛼 R = 1/2 we can use the definition 𝜂
sinh 𝑥 ≡ 12 (e𝑥 − e−𝑥 ) of the hyperbolic sine, to write Eq. (1.32) as 𝑗⊥ = 2𝑗 ∗ sinh 𝑏 using 𝑏 = 2R𝑇 𝐹 . This may be referred to as the symmetric concentration-independent Butler-Volmer equation. Inverting it gives16 𝑗⊥ 𝜂 = 𝑏 asinh .
2𝑗∗





(1.33)

At high current densities 𝑗⊥ ≫ 𝑗∗ we have 𝑏 asinh (𝑗⊥ /2𝑗∗ ) ≈ 𝑏 ln 𝑗⊥ /𝑗∗ so that Eq. (1.33) reduces to the Tafel equation.17 Linear kinetics For most applications 𝑗⊥ ≫ 𝑗∗ so that Tafel kinetics is an appropriate approximation.
For some particularly facile reactions, like hydrogen oxidation on platinum, the exchange current density 𝑗∗ is so high that | 𝑗⊥ | < 𝑗∗ . Using the first-order Taylor expansion e𝑥 ≈ 1 + 𝑥 for 𝑥 ≪ 1, Eqs. (1.32) and (1.22) give the equation for linear kinetics 𝜂 = 𝐴𝑅 𝑗 ⊥

with

𝐴𝑅 =

R𝑇
.
𝐹 𝑗∗

(1.34)

Here, in analogy with Eq. (1.8), we introduced an area-specific resistance 𝐴𝑅. Note that linear kinetics only holds for very low activation overpotentials 𝜂 ≲ R𝑇/𝐹 ∼ 25 mV.

1.5

Simplified cell model

In this section, we integrate the various elements discussed in this chapter into a first cell model.
Figure 1.7 shows typical profiles of both the electronic and the electrolyte potentials through an electrochemical cell in equilibrium during electrolysis (𝑉cell < 𝑉eq < 0) and galvanic operation (0 < 𝑉cell < 𝑉eq ), respectively. The figure is slightly complicated by including the increase in potential inside the cables and electrodes in the direction of the electron flow, but does not have to be fully comprehended at this point. In case of negligible electronic resistance, the picture would be slightly simpler.
  √ 16Here the inverse hyperbolic sine can be written as asinh(𝑥) = ln 𝑥 + 1 + 𝑥 2 so that for 𝑥 ≫ 1 we have asinh(𝑥) ≈ ln (2𝑥) 17For low current densities 𝑗⊥ ≪ 𝑗∗ we have 𝑏 asinh (𝑗⊥ /2𝑗∗ ) ≈ 𝑏 𝑗⊥ /2𝑗∗ , which is consistent with Eq. (1.34).

CHAPTER 1. ELECTROCHEMISTRY

35

Equilibrium
𝑉cell = 𝑉eq

Electrolytic
𝑉cell < 𝑉eq < 0

Galvanic
0 < 𝑉cell < 𝑉eq

𝑉eq

𝑉cell

𝑉cell

e−

𝐸a

𝜂a

𝜙eq
𝐸eq,c

Galvanic

𝜙
Electrolytic

𝜂a

𝜂c

𝑗⊥
𝜂c
𝐸 − 𝜙 eq




Figure 1.7: The electronic potential (black for an electrolytic cell, grey for a galvanic cell) and electrolyte potential 𝜙 (blue) profiles throughout the cell and inside the external circuit. In an electrolytic cell 𝑉cell < 𝑉eq < 0. The electronic potential is locally multi-valued to show both variations inside the porous electrode and the cables. The electrolyte potential 𝜙 shows a constant slope in the electrolyte, which is assumed to have a constant conductivity. The “entrance” of the porous electrodes is indicated in the middle picture with the red vertical dashed lines to guide the eye. Beyond this point, a gradually decreasing slope inside the (porous) electrodes can be seen towards the back where the ionic current and hence this slope vanishes.
For completeness, and to anticipate what lies ahead in chapter 3, we also considered the electrodes to be porous. In this case, the overpotential 𝜂 = 𝐸 − 𝜙 − (𝐸 − 𝜙)eq is not a constant, but varies throughout the electrode, further complicating the picture.
Often, the electronic conductivity of the electrodes is high enough to assume constant electrode potentials 𝐸a and 𝐸c . However, for generality, we include here an ohmic potential drop Δ𝑉, which receives contributions from electrons moving through the electrode material, current collectors, and cables before arriving at the load or source. Note that in a galvanic cell Δ𝑉 decreases the positive cell voltage, while in an electrolytic cell Δ𝑉 makes the negative cell voltage more negative. Both efficiencies thus decrease as a result of the electronic ohmic losses ∆𝑉.
We will define the electrolyte potentials 𝜙a and 𝜙c at the electrode-electrolyte interface so that ∆𝜙 ≡ 𝜙a − 𝜙c > 0 gives the ohmic drop between the electrodes, which vanishes in equilibrium ∆𝜙eq = 0. With the definition of Eq. (1.26) and 𝑉eq = 𝐸eq,c − 𝐸eq,a we can then write 𝑉cell = 𝑉eq + 𝜂c − 𝜂a − ∆𝜙 − ∆𝑉.

(1.35)

This important result gives the cell potential as a sum, or an equivalent series circuit,

36
of the equilibrium potential. All terms on the right act to reduce 𝑉eq and may be ref