# Novel materials for electrochemical energy storage in supercapacitors.

Sergey N. Pronkin

► To cite this version:

Sergey N. Pronkin. Novel materials for electrochemical energy storage in supercapacitors. Material chemistry. Université de Strasbourg, 2021. tel-03836909

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Distributed under a Creative Commons Attribution 4.0 International License
Novel materials for electrochemical
energy storage in supercapacitors

Garant: Dr. Cuong Phamm-Huu (ICPEES)

Sergey N. Pronkin

A thesis presented for the degree of
Habilitation of Research

Institute of Chemistry and Processes for Energy, Environnement,
and Health (ICPEES), UMR-7515
University of Strasbourg, France

JURY

Prof. Aziz Dinia, IPCMS, Strasbourg

Dr. Frederick Favier, ICGM, Montpellier

Dr. Camélia Matei Ghimbeu, IS2M, Mulhouse

Prof. Philippe Serp, ENSIACET, Toulouse

Dr. Katia Araujo Da Silva, ICCF, Clermont-Ferrand
*Hat man sein warum? des Lebens,
so verträgt man sich fast mit jedem wie?*

*One who has his **Why?** in life,
will overcome any **How?***

FREIDRICH NIETZCHE, 1888
# Contents

<table><tr><td>Summary of the work</td><td>4</td></tr><tr><td>Lists of variables, constants, and abbreviations</td><td>8</td></tr><tr><td>Author's Curriculum Vitae</td><td>11</td></tr><tr><td>Introduction</td><td>19</td></tr><tr><td>1 Electrochemical energy storage</td><td>24</td></tr><tr><td>1.1 Electrochemical energy storage devices</td><td>24</td></tr><tr><td>1.2 Supercapacitors</td><td>28</td></tr><tr><td>1.2.1 Electrode-electrolyte interface</td><td>29</td></tr><tr><td>1.2.2 Electrochemical double-layer capacitors</td><td>34</td></tr><tr><td>1.2.3 Phenomena of pseudocapacitance</td><td>37</td></tr><tr><td>1.2.4 Pseudocapacitors and ion-insertion batteries</td><td>43</td></tr><tr><td>1.3 Impedance of electrochemical electrode/electrolyte interface</td><td>45</td></tr><tr><td>1.3.1 Impedance of the flat electrode/electrolyte interface</td><td>47</td></tr><tr><td>1.3.2 Models of impedance of rough electrode/electrolyte interface</td><td>49</td></tr><tr><td>1.3.3 Generalized staircase model of impedance of porous electrode materials</td><td>53</td></tr><tr><td>1.4 Conclusions</td><td>70</td></tr><tr><td>2 Carbon electrode materials</td><td>74</td></tr><tr><td>2.1 0D-carbon</td><td>75</td></tr><tr><td>2.2 1D-carbon</td><td>76</td></tr><tr><td>2.3 2D-carbon</td><td>77</td></tr><tr><td>2.3.1 Chemical exfoliation of graphite</td><td>78</td></tr><tr><td>2.3.2 Sonochemical exfoliation of graphite</td><td>80</td></tr><tr><td>2.3.3 Graphene-like carbon in supercapacitors</td><td>81</td></tr><tr><td>2.3.4 Case study: Graphene-like carbon from sugar/salt mixture</td><td>82</td></tr><tr><td>2.4 3D-carbon</td><td>85</td></tr><tr><td>2.4.1 Case study: activated carbon from ripe fig fruit</td><td>87</td></tr><tr><td>2.5 Conclusions</td><td>91</td></tr></table>
<table>
  <tbody>
    <tr>
      <td><b>3</b></td>
      <td><b>Transition Metal Oxide (TMO) electrode materials</b></td>
      <td><b>93</b></td>
    </tr>
    <tr>
      <td>3.1</td>
      <td>General aspects</td>
      <td>93</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>Conductivity of TMO</td>
      <td>98</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>Electrochemical stability and structural integrity of TMO</td>
      <td>106</td>
    </tr>
    <tr>
      <td>3.3.1</td>
      <td>TMO stable in wide pH range: RuO<sub>2</sub>, TiO<sub>2</sub></td>
      <td>108</td>
    </tr>
    <tr>
      <td>3.3.2</td>
      <td>TMO stable in acidic pH: oxides of W and Mo</td>
      <td>110</td>
    </tr>
    <tr>
      <td>3.3.3</td>
      <td>TMO stable in neutral and weak alkaline pH: oxides of Mn, Fe, Co, Ni</td>
      <td>112</td>
    </tr>
    <tr>
      <td>3.3.4</td>
      <td>Impedance of porous electrode/electrolyte interface with pseudo-capacitive behavior</td>
      <td>129</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>Conclusions</td>
      <td>131</td>
    </tr>
    <tr>
      <td><b>4</b></td>
      <td><b>Future projects</b></td>
      <td><b>134</b></td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Bio-sourced activated carbon</td>
      <td>135</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>TMO/C composite materials</td>
      <td>136</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Electrochemical study of supercapacitors performance</td>
      <td>138</td>
    </tr>
    <tr>
      <td colspan="2"><b>Acknowledgements</b></td>
      <td><b>140</b></td>
    </tr>
  </tbody>
</table>
# Summary of the work

## Abstract (English)

Transition toward energy systems based on the renewable primary sources, such as solar energy and wind, is a pressing challenge of our time. Considering an intermittent character of these sources, the efficient and reversible energy storage devices are the crucial components of future energy grids. Electrochemical energy storage devices (EESD), namely batteries, supercapacitors, and hybrid cells, offer high reversibility and efficiency of energy storage, compact structure, and zero operation emission. The performance of EESD strongly depends on the properties of electrode materials, in particular on their specific energy density, electronic and ionic conductivities, specific surface area (SSA), its surface interaction with an electrolyte. Electrode materials should be electrochemically stable and keep structural integrity upon charging/discharging cycles. Finally, from the viewpoint of sustainable development, the main components of these materials should be fabricated from relatively abundant and preferably non-toxic sources.

Two classes of materials play a key role for EESD: carbon materials and transition metal oxides (TMO). The present work has a goal to analyze the dependence of the performance of these materials as electrodes for supercapacitors on their structural properties in order to (i) point out the key determining parameters and (ii) offer a guidelines for further development of electrode materials.

After brief introduction in the Chapter, the principles of electrochemical energy storage in supercapacitors are considered in the Chapter 1. Main types of mechanisms of charge storage, namely, charging of interfacial double layer and pseudocapacitance are discussed. The differences between two main classes of supercapacitors, namely, electrochemical double layer capacitors (EDLCs) and pseudocapacitors, are presented. The criteria discerning the performance of electrode materials in supercapacitors and ion intercalation batteries are outlined. Finally, the model of an impedance of porous electrode materials is presented, and influence of various structural parameters on the
electrochemical properties of materials is evaluated.

In the Chapter 2, carbon electrode materials are considered for supercapacitor application. Their main advantages (electronic conductivity, chemical stability, high SSA) and shortcomings (low surface specific capacitance) are highlighted. Carbon materials are classified by their geometry as 0D (nano-onions), 1D (nanotubes, nanofibers), 2D (graphene based) and 3D (activated carbon). Last two classes are considered in more details as the most promising candidates for supercapacitors. In particular, it is concluded that open porous structure with strong contribution of micropores and easy accessibility of electrolyte, as well as the presence of graphitic domains capable to host intercalated ions, are favorable for high capacitance of carbon electrodes.

In the Chapter 3, transition metal oxides (TMO) are presented as a materials for supercapacitors. The influence of their conductivity (both electronic and ionic) and electrochemical stability on their performance is evaluated. By analyzing Pourbaix diagrams of different metals, TMO are classified by their stability at different pH: oxides, stable at any pH (RuO₂, TiO₂), stable at acidic pH (WO₃, MoO₃), and stable at neutral/alkaline pH (oxides of Co, Fe, Mn). Particular attention is paid to the spinel oxides of Fe (Fe₃O₄) and Mn (Mn₃O₄). The strategies to improve the performance of TMO/C composites, namely preparation of substituted spinel oxides, are discussed.

Finally, in the chapter 4, main directions of further work in the field of electrode materials for supercapacitors are proposed.

* Development of bio-sourced and waste-sourced activated carbon materials with controlled porous structure.

* Solid state synthesis of ternary spinel oxides of type MₓFe₂-xO₄, where M=Mn²+, Ca²+, Cu²+, Zn²+.

* Development of detailed model of an impedance of porous electrodes considering both micro- and macrostructure of electrode material.
## Resume (Français)

La transition vers des systèmes énergétiques basés sur des sources primaires renouve-
lables, telles que l'énergie solaire et l'énergie éolienne, est un défi urgent de notre époque.
Compte tenu du caractère intermittent de ces sources, les dispositifs de stockage d'énergie
efficaces et réversibles sont les composants essentiels des futurs réseaux énergétiques. Les
dispositifs de stockage d'énergie électrochimique (EESD), à savoir les batteries, les su-
percondensateurs et les cellules hybrides, offrent une réversibilité et une efficacité du
stockage d'énergie élevées, une structure compacte et une zero émission de fonction-
nement. Les performances de ces dispositifs dépendent fortement des propriétés des
matériaux d'électrode, en particulier de leur densité d'énergie spécifique, conductivités
électronique et ionique, surface spécifique (SSA) et l'interaction de surface avec un élec-
trolyte. Les matériaux d'électrodes doivent être électrochimiquement stables et conserver
leur intégrité structurelle lors des cycles de charge/décharge. Enfin, du point de vue du
développement durable, les principaux composants de ces matériaux doivent être fab-
riqués à partir de sources relativement abondantes et de préférence non toxiques.

Deux classes de matériaux jouent un rôle clé pour les EESD : les matériaux en carbone et les oxydes de métaux de transition (TMO). Le présent travail a pour but d'analyser la dépendance des performances de ces matériaux en tant qu'électrodes pour supercondensateurs par rapport à leurs propriétés structurelles afin (i) de mettre en évidence les paramètres déterminants et (ii) d'offrir des lignes directrices pour le développement futur des matériaux d'électrode.

Après une brève introduction dans le chapitre, les principes du stockage d'énergie électrochimique dans les supercondensateurs sont examinés dans le chapitre 1. Les principaux types de mécanismes de stockage de charge, à savoir la charge de la double couche interfaciale et la pseudocapacitance, sont discutés. Les différences entre les deux principales classes de supercondensateurs, à savoir les condensateurs à double couche électrochimique (EDLC) et les pseudocondensateurs, sont présentées. Les critères permettant de discerner les performances des matériaux d'électrode dans les supercondensateurs et les batteries à intercalation ionique sont exposés. Enfin, le modèle d'impédance des matériaux d'électrode poreux est présenté, et l'influence de divers paramètres structurels sur les propriétés électrochimiques des matériaux est évaluée.

Dans le chapitre 2, les matériaux d'électrode en carbone sont considérés pour l'application de supercondensateurs. Leurs principaux avantages (conductivité électronique, stabilité chimique, SSA élevée) et inconvénients (faible capacité spécifique de surface) sont mis en évidence. Les matériaux en carbone sont classés selon leur géométrie : 0D (nano-
oignons), 1D (nanotubes, nanofibres), 2D (carbon à base de graphène) et 3D (carbon actif). Les deux dernières classes sont considérées plus en détail comme les candidats les plus prometteurs pour les supercondensateurs. En particulier, il est conclu que la structure poreuse ouverte avec une forte contribution des micropores et une accessibilité facile de l'électrolyte, ainsi que la présence de domaines graphitiques capables d'accueillir des ions intercalés, sont favorables à une capacité élevée des électrodes de carbone.

Dans le chapitre 3, les oxydes de métaux de transition (TMO) sont présentés comme des matériaux pour les supercondensateurs. L'influence de leur conductivité (électron-ique et ionique) et de leur stabilité électrochimique sur leurs performances est évaluée. En analysant les diagrammes de Pourbaix de différents métaux, les TMO sont classés selon leur stabilité à différents pH : oxydes, stables à tout pH (RuO₂, TiO₂), stables à pH acide (WO₃, MoO₃), et stables à pH neutre/alcalin (oxydes de Co, Fe, Mn). Une attention particulière est accordée aux oxydes spinelles de Fe (Fe₃O₄) et de Mn (Mn₃O₄). Les stratégies visant à améliorer les performances des composites TMO/C, à savoir la préparation d'oxydes de spinelle substitués, sont discutées.

Enfin, dans le chapitre 4, les principales directions des travaux futurs dans le domaine des matériaux d'électrode pour les supercondensateurs sont proposées.

* Développement de matériaux en charbon actif bio-s源cés et issus de déchets avec une structure poreuse contrôlée.

* Synthèse à l'état solide d'oxydes spinelles ternaires de type MₓFe₂₋ₓO₄, où M=Mn²⁺, Ca²⁺, Cu²⁺, Zn²⁺.

* Développement d'un modèle détaillé de l'impédance des électrodes poreuses prenant en compte la micro et la macrostructure du matériau de l'électrode.
# Preamble

## List of general variables

$A$ - surface area, $cm^2$

$C_g$ - mass specific capacitance, F/g

$C_s$ - surface specific capacitance, $\mu$F/cm$^2$

$C_V$ - volume specific capacitance, F/cm$^3$

$C_{dl}$ - capacity of interfacial double layer, F

$C_i$ - concentration of $i^{th}$-component in solid or in electrolyte, mol/cm$^3$

$C_p$ - specific heat capacitance, J·mol$^{-1}$·K$^{-1}$

$d$ - diameter of particles, or pores, m

$D$ - diffusion coefficient, cm$^2$/s

$m$ - mass of an electrode material, g

$Q_c$ - maximal theoretical charge storage, C/g

$E$ - mass specific energy density, W.h/kg

$E$ - electrode potential, V

$f$ - AC modulation frequency, Hz

$I$ - current passing through electrode, A

$\vec{I}$ - AC modulated interfacial current, A

$i$ - surface specific current density, A/cm$^2$ (1)

$i_m$ - mass specific current density, A/g

$l$ - length of a pore, m

$\vec{E}$ - AC-modulated electrode potential, V

$P$ - mass specific power density, W/kg

$R$ - serial resistance, Ohm

$SSA$ - mass specific surface area, $m^2$/g

<footnote n="1">¹i is also an imaginary unit in the equations related to electrochemical impedance</footnote>
$T$ - temperature, K

$V$ - voltage of operation of a superacapacitor device, V

$v$ - linear potential modulation rate, mV/s

$\vec{Z}$ - electrochemical interfacial impedance, Ohm

$\delta$ - intercalation factor (stoichiometric to oxide structural unit)

$\rho$ - material density, g/cm$^3$

$\omega$ - angular frequency, rad/s

$\sigma$ - conductivity, Ohm$^{-1}.cm^{-1}$

$\phi$ - phase shift angle between potential stimulus $\vec{E}$ and signal $\vec{I}$

## List of constants

$F = 96498 \text{ C/mol}$ - the Faraday constant

$\varepsilon_0 \approx 8.854187 \cdot 10^{-12} \text{ F/m}$ - the vacuum permittivity

$R = 8.314 \text{ J/mol.K}$ - the universal gas constant

## List of abbreviations

AC - Activated Carbon

BET - Brunauer-Emmet-Teller (referred to the method of SSA measurements based on BET adsorption isotherm)

CV - Cyclic Voltammetry

EDL - Electrical Double Layer

EDLC - Electrochemical Double Layer Capacitor

EESD - Electrochemical Energy Storage Devices

EIS - Electrochemical Impedance Spectroscopy

GCPL - Galvanostatic Cycling with Potential Limits

IHP - Inner Helmholtz Plane

NHE - Normal Hydrogen Electrode

OHP - Outer Helmholtz Plane

ORR - Oxygen Reduction Reaction

PEDOT:PSS - poly(3,4-ethylenedioxythiophene) polystyrene sulfonate polymers mixture

PZC - Potential of Zero Charge

PVA - Polyvinyl alcohol
SEM - Scanning Electron Microscopy

SSA - Specific Surface Area

TMO - Transition Metal Oxides

TEM - Transmission Electron Microscopy

XPS - X-ray Photoelectrons Spectroscopy

XRD - X-ray diffraction
# Author's Curriculum Vitae

<table><tr><td>Academic degree:</td><td>Dr.</td></tr><tr><td>Academic rank:</td><td>Assistant Professor (Maître de Conférence)</td></tr><tr><td>Name:</td><td>Sergey Pronkin</td></tr><tr><td>E-Mail:</td><td>sergey.pronkin@unistra.fr</td></tr><tr><td>Date and place of birth:</td><td>17.05.1976 in Moscow, Russia</td></tr><tr><td>Citizenship:</td><td>Russian</td></tr><tr><td>Current position:</td><td>Assistant Professor (Maître de Conférence) in the European Engineering School of Chemistry, Polymers and Materials, University of Strasbourg (ECPM-Unistra), Strasbourg, France</td></tr><tr><td>Current teaching affiliation:</td><td>European Engineering School of Chemistry, Polymers and Materials, University of Strasbourg (ECPM-Unistra), Strasbourg, France</td></tr><tr><td>Current research affiliation:</td><td>Institute of Chemistry and Processes for Energy, Environment and Health (ICPEES), Strasbourg, France</td></tr></table>

## Professional experience

<table><tr><td>September 2009 –<br>now</td><td>Assistant Professor in the European Engineering School of Chemistry, Polymers and Materials, University of Strasbourg (ECPM-Unistra) / Institute of Chemistry and Processes for Energy, Environment and Health (ICPEES), Strasbourg, France</td></tr></table>
<table><tr><td>February 2007 –<br/>August 2009</td><td>Researcher in the group of Prof. Dr. Elena Savinova, Laboratory of Materials, Surfaces and Processes for Catalysis (LM-SPC) (now Institute of Chemistry and Processes for Energy, Environment and Health (ICPEES)), Strasbourg, France<br/>Research topic: "Spatially ordered arrays for fuel cell reactions"</td></tr><tr><td>May 2005 –<br/>January 2007</td><td>Post-doctoral researcher in the group of Electrocatalysis and Fuel Cells of Prof. Dr. Elena Savinova, Boreskov Institute of Catalysis, Novosibirsk, Russia<br/>Research topic: "Electrochemistry and electrocatalytic properties of supported Pd mono- and bimetallic nanoparticles"</td></tr><tr><td>October 2001 –<br/>March 2005</td><td>Post-doctoral researcher (Humboldt fellowship 2001-2003) in Electrochemistry group of Priv. Doz. Th. Wandlowski, Institute of Thin Films and Interfaces, Research Center Jülich, Jülich, Germany<br/>Research topic: "Nanostructured Pd, Rh and Pt electrodes as model systems for the search of structural effects in electrocatalysis"</td></tr></table>

## Highest educational degree

PhD in Electrochemistry, 2000, Department of Electrochemistry, Moscow State University, Moscow, Russia. Thesis "The nanostructure of platinum electrocatalysts and their electrochemical properties".

Supervisor: Prof. Galina A. Tsirlina

## Other educational degree

University degree in higher education and teaching (Diplôme d'Université Pédagogie dans l'Enseignemen Supérieur), 2018, University of Strasbourg. Thesis "Modification of a course of chemical kinetics to hybrid format: advantages and challenges".
## Scientific publications

Book chapters 2

Scientific papers 40

Current h-factor: 23 (Web of Science)

Web of Science ResearcherID: A-7395-2010

ORCID ID: 0000-0002-6981-7831

## Scientific interests

### Electrochemical material science:

preparation and electrochemical characterization of composite electrode materials,
electrodeposition,

anodization,

impedance spectroscopy

### Electrochemical energy storage devices:

supercapacitors,

metal oxide / carbon / polymer composites

### Electrocatalysis:

bimetallic catalysts,

metal-support interaction,

kinetics of electrocatalytic processes,

FTIR spectroelectrochemistry

## Tutorship

### Co-tutorship of PhD students:

Xiong Zhang, ICPEES, 2019 – now, co-tutorship with Dr. Pham-Huu Cuong
Wei Wang, ICPEES, 2016-2019, co-tutorship with Dr. Pham-Huu Cuong

Pierre-Alexandre Gross, ICPEES, 2010-2013, co-tutorship with Dr. Valerie Keller and
Prof. Elena Savinova

Tutorship

Tutorship of master students:

Emilie Rojtmann, National High School of Chemistry of Mulhouse, Mulhouse, France,
April 2021 – August 2021 (Project COM-Gra, M1 internship)

Boya Wu, University of Strasbourg, Strasbourg, France, February 2020 – July 2020
(Project COM-Gra, M2 internship)

Omar Rahmouni, Mohamed V University, Rabat, Morocco, February 2019 – July 2019
(Project ERASMUS-MediSol, M2 internship)

Ayoub Mahdi Elyaagoubi, Mohamed V University, Rabat, Morocco, February 2018 –
July 2018 (Project ERASMUS-MediSol, M2 internship)

Mohhamed Salloma, Helwan University, Helwan, Egypt, September 2017 – February 2018 (Project ERASMUS-MediSol, M2 internship)

Pierre-Alexandre Gross, University of Strasbourg, Strasbourg, France, March 2009 –
July 2009 (M1 internship), February 2010 – June 2010 (M2 internship)

Yeuk-Ting Law, University of Strasbourg, Strasbourg, France, March 2007 – July 2007
(M1 internship), February 2008 – June 2008 (M2 internship)

Authorship of Courses

Courses in European Engineering School of Chemistry, Polymers and Materials (ECPM), Uni-
versity of Strasbourg, Strasbourg, France:
* "Mathematics for chemists" for Classe Preparatoire Intégré (integrated preparatory class)

* "Chemical kinetics", online (Moodle) course

## Fellowships

2001 – 2003

Alexander von Humboldt foundation fellowship

## Selected publications

### Book chapters

1. Maillard F., Pronkin S.N., Savinova E.R. Influence of size on the electrocatalytic activities of supported metal nanoparticles in fuel cell-related reactions. In: Handbook of Fuel Cells (eds. Vielstich W., Gasteiger H.A., Yokokawa H.), Vol. 5, Chapter 7. Wiley, 2009

2. Maillard F., Pronkin S.N., Savinova E.R. Size Effects in Electrocatalysis of Fuel Cell Reactions on Supported Metal Nanoparticles. In: Fuel Cell Catalysis: a Surface Science Approach (eds. Koper M., Wieckowski A.). Wiley, 2009

### Selected scientific papers

1. Ba H., Wang W., Pronkin S., Romero T., Baaziz W., Nguyen-Dinh L., Chu W., Ersen O., Pham-Huu C. Biosourced foam-like activated carbon materials as high-performance supercapacitors. Advanced Sustainable Systems 2018, 2, 1700123. https://doi.org/10.1002/adsu.201700123

2. Kéranguéven G., Faye J., Royer S., Pronkin S.N. Electrochemical properties and capacitance of Hausmannite Mn3O4 – carbon composite synthesized by in situ autocombustion method. Electrochimica Acta 2016, 222, 755-764. https://doi.org/10.1016/j.electacta.2016.11.032
3. Lopez L., Kim Y., Jierry L., Hemmerle J., Boulmedais F., Schaaf P., Pronkin S., Kotov N.A. Electrochemistry on stretchable nanocomposite electrodes: dependence on strain. ACS Nano 2018, 12, 9, 9223-9232. https://doi.org/10.1021/acsnano.8b03962

4. Pardieu E., Pronkin S., Dolci M., Dintzer T., Pichon B.P., Begin D., Pham-Huu C., Schaaf P., Begin-Colin S., Boulmedais F. Hybrid layer-by-layer composites based on a conducting polyelectrolyte and Fe3O4 nanostructures grafted onto graphene for supercapacitor application. Journal of Materials Chemistry A 2015, 3, 22877-22885. https://doi.org/10.1039/C5TA05132K

5. Ba H., Wang W., Tuci G., Pronkin S.N., Weinberg C., Nguyen-Dinh L., Giambastiani G., Pham-Huu C. Tridimensional few-layer graphene-like structures from sugar-salt mixtures as high-performance supercapacitor electrodes. Materials Today Energy 2018, 10, 118-125. https://doi.org/10.1016/j.mtener.2018.08.015

Scientific projects (2009 - now)

As project leader:

<table><thead><tr><th>Project code</th><th>Project title</th><th>Funding agency</th><th>Duration</th><th>Total funding / funding of ICPEES</th></tr></thead><tbody><tr><td>INFINE</td><td>"Synthesis and control of interactions in new fluorinated inorganic oxide-carbon nanocomposites for electrode materials design"</td><td>ANR, France</td><td>2021-2025</td><td>501.6 kEuro / 155.9 kEuro ICPEES</td></tr><tr><td>MMO-Grace</td><td>"Mixed metal oxide / graphene composite electrode materials for electrochemical storage devices"</td><td>CNRS, France</td><td>2019</td><td>10 000 Euro (all ICPEES)</td></tr></tbody></table>
<table><tr><td>COM-Gra</td><td>"Metal oxide – graphene composites: new materials for supercapacitor electrodes"</td><td>MICA-Carnot, Alsace, France</td><td>2020</td><td>48 000<br>Euro /<br>26 000 Euro<br>ICPEES</td></tr></table>

As project participant:

<table><thead><tr><th>Project code</th><th>Project title</th><th>Funding agency</th><th>Duration</th><th>Total funding / funding of ICPEES</th></tr></thead><tbody><tr><td>ANR-06-CEXC-004-01</td><td>"Novel spatially ordered nanomaterials for the energy conversion"</td><td>ANR, France</td><td>2006-2009</td><td>799 000<br>Euro</td></tr><tr><td>EU-DEMMEA</td><td>"Understanding the dergadation mechanisms of membrane-electrodeassembly for high temperature PEMFCs and optimization of individual components"</td><td>European Union</td><td>2008-2012</td><td>220 000<br>Euro</td></tr><tr><td>STEPPES</td><td>"Nanotubes de TiO2 ordonnes pour le development de photoanodes pour la production d'hydrogéne sous illumination solaire"</td><td>CNRS, France</td><td>2008-2011</td><td>115 000<br>Euro</td></tr><tr><td>NIE</td><td>"Nanostructures en Interaction avec leur Environnement"</td><td>FRC, France</td><td>2012-2014</td><td>119 000<br>Euro</td></tr><tr><td>HELACAM</td><td>"Hydrogen electrocatalysis in alkaline media"</td><td>ERA-NET.RUS</td><td>2012-2014</td><td>30 000<br>Euro</td></tr></tbody></table>
<table><tr><td>PMNA</td><td>"Développement de la pile combustible H2/O2 à membrane échangeuse d'anions à base de catalyseurs sans platine à basse température (30° – 80° C)"</td><td>Région Alsace, France</td><td>2013-2015</td><td>103 000 Euro</td></tr><tr><td>BioSenseDress</td><td>"Développement de biocapteurs enzymatiques nanofibreux: vers les pansements intelligents"</td><td>MICA, Alsace, France</td><td>2018</td><td>50 000 Euro</td></tr></table>
Introduction

Electrochemical energy storage devices (EESD), namely, batteries and supercapac-
itors, play ever growing role in the local and global energy networks. Their advan-
tages are high reversibility and storage efficiency, compact versatile size and func-
tioning with zero emission. Comparing to supercapacitors, batteries have higher
energy storage capacities up to few 100 W.h/kg. It is due to a charge storage
mechanism involving electrochemical reactions and cation intercalation into the
bulk of the material of electrodes. In supercapacitors, a charge is stored at the
interface between electrode and electrolyte or in its vicinity. This results in lower
energy densities of supercapacitors, i.e. ca. 10-50 W.h/kg [1]. However, as inter-
facial charge storage process is significantly faster and more reversible than bulk
electrochemical reaction, supercapacitors have (in comparison to batteries):

• faster charging time (from few min to few s),

• higher power output (ca. $1 - 10^2$ kW/kg),

* much better capacity retention in charging/discharging cycles (above 80% for more than $10^5$ cycles),

• no memory effect.

Due to these advantages, supercapacitors are used as power buffers in electrical
grids, kinetic energy recuperation devices in hybrid cars, power sources for small
rechargeable electrical tools and defibrillators, and have many other applications
[2].

A crucial strategy for performance improvement of supercapacitors is a devel-
opment of novel electrode materials with optimized composition and structure.
For high performance, electrode materials have to provide
* high electronic conductivity,

* high specific surface area of electrode/electrolyte interface,

* high accessibility of electrode surface for electrolyte species,

* high mechanical and (electro-)chemical stability.

<figure><img src="image_2.png" /><figcaption>Figure 1: The mechanism of charge storage in supercapacitors (A) by ions adsorption at interface, (B) by ion adsorption and insertion into the electrode material.</figcaption></figure>

Nowadays, carbon-based materials are most commonly used due to their highly adjustable morphology, high abundance and potential of scale-up production. The charge is stored on carbon/electrolyte interface via formation of electrical double layer by physisorption of electrolyte ions and reversible reduction/oxidation of carbon surface functional groups [3] (Fig.1A).

The supercapacitors with carbon electrodes are often referred to as electrochemical double-layer capacitors or EDLCs. Carbon is an abundant and non-toxic material. The development of bio-sourced carbon materials is especially beneficial for sustainable development [4]. The limiting factor of carbon-based electrodes performance is a relatively low surface specific charge capacitance (ca. 15 µF/cm²). The highest theoretical specific surface area of single-sheet graphene is ca. 2630 m²/g [5], which translates into highest achievable mass specific capacitance of carbon-based electrode $C_g \approx 500$ F/g, or ca. 35 W.h/kg (for symmetrical capacitor with 1 V voltage range). However, even model laboratory carbon electrodes demonstrate specific capacitance in the range 100-200 F/g due to lower surface utilization compared to ideal single-sheet graphene electrode [6]. Scaling up materials to real-type capacitor devices results in further losses of performance due to inhomogenous geometry of electrodes and imperfect contact with electrolyte. Nevertheless, careful adjustment of the morphology of carbon materials may provide the values of energy capacitance above 10 W.h/kg [7].

Transition metal oxides (TMOs) are used separately or together with carbon as another type of materials for supercapacitors. The main advantage of TMOs is that a charge can be stored by reversible surface and sub-surface redox reactions and intercalated electrolyte cations into sub-surface layer of TMOs (Fig.1B). These
mechanisms, often referred to as “pseudocapacitance”, provide higher interfacial and mass specific capacitancies [8]. Hence, TMO-based supercapacitors are also called *pseudocapacitors*. In particular, RuO₂ electrodes show specific capacitance above 900 F/g; however, RuO₂ is a toxic and expensive material. Relatively high mass specific capacitance was also reported for oxides of Mn (above 600 F/g) and Ni (up to 1000 F/g in hydrated amorphous form) [9]. The most common obstacles for TMO electrodes utilization in supercapacitors are their low electronic conductivity and degradation of performance due to loss of conductivity during oxidation/reduction cycles.

The most promising strategy toward novel electrode materials is a development of composites of TMOs with high capacitance and carbon materials with high conductivity (TMO/C composites). Performance of the composite materials depends on the properties (e.g. composition, crystalline structure, morphology) of both TMO and carbon, as well as on the efficiency of mixing of the components.

The present work is dedicated to the recent advancements in the development of new types of electrode materials for supercapacitors. Particularly, the electrode materials, developed in ICPEES with active participation of the author, are discussed in details. The goals of the work are:

* to outline the factors determining the performance of different classes of electrode materials for supercapacitors, namely, carbon materials and TMO/C composites,

* to propose the guidelines for further improvement of electrochemical performance of carbon materials and TMO/C composites.

For carbon materials, this work is focused on graphene-based (2D) carbons, and activated (3D) carbon materials. In particular, the bio-sourced activated carbon with high specific surface area (SSA) above 2000 m²/g and specific capacitance of 340 F/g developed in ICPEES [10] is presented. A facile synthesis of the few layers graphene (FLG), worked out in ICPEES [11], is also discussed in details. This synthesis results in a material with lower SSA but comparable specific capacitance of 314 F/g due to optimized porous and graphitic structure.

For TMO/C composites, the work focuses on application of spinel oxides of Fe and Mn. Namely, a composite Fe₃O₄@graphene:PEDOT/PSS, developed in
ICPEES-ICS-IPCMS<fnref n="2" /> is described. This is a composite of Fe₃O₄ magnetite nano-particles on graphene as storage component and PEDOT/PSS (poly(3,4-ethylene-dioxythiophene) / polystyrene sulfonate polymers mixture) layers as structurizing component [12]. The specific capacitance of 153 F/g was obtained even for thick layers of this material (up to 20µm), demonstrating efficient utilization of storage components due to hierarchial layered structure. The other type of composite materials, discussed in the work, was developed in ICPEES [13], and consists of 10-25 nm nanoparticles of hausmannite Mn₃O₄ oxide uniformly mixed with carbon XC-72. This material demonstrated outstanding capacitance above 250 F/g.

The current work is organized as follows. Chapter 1 gives a general description of electrochemical supercapacitors. In particular, the main principles, characteristics, and parameters, determining supercapacitor performance are presented. The formation of an electrical double layer and the pseudocapacitance phenomenon are discussed. Special attention is given to the impedance spectroscopy as a tool for characterizing interfacial processes. The mathematical models of an impedance of porous electrodes are described.

Chapter 2 deals with the carbon electrodes for supercapacitors. Various types of carbon materials are reviewed and classified by their geometry. Graphene type carbon and porous bio-sourced activated carbon are described in more details, focusing on the materials developed in ICPEES with participation of the author [11, 10].

Chapter 3 is devoted to the TMO and TMO/C composites which are utilized in supercapacitor electrodes. A review of various types of TMO is given with the focus on binary oxides and spinel oxides. The composite materials with oxides of Fe and Mn developed in ICPEES with participation of the author [12, 13] are described.

In Chapter 4 the main directions of further development of the present work are discussed. In particular, the synthesis of activated and porous carbon materials from biological sources and domestic wastes is presented as a prospective approach for sustainable development. The synthesis of TMO/C materials with substituted spinel oxides, MₓFe₂-ₓO₄, where M = Mn<fnref n="2" />+, Cu<fnref n="2" />+, Zn<fnref n="2" />+, Ca<fnref n="2" />+, is proposed as a promising approach toward composite electrode materials for pseudocapacitors. Finally, further development of electrochemical techniques for characterization of

<footnote n="2">²ICS : Institut de Charles Sadron (Strasbourg); IPCMS : Institut de Physique et Chimie de Matériaux de Strasbourg</footnote>
electrode materials, based on the model of interfacial impedance, is discussed.
# Chapter 1

# Electrochemical energy storage

## 1.1 Electrochemical energy storage devices

Nowadays more than 40% of primary energy sources are consumed by the production of *electricity*. In 2015 world electricity was predominantly (>81%) produced from fossil sources: coal (41%), natural gas (21%), oil (5.4%), and nuclear fuels (13.5%) [14]. The reserves of these sources allow to generate relatively cheap electricity in amount that can be to a large extent adjusted according to immediate needs. Nevertheless, there is a global trend driven by the need of sustainable development to replace these sources by renewable primary energy sources, such as hydroplant, solar, and wind energy. This trend is caused by limited availability and uneven mondial distribution of fossil fuels, as well as strong negative environmental impact of electricity production from them.

Renewable energy sources are obviously more abundant, if not unlimited, comparing to fossil fuels. However, wide scale electricity generation from renewable sources is yet held back by high production price including costs of materials and installations, maintenance and repairs [15]. However, these obstacles will be overcome by continuously improving technologies and "merit-order" effect. The latter is related to a constant increase in the share of renewable energy production [16, 15, 17].

Regardless of electricity production sources, either fossil or renewable, ever-growing world demands of electricity require development of efficient electricity storage technologies and devices. Indeed, electricity production from fossil fuels is predominantly concentrated in large power plants, thus requiring transport of
electricity to final users. The domestic demands of the electricity fluctuate strongly over the course of the day and throughout the week. Thus, an electricity grid between a production plant and final users requires efficient electricity storage devices. Same is true for electricity production at hydroplants. On the other hand, electricity production from solar or wind energy can be more localized, even to the scale of a single household, thus easing up the restrictions of complex electricity grids. However, both solar and wind energy availability is also fluctuating strongly, both geographically and temporarily (over the course of a day, a season, a year), thus, requiring reversible electricity storage facilities.

Electrochemical energy storage devices (EESD), namely, supercapacitors (also known as ultracapacitors) and batteries, provide a possibility of reversible electricity storage by electrochemical phenomena occurring at electrode/electrolyte interface and in the bulk of electrode and electrolyte. Comparing to mechanical energy storage devices (e.g. flywheel), electrochemical storage devices provide higher energy density, better reversibility. In comparison to compressed air storage and pumped hydropower storage devices, EESD have the advantage of zero emission operation [18].

Most important characteristics of performance of an electrochemical storage device are following:

* mass specific or gravimetric, charge capacitance $C_g$ of a device (in A.h/kg, or mA.h/g for batteries, in F/g for supercapacitors),

* maximal operational voltage $V$ (in V),

* mass specific energy density $E$ (in W.h/kg), defined as an amount of electrical energy that can be stored in a device per its mass unit,

* mass specific power $P$ (in W/kg), defined as an amount of energy that can be provided by a device in a unit of time per its mass unit,

* retention or cyclability (in %), defined as a loss of specific energy capacity of a device after a given number of full charge/discharge cycles performed at a given charge/discharge rate,

* price of energy stored (for example, in Euro/kW.h).

Mass specific characteristics presented above are most often used for comparison of the performance of different electrode materials in EESD. Besides electrodes,
EESD also contain components, such as electrolyte, separator, current collector, which contribute to the mass of device without adding energy storage density. Thus, the mass contribution of electrode materials should be predominant, requiring utilization of relatively thick electrode layers. For proper evaluation of the performance of thick layers of electrode materials in real devices, it is also important to study volume specific characteristics (C<sub>V</sub>, E<sub>V</sub>, P<sub>V</sub>), i.e. characteristics calculated per volume unit [19].

Optimal characteristics of electrochemical storage devices varies significantly according to their type of application [20].

* Mobile devices (telephones, computers) require relatively low power output, but high specific energy for long autonomy. On the other hand, for these applications the price acceptance range is relatively high: 2-10 Euro/kW.h, which is 20-100 times higher than the regular price of kW.h in an electrical grid.

* Autonomous means of transportation (cars, buses, trains) require high specific energy capacitance in the range of few tens of kW.h/kg in order to provide few hundreds of kilometers of transportation.

* For stationary (domestic or industrial) applications, a wide range of power output is required in order to ensure uninterrupted functioning of facilities: from few hundreds of W to few tens of MW.

There is no single type of electrochemical storage device that corresponds to this wide variety of demands. In many cases, including automotive transportation, a combination of different types of EESD is very beneficial. For example, combining high energy density batteries with high power output capacitors improves efficiency of energy storage in electrical cars and smart energy grids [21]. Table 1.1 summarizes the range of the typical characteristics for various types of electrochemical energy storage devices. More details can be seen in Fig.1.1 [1].

In this work we focus on novel electrode materials for a particular type of EESD, namely, supercapacitors. In the following section, the principles of supercapacitors, and their similarities and differences comparing to other types of EESD are discussed.
<table><caption>Table 1.1: Typical range of characteristic parameters of electrochemical storage devices.</caption><thead><tr><th>Storage device</th><th>Voltage, V</th><th>Specific Energy, W.h/kg</th><th>Specific Power, W/kg</th><th>Typical charge / discharge time</th><th>Life span</th></tr></thead><tbody><tr><td>Capacitors</td><td>up to 10<sup>3</sup></td><td>10<sup>-1</sup> – 10<sup>-2</sup></td><td>> 10<sup>5</sup></td><td>ms</td><td>unlimited</td></tr><tr><td>Supercapacitors</td><td>1-3</td><td>≈ 10</td><td>≈ 10<sup>4</sup></td><td>s - ms</td><td>at least 80% retention in 10<sup>5</sup> cycles</td></tr><tr><td>Batteries</td><td>1-3</td><td>> 10<sup>2</sup></td><td>≈ 10 – 10<sup>2</sup></td><td>h - s</td><td>at least 80% retention in 10<sup>3</sup> cycles</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>Figure 1.1: Ragone plots for electrochemical energy storage devices [1].</figcaption></figure>
## 1.2 Supercapacitors

Supercapacitors (also known as ultracapacitors) are the devices with specific energy and power characteristics intermediate between solid state electrostatic capacitors and batteries. Supercapacitors are capable to store significantly more energy comparing to solid state capacitors. On the other hand, in comparison with batteries, they can provide much higher power outputs, being charged and discharged much more rapidly, and sustain up to 100 times more charge/discharge cycles without significant loss of performance. Another advantages of supercapacitors over batteries are the wider operational temperature range (down to -40°C) and safer operation [22]. Outstanding reversibility of supercapacitors, short charging time, and high power output make them optimal storage devices for following applications:

* energy backup systems,

* kinetic energy recovery systems,

* portable high power electric tools,

* defibrilators.

A capacitor is an electric device which under the application of voltage $V$ stores electrical charge $Q$ between two conducting plates separated by an insulator. Capacity $C$ is defined as a charge stored per applied voltage:

$$C = \frac{Q}{V}. \qquad (1.1)$$

Capacity $C$ depends on the area of electrodes $A$, distance $d$ between electrodes, and the dielectric constant $\varepsilon$ of the insulator between the electrodes:

$$C = \frac{\varepsilon\varepsilon_0}{d}A, \qquad (1.2)$$

where $\varepsilon_0 = 8.854 \cdot 10^{-12}$ F/m is the electric constant (vacuum permittivity).

As mentioned above, from the practical point of view it is also important to define mass-specific capacitance of the device: $C_g = C/m$ (in F/g), where $m$ is a mass of the capacitor. Mass specific energy $E$ and specific power $P$ are given by
(1.3) and (1.4):

$$E = \frac{C_g V^2}{2}, \qquad (1.3)$$

$$P = \frac{V^2}{4mR_{ESR}}. \qquad (1.4)$$

Here $R_{ESR}$ is an equivalent serial resistance of a capacitor.

Table 1.2 gives some typical values of $\varepsilon$ for various solid dry insulators, utilized in solid state capacitors, and water. Due to mechanical properties of solid insulators, smallest practical distance $d$ between the electrodes of solid state capacitors has an order of $\mu$m. Thus, solid state capacitors offer surface specific capacitance $C_s$, defined as a capacity per surface area of electrode $C_s = C/A$, in the range of $10^{-9}$ F/cm$^2$.

In contrast to solid state capacitors, in electrochemical capacitors the electrodes are separated by an electrolyte, for example, an aqueous solution of an inorganic salt. In this case, the applied voltage is limited by the potential range of stability of a solvent; for example, for water this range is 1.23 V. Thus, electrochemical capacitors have much lower working voltage range $V$, and, lower specific power $P$, see eq.(1.4), than solid state capacitors. However, electrochemical capacitors have significantly higher specific capacitance, providing higher range of specific energy $E$ despite of lower $V$. High specific capacitance of electrochemical capacitors is related to the formation of *electrical double layer* at the interface between electrode and electrolyte, which will be explained in the next subsection.

### 1.2.1 Electrode-electrolyte interface

Physical models of an interface between an electrode (electronic conductor) and a liquid electrolyte (ionic conductor) have been developed during the last century

<table><caption>Table 1.2: Typical dielectric constants $\varepsilon$ for some insulators.</caption><thead><tr><td>Insulator</td><td>ε</td></tr></thead><tbody><tr><td>Dry air</td><td>1</td></tr><tr><td>Paper</td><td>≈2-3</td></tr><tr><td>Resins</td><td>≈4-5</td></tr><tr><td>Glass</td><td>≈10</td></tr><tr><td>Ceramics</td><td>≈20-30</td></tr><tr><td>Water</td><td>≈80</td></tr></tbody></table>
[23, 24]. Electrolyte is considered as a homogeneous media containing cations and anions, as well as dipoles of a solvent. Electrolyte species are interacting with electrode via electrostatic force (physisorption) or chemical interaction (chemisorption). This results in a potential-dependent adsorption of electrolyte species at the interface. At a certain potential of electrode, anions and cations are adsorbed in the quantities which provide equal values for negative and positive charges. Therefore, total charge at the interface is equal to 0, and this particular potential is called *potential of zero charge* $E_{pzc}$. The value of $E_{pzc}$ depends on the electrode properties (in particular, on $E_F$ - the Fermi level of electrons in the electrode material) and the interaction of electrolyte species with the electrode surface. It means that the value of $E_{pzc}$ is specific for a given electrode/electrolyte system.

If the electrode potential is different from $E_{pzc}$, for example if $E > E_{pzc}$, then the adsorption of anions and the negative-end orientation of solvent dipoles are predominant, resulting in a negative charge accumulated at the electrolyte side of the interface. This negative charge is compensated by the same value of excessive positive charge inside the electrode. The electrolyte species distribution at the interface is depicted in Fig.1.2.

A distribution of electrolyte species on the electrode/electrolyte interface results in the formation of the *electrical double layer* (EDL) which consists, as the name suggests, of 2 layers of adsorbed species.

First, closest to the electrode surface, there is a layer of adsorbed species which have partially lost their mobility due to an interaction with charged electrode surface. This is the so-called *dense layer*, or the *Helmholtz layer*. If the interaction is strong enough, electrolyte ions partially lose their solvation shells and approach electrode surface to a very short distance of few Å, thus forming the *inner Helmholtz plane*, IHP. This phenomena results from the *specific adsorption* of ions, which is more common for anions than for cations, because the latter are in general more strongly solvated. Specific adsorption may also involve partial transfer of a charge to/from an adsorbed ion. This phenomena is important for supercapacitors electrodes, because it is a part of the *pseudocapacitance* phenomena, which will be considered in more details in the Subsection 1.2.3.

If energy of ion interaction with electrode material is not strong enough for partial loss of solvation shell, then the solvated ions are located at the plane which is distanced further from the electrode surface. This plane is called *outer Helmholtz plane* (OHP). The distance between the electrode surface and the OHP ("thick-
<figure><img src="image_4.png" /><figcaption>Figure 1.2: A model of metal/electrolyte interface [24].</figcaption></figure>

ness" of the Helmholtz layer) depends on the nature of the electrode and the electrolyte. This thickness is determined mostly by the size of solvated ions and has values in the range of 0.4-0.8 nm [23].

The electric field created by excess charge at the electrode side of interface is, in general, not confined within the OHP and propagates further into the electrolyte. The propagation of the field results in preferential adsorption of ions with an opposite charge to the electrode. These ions are solvated and mobile due to thermal movement, and they form the second part of the EDL: the *diffuse layer* (the *Gouy-Chapman layer*). Contrary to the Helmholtz layer, the thickness of the Gouy-Chapman layer depends strongly on the electrolyte concentration and temperature. The thickness of Gouy-Chapman layer spans from ca. 1 nm for concentrated electrolytes up to few tens of nm for diluted solutions.

Summing up, a modulation of electrode potential results in adsorption/desorption of electrolyte species and accumulation of a certain charge at the electrode/electrolyte interface. Thus, from the electrochemical point of view, the EDL acts as a capacitor, where electrode makes one plate and the ions of the electrolyte form another
<figure><img src="image_5.png" /><figcaption>Figure 1.3: Dependence of the EDL capacitance formed on the Bi electrode in aqueous KF solutions with different concentrations [23].</figcaption></figure>

plate. As the EDL is formed by serial connection of the Helmholtz and Gouy-Chapman layers, its capacity $C_{dl}$ is given as follows:

$$ \frac{1}{C_{dl}} = \frac{1}{C_{H}} + \frac{1}{C_{GC}}, \qquad (1.5) $$

where $C_H$ and $C_{GC}$ are the capacities of the Helmholtz and Gouy-Chapman layers correspondingly.

As one can see in Fig.1.3, the capacity of the Gouy-Chapman layer $C_{GC}$ depends on the electrode potential: $C_{GC}$ is the lowest at $E = E_{pzc}$ due to the shortest electric field propagation from the electrode to the electrolyte. $C_{GC}$ also depends on the electrolyte concentration: the thickness of the Gouy-Chapman layer is larger in less concentrated solutions, resulting in the lower $C_{GC}$ (see (1.2)). The equation (1.5) indicates that the total capacity of the EDL is determined by its smallest part, which is $C_{GC}$ for diluted solutions and $E \approx E_{pzc}$. Thus, the interfacial capacitance $C_{dl}$ is higher in concentrated electrolytes. The dependence of $C_{dl}$ on the potential is weaker in concentrated electrolytes, but nevertheless observable.

This dependence of electrochemical interfacial capacitance on the electrode potential and electrolyte concentration is reflected in cyclic voltammograms (CV). For solid state capacitors, the diagram of the charging current $I$ depending on the linearly changing applied voltage $V$ (i.e. cyclic voltammogram $I-V$) is square-like. This shape of CV reflects the fact that the capacitance of a solid state capaci-
<figure><img src="image_6.png" /><figcaption>Figure 1.4: Dependence of the interfacial capacitance (top) and corresponding CV curves (bottom) of Au single crystal electrodes: (A) Au(111) and (B) Au(100), measured in 1 mM HClO₄ + 0.1 M KClO₄ (solid lines), in 0.1 M KClO₄ (dashed lines) [25].</figcaption></figure>

for *C* does not depend on the applied voltage *V*. However, as a capacitance of an electrochemical interface depends on an electrode potential *E* (Fig.1.4(A,B)), the resulting CV have more complex shapes. Thus, the charging current *I* depends on the potential *E*, as well as on the composition of electrolyte and even crystallographic orientation of the solid electrode surface (Fig.1.4(C,D) [25]).

For the analysis of interfacial capacitance of electrode materials with complex structure, we can assume that in concentrated electrolyte solutions the thickness of EDL does not strongly depend on the potential, and can be estimated to be ca. 1-2 nm. Considering a high value of water dielectric constant $\epsilon \approx 80$, the surface specific capacitance of an electrochemical electrode/electrolyte interface can be estimated using (1.2) as $C_s \approx 2 \cdot 10^{-5} \text{ F.cm}^{-2} = 20 \text{ } \mu\text{F.cm}^{-2}$. This value is in a good agreement with typical values experimentally observed for metal/electrolyte interfaces (Fig. 1.3, 1.4). It is important to emphasize that this value is more than 3 orders of magnitude higher than the value of $C_s$ for solid-state capacitors. Thus, electrochemical capacitors offer several orders higher energy density (see Fig.1.1), which explains the name "supercapacitors". Depending on the mechanism of charge storage and involved interfacial processes, two classes of supercapacitors are generally distinguished: electrochemical double layer capacitors (EDLC) and pseudocapacitors.
<figure><img src="image_7.png" /><figcaption>Figure 1.5: The scheme of a distribution of the electrical potential between the electrodes of a solid state capacitor (A) and an EDLC (B).</figcaption></figure>

### 1.2.2 Electrochemical double-layer capacitors

Fig.1.5 depicts the distribution of the electrical potential between two parallel electrodes of a solid state capacitor and an EDLC. Contrary to a solid state capacitor, an EDLC device is in fact a serial combination of two capacitors, corresponding to the EDL formed at both of electrodes (see equivalent circuits in Fig.1.5(A) and 1.5(B)). The capacity C of the circuit shown in Fig.1.5(B) is given by a serial combination of capacitors:

$$ \frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}. \qquad (1.6) $$

In a symmetric EDLC, the identical electrodes are used at both sides, i.e. $C_1 = C_2$, therefore the total capacity is equal to the half of the capacity of one electrode $C = C_1/2$.

The supercapacitor circuit contains also $R_e$ element, which is the electrical resistance of an electrolyte. The electrolyte is confined in a solid porous separator (polymer, fiber (paper, cotton) or ceramic) to prevent short-circuiting of the capacitor. In general, $R_e$ has a strongest and predominant contribution to the equivalent serial resistance $R_{ESR}$ of the capacitor. The contributions from the
other components, such as the electrodes and contacts, are optimized to have much lower values. The electrolytes with high ionic concentrations are generally used to ensure high ionic conductivity and sufficient amount of ions to form the EDL at the large surface area interface. From the practical point of view, preferable electrolytes should have low viscosity and volatility of solvent, low toxicity, and low cost [22]. There are 3 types of electrolytes generally used in supercapacitors.

* Aqueous electrolytes. Apart from obvious advantages of aqueous electrolytes, such as easy handling and low costs, water has very high dielectric constant $\epsilon$, which results in high interfacial capacitance and high solubility of inorganic ionic compounds. It allows to use highly concentrated electrolytes with high ionic conductivity in the range of ca. $1 \text{ Ohm}^{-1} \cdot \text{cm}^{-1}$. Viscosity of aqueous electrolytes is often decreased by adding soluble polymers (for example, polyvinyl alcohol (PVA)) in order to prevent leakage and avoid water evaporation. Apparently, the aqueous electrolytes cannot be utilized below freezing point, which limits the operational temperature range of a supercapacitor.

The main disadvantage of aqueous electrolytes is relatively narrow potential range of solvent stability limited thermodynamically to 1.23 V. Thus, the EDLC with aqueous electrolytes usually have operational voltage range $V$ equal or below 1 V. Considering that specific energy (see (1.3)) is proportional to $V^2$, the limited voltage operation window is a significant disadvantage of utilization of aqueous electrolytes in supercapacitors.

* Organic electrolytes. Acetonitrile (AN) and propylene carbonate (PC) are two most commonly used organic solvents in supercapacitors. Their physicochemical properties are compared with water in the Table 1.3. Organic solvents allow to extend the working temperature range, and, what is more important, the voltage range, which can be as high as 3.5 V [9]. However, organic solvents have lower dielectric constants, resulting in the lower interfacial capacities. Tetrafluoroborate salts of tetraalkylammonium are commonly used as ionic components of electrolyte with organic solvents due to their relatively high solubility.

Ionic conductivity of these electrolytes is ca. $\approx 0.01-0.06 \text{ Ohm}^{-1} \cdot \text{cm}^{-1}$, which is lower than that of aqueous electrolytes due to lower mobility and solubility of ionic compounds. In organic electrolytes the ionic transport is slower comparing to aqueous electrolytes. This results, in particular, in much
<table><caption>Table 1.3: Physicochemical properties of solvents most commonly used in supercapacitors</caption><thead><tr><th>Solvent</th><th>Formula</th><th>Dielectric constant ε</th><th>Melting point t<sub>m</sub>, °C</th><th>Boiling point t<sub>b</sub>, °C</th></tr></thead><tbody><tr><td>Water</td><td>H<sub>2</sub>O</td><td>80</td><td>0</td><td>100</td></tr><tr><td>Acetonitrile</td><td>H<sub>3</sub>C–CN</td><td>38.8</td><td>-46</td><td>81.3</td></tr><tr><td>Propylene carbonate</td><td>H<sub>3</sub>C–C<sub>3</sub>H<sub>3</sub>O<sub>3</sub></td><td>64</td><td>-48.8</td><td>242</td></tr></tbody></table>

stronger decrease in volumetric capacitance of the material with increase of its thickness [26]. Thus, despite having significantly higher specific energy for thin (lab-scale) film electrodes, supercapacitors with organic electrolytes often have lower specific energy and power. Also, organic electrolytes are significantly more difficult to handle due to their strong hydroscopicity: water content below 3-5 ppm is needed to maintain the properties of organic electrolytes.

* Ionic liquids (IL). IL are organic salts with a high low melting temperature. In particular, the room temperature ionic liquids remain in liquid state at ambient conditions and can be used as an electrolyte for supercapacitors. The advantages of IL as electrolytes are: (i) the wide voltage operation range (up to 6V [22]), (ii) the low volatility, (iii) the wide temperature range (down to -95°C), and (iv) the adjustable properties by design of chemical structure of cation and/or anion [27].

The main disadvantages are: (i) low ionic conductivity comparing to aqueous electrolytes (similar to concentrated organic electrolytes), and (ii) high costs of synthesis and handling.

Short thickness of the EDL allows to use the materials with highly developed surface as the electrodes for EDLC, for example, highly porous carbon materials (see Fig.1A). Carbon materials offer mass specific surface area (SSA) as large as 3000 m²/g [7]. On the other hand, carbon/electrolyte interface has lower surface specific capacitance in comparison to metal/electrolyte interface, i.e. below 20 μF.cm⁻² [28]. For a single-layer sp² carbon graphene sheet, the value of 13.5 μF.cm⁻² is predicted and expected to decrease with the thickness of the carbon stack down to ca. 5 μF.cm⁻² for graphite structure. Thus, for carbon materials with highly developed SSA, the specific capacitance is expected to be in the range
<figure><img src="image_8.png" /><figcaption>Figure 1.6: Schematic equation of quinone - hydroquinone surface redox transformation.</figcaption></figure>

150-675 F/g. In the Chapter 2 different types of carbon materials utilized in EDLC will be discussed.

### 1.2.3 Phenomena of pseudocapacitance

The pseudocapacitors are the capacitor devices relying on the pseudocapacitance as a main mechanism of a charge storage. The phenomena of pseudocapacitance is explained below.

In EDLC the charge is stored at the electrode/electrolyte interface due to the electrochemical adsorption of electrolyte ions, i.e. due to the formation of interfacial electrochemical double layer. This is the case when the material of an electrode is electrochemically inert under the applied potential, i.e. no change in oxidation state of any component of electrolyte or electrode occurs during the potential modulation. As the interfacial capacitance in concentrated solutions does not strongly depend on the potential (Fig.1.3), the shape of the CV curves for these electrodes is close to rectangular. This indicates that the current of charging/discharging of double layer is nearly constant, i.e. is practically independent from potential.

However, many of electrode materials undergo surface electrochemical transformations at a certain potential. The following electrochemical phenomena can occur: oxidation/reduction of surface oxides, partial charge transfer to/from adsorbed electrolyte species (for example, electrochemical adsorption/desorption of atomic hydrogen).

For carbon surfaces, the characteristic surface redox transformation is related to the presence of quinone-type groups on partially oxidized surface [29] (Fig.1.6). These groups are responsible for the reversible quinone-hydroquinone electrochemical transition [3], reflected by an appearance of distinct peaks on the CV curves around $E_0 = 0.668$ V (NHE). This potential is claimed to be the equilibrium
<figure><img src="image_9.png" /><figcaption>Figure 1.7: CV curves of carbon Vulcan-2700 measured in 1M H₂SO₄ at 14 mV/s, before (solid lines) and after (dashed lines) electrochemical oxidation in 96% H₃PO₄ at 160°C for 45 hours [29].</figcaption></figure>

potential for surface quinone-hydroquinone redox couple (Fig.1.7). The presence of this reversible redox transition results in the potential-dependent charge transfer to/from the electrode. As this redox reaction is reversible, it provides another mechanism of charge storage upon potential modulation similar to the double layer interfacial capacitance. Thus, this phenomenon is referred to as "pseudocapacitance".

Psuedocapacitance increases the total capacitance of electrode/electrolyte interface. For example, redox reaction (1.6) requires $1 e^-$ per 1 quinone-type surface group. For various types of Vulcan carbon, the surface density of the quinone groups has been estimated as 0.09-1.11 %wt. [29] or $10^{-10}-10^{-11} \text{ mol} \cdot \text{cm}^{-2}$ (by the author's estimations), depending on the oxidation state of carbon surface. Thus, according to the Faraday law, Vulcan carbon pseudocapacitance, related to presence of quinone surface groups, provides additional $1-10 \mu F \cdot \text{cm}^{-2}$ as calculated for 1 V potential range. This contribution is comparable with the EDL capacitance value of carbon surface.

A low surface specific capacitance of $sp^2$ carbon surface can be increased in the presence of heteroatoms of non-metal elements at the interface, such as N, S, B. The effect of N-doping on the properties of carbon materials, in particular, the improvement of charge storage capacitance, is studied most thoroughly [30].
<figure><img src="image_10.png" /><figcaption>B: pyrrole</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>C: pyridine</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>D: pyridone</figcaption></figure>

<figure><img src="image_13.png" /></figure>

Figure 1.8: Some functional N-containing groups that can be formed at the carbon/electrolyte interface (Fig. adapted from [31]).

Nitrogen can be incorporated into the carbon structure in various coordination states. These states are characterized by their specific electronic configurations as reflected by N1s XPS BE (binding energy) values: pyridine (398.4 ± 0.2 eV), imine (398.8 ± 0.2 eV), pyrroles (400.6 ± 0.2 eV), pyridones (400.6 ± 0.2 eV) and oxidized nitrogen (nitrito/nitroso groups) (400.7 ± 0.2 eV) [31], see Fig.1.8. An increase in charge storage capacitance due to the presence of N atoms can be caused by the following effects.

* **Better wettability of carbon surface.** Nitrogen increases the polarization of surface bonds [30], and, in general, increases hydrophilicity of the carbon surface. This effect is, however, claimed to be minor [32]: a higher capacitance of a B-doped carbon electrode comparing of an N-doped electrode is reported, despite better wettability of the latter.

* **Better conductivity.** Nitrogen is more electronegative than carbon and provides additional electron into π-conjugated electron state, resulting in increased conductivity. However, nitrogen disturbs otherwise uniform carbon structure, resulting in possible decrease in conductivity. Thus, N-doping of carbon can have both positive and negative influence on conductivity, and, in general, an optimal composition and type of functionalization must be found.
For example, N-doping of carbon nanofibers up to 3.1%$_{wt.}$ N resulted in an increase of their conductivity by more than 50%. However, further increase in N-content up to 8.2%$_{wt.}$ resulted in ca. 33% lower conductivity in comparison to the pristine material [33]. It is important to note that in general carbon materials in supercapacitors already have high conductivity, and an increase in conductivity due to N-doping does not significantly influence their performance [32].

* **Pseudocapacitance.** The N-doped carbon electrodes are known to have strongly enhanced activity in the electrocatalytic oxygen reduction reaction (ORR) [34, 35, 36] comparing to the bare carbon electrodes. The exact nature of active sites on the surface of N-doped carbon electrocatalysts is still debated. However, the enhancement of the activity by N-doping has been attributed to (i) increased polarisation of chemical bonds at electrode/electrolyte interface and (ii) adjustment of the conjugated carbon π-electrons density [35]. Altogether, these effects result in stronger interaction of electrolyte components with carbon surface, facilitating their adsorption and further redox transition. It has been argued [32] that these effects are also responsible for the enhanced intercalation of electrolyte components ($H^+OH^-$) in the sub-surface interfacial region and further into the bulk of carbon materials. The enhanced intercalation results in increase of the charge related to pseudocapacitance phenomena. In fact, it has been claimed that the enhanced interaction between electrolyte species and carbon surface is the main reason for increasing charge capacitance for the N-doped carbon electrodes.

Contrary to the EDL capacitance, the pseudocapacitance is not confined to the electrode/electrolyte interface but can propagate further into the bulk of the electrode. An impressive example is a hydrogen cation intercalation into carbon in acidic electrolytes, which was studied in details on activated carbon CH900-200 ACC (Japan), SSA = 1520 m²/g) [37, 38]. It was concluded that the maximal capacitance of intercalation, obtained by prolonged (several hours) galvanostatic cathodic polarization, corresponds to the reversible formation of a stoichiometric compound "hydrogen carbide" $C_6H$, i.e. to the storage of 1 e⁻ per 6 carbon atoms:

$$C_6 + H^+ + e^- \rightleftharpoons C_6H \quad (1.7)$$
This intercalation provides the maximal contribution of pseudo-capacitance for the studied CH900-200 carbon equal to 942.8 F/g. This results in outstanding total specific capacitance 1114.3 F/g, which is, to the best of author's knowledge, is the highest specific capacitance reported for carbon electrodes. However, the hydrogen intercalation into carbon requires the acid-induced swelling of carbon graphite structure, which is a slow process. In [38] a model of charge storage pseudocapacitance in activated carbon limited by slow solid-state diffusion of intercalated hydrogen was proposed. The estimated diffusion coefficient of hydrogen in carbon has a value $D \approx 5 \cdot 10^{-17}$ cm$^2$/s. This value is few orders of magnitude lower than the one reported for solid-state diffusion coefficients of Li$^+$ cations in the graphite anode of Li batteries [39, 40]. Such slow diffusion makes the above mentioned high capacitance of the CH900-200 ACC impractical.

Another important difference between the EDL capacitance and pseudocapacitance is the strong dependence of the latter on the electrode potential. The pseudocapacitance is the strongest at the potential corresponding to the reversible potential of redox transformation $E_0$. This fact is reflected by the pronounced pair of redox peaks on the CV curve around $E_0$. This pair of peaks is centered around $E_0$, and the peaks separation depends on the rate of the process [41]. The CV peaks separation is equal to ca. 60 mV for the fast redox transformation which is not limited by the solid-state diffusion, e.g. the surface redox transitions (Fig.1.7). However, if the process is strongly controlled either by the slow kinetics of the interfacial charge transfer or by the solid-state diffusion, then the peaks separation can be as strong as few hundreds of mV (Fig. 1.9). The strong dependence of $C_g$ on the electrode potential makes misleading its utilization as a characteristic for the performance of a supercapacitor electrode material [42]. At the potential range around $E_0$, $C_g$ has much higher values than expected for the EDL capacitance. However, these high $C_g$ values are observed only in a limited range of the electrode potential. According to (1.3), a limited potential range provides a low energy density despite the high $C_g$ values. Thus, in general, the energy density is a better characterization parameter for the efficiency of pseudocapacitance materials.

The pseudocapacitance is a main phenomenon involved in the charge storage in metal oxide materials, which form the second biggest class of materials for supercapacitors electrodes [1, 22, 9, 43]. Similarly to the oxidized carbon surface, the atoms of metal at or in the vicinity of the electrode surface can undergo a reversible
<figure><img src="image_14.png" /><figcaption>Figure 1.9: The capacitance-potential curves of the activated carbon CH900-200 in 48% H₂SO₄ measured at the potential sweep rates 0.5 mV/s (1), 1 mV/s (2), 2 mV/s (3,4). The pairs of peaks in the range of -0.4 .. 0.4 V correspond to the hydrogen intercalation, according to (1.7) [37].</figcaption></figure>

redox transformation due to the variability of oxidation states for transition metals. In comparison to the carbon, the percentage of metal sites available for the redox transformation is much higher: every metal atom is capable to change its oxidation degree and participate in the charge storage. This phenomenon provides significantly higher pseudocapacitance values for metal oxides comparing to carbon materials. The values above 900 F/g are routinely observed for the RuO₂-based electrode materials, which are the ones of the best performing oxide materials for supercapacitors. As for the carbon electrodes, different mechanisms of pseudocapacitance of the metal oxides can be proposed: the surface redox-transitions and the intercalation. However, these mechanisms are related for the metal oxides. For example, for RuO₂, which is the most studied metal oxide supercapacitor material, the redox transition is described by the reaction (1.8) [44]:

$$ \mathrm{RuO_2} + \delta\mathrm{H^+} + \delta\mathrm{e^-} \rightleftharpoons \mathrm{H_\delta RuO_2} \quad (1.8) $$

The transformation degree and the stored charge are characterized by the intercalation parameter $\delta$, $0 < \delta < 1$. The reaction starts as the electroreduction of the RuO₂ surface and then propagates into the bulk of oxide. According to (1.8), this process can be also considered as the H-intercalation. There is also an ongoing discussion on the possibility of the intercalation of another electrolyte cations beside H⁺, for example, K⁺. The exact nature of the pseudocapacitance phenomena for various types of oxide materials is discussed in details in the Section 3.
Before starting the discussion on the particular types of electrode materials for supercapacitors, the important difference between batteries and pseudocapacitors is outlined in the next Subsection.

### 1.2.4 Pseudocapacitors and ion-insertion batteries

The cation intercalation is an important mechanism of pseudocapacitance in the materials for supercapacitor. The cation intercalation is also a charge storage mechanism in the batteries: in particular, the $Li^+$ cations are intercalated into the graphite cathodes in the conventional Li batteries [45]:

$$C_6 + Li^+ + e^- \rightleftharpoons LiC_6 \quad (1.9)$$

Apparent similarities between redox intercalation reactions (1.7) and (1.9) require the definition of the criterion to distinguish between the electrodes behavior of batteries and supercapacitors [8, 42]. It is important to emphasize that the main differences in the performance of batteries and supercapacitors are the higher power output and the cyclability of the latter. Thus, the intercalation reaction in supercapacitors should be relatively fast and not induce the significant structural changes of the electrode material.

In general, the intercalation reaction can be expressed as:

$$\delta Ct^+ + \delta e^- + M \rightleftharpoons M - Ct_{\delta} \quad (1.10)$$

Here $Ct^+$ is an electrolyte cation, $\delta$ is an intercalation parameter, M is an intercalated host material, and $M-Ct_{\delta}$ is an interfacial layer of M with the intercalated cation. The rate of this reaction can be expressed by the equation [40], which is similar to the general Butler-Volmer equation combined with the Frumkin isotherm:

$$i = i_0 \left( (1 - \delta)e^{-(1-\alpha)g\delta}e^{(1-\alpha)f(E-E_0)} - \delta e^{\alpha g\delta}e^{-\alpha f(E-E_0)} \right). \quad (1.11)$$

Here $i$ is a current density of the intercalation reaction, $i_0$ is an exchange current density of the intercalation reaction, $\alpha$ is a symmetry factor ($\alpha = 0.5$ for reversible intercalation [40]), $g$ is the constant of the Frumkin isotherm, characterizing the lateral interaction between the intercalated ions ($g<0$ for attractive interaction and $g>0$ for repulsive interaction), $f = F/RT$, $E$ is an electrode potential, and
$E_0$ is a equilibrium potential of the reaction (1.10).

If the potential is changed relatively slow (slow sweep rate or slow charging current), then $\delta$ corresponds to the potential-dependent equilibrium between intercalation and de-intercalation according to the Frumkin isotherm:

$$ \frac{\delta}{1 - \delta} = e^{f(E - E_0)} e^{-g(\delta - 0.5)}. \qquad (1.12) $$

Let us note that the equations (1.11) and (1.12) are given for the electrode/electrolyte interface. These equations are valid only if the solid state diffusion is sufficiently fast, so that the gradient of $\delta$ within the bulk of host material is negligible and is not influencing interfacial processes. In [40] it is shown that for the Li intercalation into $Li_xCoO_2$ the model is valid at the sweep rate range of 10-50 $\mu$V/s. At a faster potential change the solid phase diffusion influences and controls the rate of interfacial process. Therefore, the equations (1.11) and (1.12) are no longer valid.

In order to evaluate the reversibility of an intercalation reaction, Simon et al. [8] proposed to analyze the dependence of the CV peaks current density $j$ on the potential sweep rate $v$: $j = av^b$. The power $b$ is an experimental quantitative criteria of the reversibility. The value of $b$ is changing, depending on the rate-determining step of intercalation process. Namely, $b = 1$ for the reversible interfacial reaction and $b = 0.5$ for the reaction limited by the semi-infinite diffusion of the reagents. For example, the data of [37] for the capacitance of CH900-200 shows the constancy of the charging current normalized by the potential sweep rate $v$, i.e. $b = 1$. Thus, according to the criteria [8], the studied carbon electrode acts as a supercapacitor electrode.

However, it should be noticed that even the battery electrode materials can demonstrate reversible electrochemical behavior in agreement with the equations (1.11) and (1.12) at sufficiently low rate of the potential change. As it has been mentioned above, this was observed experimentally for the $Li^+$ intercalation to $Li_xCoO_2$ [40] at the sweep rates below 50 $\mu$V/s and to the graphite at the sweep rates below 4 $\mu$V/s.

Thus, the reversibility of the intercalation reaction cannot be the only criterion that allows to classify a material as a supercapacitor electrode material. As mentioned above, the high power output and capacitance retention are the main performance criteria to distinguish supercapacitors from batteries. Thus, the tests of reversibility of charging/discharging have to be performed at the conditions,
which are expected for the supercapacitor operations: i.e. the charging time of the order of a minute (60C in terms of the Li batteries characterisation). For example, for a capacitor with the operational potential range of 1V, such charging time suggests the rate of potential cycling of ca 16.7 mV/s; or, for the case of galvanostatic cycling and the specific capacitance of 120 F/g it suggests the charging current of 2 A/g. If the characterization of electrode performance is done under these conditions and reversible behavior is observed, then it can be concluded that the electrode material behaves as a capacitor, even though if the mechanism of the charge storage involves the intercalation reaction.

The performance of the materials as the supercapacitor electrodes can be studied in various configurations (cells with 2 or 3 electrodes, membrane separator or liquid electrolytes) and by various electrochemical techniques. The most commonly used techniques are the cyclic voltammetry (CV) and the galvanostatic charging with potential limits (GCPL). Another important electrochemical technique allowing detailed study of processes on the electrode/electrolyte interface is the electrochemical impedance spectroscopy (EIS). A comparison of the capacitance analysis using these 3 groups of methods (CV, GCPL, EIS) was given in [46]. It was concluded that the methods provide consistent data, however a special care (specific to each of the methods) should be taken to ensure the validity of the results. The EIS, in particular, requires a relatively small potential modulation (few mV) and, in general, a multi-parameter model to extract the relevant data. Under these conditions, the EIS provides the most detailed insight into the interfacial processes. Thus, in the next section, the EIS is discussed in details. Particularly, a model of the impedance of porous electrodes is presented. The model is applied to predict the optimal structures of electrode materials for the best performance in supercapacitors.

## 1.3 Impedance of electrochemical electrode/electrolyte interface

The electrochemical impedance spectroscopy is a powerful tool for studying the processes on the electrode/electrolyte interface [47]. The interfacial impedance is measured as a function relating the modulating electrode potential $\vec{E} = E_0 \cdot \sin(\omega t)$ and the current passing through the electrode/electrolyte interface $\vec{I} = I_0 \cdot \sin(\omega t+$
$\phi_0$):

$$ \vec{Z}(\omega) = \frac{\vec{E}(\omega)}{\vec{I}(\omega)}. \qquad (1.13) $$

Here $\omega = 2 \cdot \pi f$ is an angular frequency, $f$ is a modulation frequency, and $\phi$ is a phase shift angle between $\vec{E}$ and $\vec{I}$. As discussed above, a modulation of the electrode potential may result in a number of interfacial processes, which are generally classified as non-faradaic and faradaic. In a non-faradaic process neither electrolyte species, nor electrode components change their oxidation states, i.e. there is no redox (faradaic) process. Generally, 3 types of interfacial processes may occur.

* **Adsorption/desorption.** The adsorption of ions on the electrode surface results in the formation of the EDL [23] and accumulation of the charge on the interface, similar to the charge accumulation in a capacitor. Thus, the interfacial impedance of the process is given as

$$ \vec{Z}_C = \frac{1}{i\omega C_{dl}}, \qquad (1.14) $$

where $C_{dl}$ is an EDL capacitance. $C_{dl}$ can be calculated as $C_{dl} = C_s \cdot A$, where $A$ is a surface area of the electrode/electrolyte interface. $C_s$ is a surface specific interfacial capacitance, which depends on the nature of the interface and is generally found in the range of 5-30 $\mu F/cm^2$. According to (1.14), the phase angle of $\vec{Z}_C$ is $\phi = 90^\circ$ (Fig.1.10(1)).

* **Interfacial charge transfer.** If at least one component of an electrolyte or an electrode changes its oxidation state due to the potential modulation, then the redox faradaic process occurs at the electrode/electrolyte interface:

$$ \mathrm{Ox} + \mathrm{ne}^{-} \overset{k_0}{\rightleftharpoons} \mathrm{Red} \qquad (1.15) $$

If the electrode potential is close to the equilibrium potential $E_{Ox/Red}^0$ and the potential modulation is small, then $\vec{I}$ changes linearly with $\vec{E}$, similarly to Ohm's law. Thus, the impedance of the interfacial charge transfer can be evaluated as the charge transfer resistance $R_{ct}$ with $\phi = 0^\circ$:

$$ R_{ct} = \frac{RT}{nF i_0}. \qquad (1.16) $$
Here $i_0$ is an exchange current rate of the process, which is related to the kinetic parameters of the reaction:

$$i_0 = F A k_0 C_{\text{Ox}}^{1-\alpha} C_{\text{Red}}^{\alpha}, \qquad (1.17)$$

where $C_{Ox}$ and $C_{Red}$ are the concentrations of Ox and Red in the electrolyte, $\alpha$ is a symmetry factor of a given electrochemical reaction (1.15), $0 < \alpha < 1$, $k_0$ is a heterogeneous rate constant of the reaction (1.15).

* **Diffusion to/from the interface.** According to the equations (1.16),(1.17), the value of $R_{ct}$ depends on the concentration of Ox and Red. If the reaction (1.15) is fast, then the concentrations of reagents at the interface are not the same as in the electrolyte. Thus, for the fast reactions, the rate of reaction (1.15) is influenced by the rate of diffusion of reagents to/from the interface. In the impedance measurements, this influence is reflected by an additional component of the impedance in a serial combination with $R_{ct}$, namely, the Warburg impedance $\vec{Z}_W$:

$$\vec{Z}_W = \frac{\sigma}{\sqrt{\omega}} + \frac{\sigma}{i\sqrt{\omega}}. \qquad (1.18)$$

The parameter $\sigma$ is the Warburg constant. It is related to the parameters of the reagents diffusion in an electrolyte:

$$\sigma = \frac{RT}{n^2 F^2 A \sqrt{2}} \left( \frac{1}{C_{\text{Ox}}\sqrt{D_{\text{Ox}}}} + \frac{1}{C_{\text{Red}}\sqrt{D_{\text{Red}}}} \right). \qquad (1.19)$$

Here $D_{Ox}$ and $D_{Red}$ are the diffusion coefficients of Ox and Red in the electrolyte. From (1.18) it can be seen that the Warburg impedance $\vec{Z}_W$ has the phase angle $\phi = 45^{\circ}$ ($\text{Re}(\vec{Z}_W) = \text{Im}(\vec{Z}_W)$).

### 1.3.1 Impedance of the flat electrode/electrolyte interface.

In the absence of redox processes on the flat electrode/electrolyte interface, its impedance can be represented by a serial combination of $R_1$ and $C_{dl}$, where $R_1$ depends on the conductivity of an electrolyte and the geometry of the electrochemical system. $R_1$ can be measured as a limit of $\vec{Z}(\omega)$ at high $f$.
<figure><img src="image_15.png" /><figcaption>Figure 1.10: The schematic representation of the Nyquist diagram for the non-faradaic impedance $\vec{Z}_{nf}$: (1) flat electrode (1.14), (2) semi-infinite cylindrical pore, (3) finite cylindrical pore, (4) an electrode with pores of different diameters.</figcaption></figure>

When a redox process occurs on the interface, the interfacial impedance includes a faradaic impedance component $\vec{Z}_f$. At a small potential modulation, $C_{dl}$ and $\vec{Z}_f$ can be considered as independent components. Therefore, the interfacial impedance can be modeled as their parallel combination (Fig.1.11(A)).

In a simplest case of a faradaic process, namely, the reversible electrochemical reaction, controlled by its kinetics and the diffusion of electrolyte components, $\vec{Z}_f$ is modeled by the Randles circuit (Fig.1.11(B)). This circuit consists of a serial combination of 2 components: $R_{ct}$ and $\vec{Z}_W$.

The Randles model predicts that $\lim_{\omega\to\infty}(\vec{Z}) = R_1$ and $\lim_{\omega\to 0}(\vec{Z}) = R_1 + R_{ct} + Z_w$, as it is observed, for example, for the reversible redox transformation of the FeII/FeIII-cyanide complex on the Au polycrystalline electrode [48]. Fitting experimental impedance spectra by the Randles model allows to determine $k_0$, $D_{Ox}$ and $D_{Red}$ for a reversible faradaic process on the flat electrode.

For more complex reactions, the model of Faradaic impedance has to be modified. For example, if the reaction involves the formation of adsorbed intermediate, it results in an accumulation of a charge on the interface and adds a capacitive component into $\vec{Z}_f$. The impedance of this process is described by the Melik-Gaykazyan model impedance [49](Fig.1.11(C)).
<figure><img src="image_16.png" /><figcaption>Figure 1.11: (A) The model equivalent circuit of the impedance of the flat electrode/electrolyte interface in the presence of a faradaic process. The models of the faradaic impedance $\vec{Z}_f$: (B) the Randles model, (C) the Melik-Gaykazyan model, (D) pseudocapacitance model.</figcaption></figure>

### 1.3.2 Models of impedance of rough electrode/electrolyte interface

The models from Subsection 1.3.1 describe the impedance of flat uniform electrode/electrolyte interface, for which the current lines are parallel and evenly distributed, and also are perpendicular to the interface. However, the materials with rough surfaces and porous materials are commonly used in the EESD in order to increase surface and volume specific energy densities.

For the rough electrode surface, the current lines, directing toward the surface, are not parallel. This results in uneven conditions of the interface polarization. In this case the frequency dependence of impedance parameters, such as $C_{dl}$ (1.14), $R_{ct}$ (1.16) and $\sigma$ (1.19), is observed [50]. For the non-faradaic impedance, this frequency dependence can be taken into account by using the model of constant phase element (CPE):

$$ \vec{Z}_{CPE} = \frac{1}{(i\omega C_{dl})^a}. \qquad (1.20) $$

Here $a$ is the CPE constant; if $a=1$, then $\vec{Z}_{CPE} = \vec{Z}_{nf} - R_1$, and the CPE acts as an ideal capacitor with the frequency-independent $C_{dl}$. For the rough electrodes, the frequency dependence of the measured $C_{dl}$ is described by $\vec{Z}_{CPE}$ with $a < 1$. For example, $a \approx 0.9$ for the polycrystalline Au / $HClO_4$ interface, while $a \approx 0.99$ for the Au single crystal electrode [51].

The detailed finite element analysis demonstrated that the effect of the capacitance frequency dispersion on the non-faradaic impedance is observed at high frequencies $f > 10$ kHz for not very rough electrodes [52, 53]. However, for the adsorption with the interfacial partial charge transfer (i.e. the specific adsorption),
the frequency dispersion of $R_{ct}$ and $C_{ads}$ is predicted even for low frequencies [54].

In general, the application of the CPE model allows to fit the experimental impedance spectra and determine, for example, the $C_{dl}$ capacitance even for rough or non-uniform electrodes. However, due to the multiple possible origins of $a < 1$, one has to be certain to understand the properties of a particular electrode/electrolyte interface and evaluate the applicability of the CPE model in each particular case. On the other hand, if the utilization of the CPE model is valid, the value of $a$ provides the additional information about the structure of the electrode/electrolyte interface. For example, if $a < 1$ value is related only to the surface roughness, then the roughness can be characterized by a fractal dimension of the surface $D_f$, which is related to $a$ [50]:

$$a = \frac{1}{D_f - 1}. \qquad (1.21)$$

The porous electrode materials have a wide range of applications in various types of electrochemical devices, including supercapacitors [1].

The formation of the EDL and the pseudocapacitance are two main mechanisms of charge storage in the supercapacitors. The impedance of the EDL formation can be modeled as a non-faradaic impedance. In the presence of the pseudocapacitance, the interfacial impedance includes the faradaic impedance, which can be modeled as a serial combination of $R_{ct}$ (related to the kinetics of interfacial reaction (1.16)) and $C_{ps}$ (related to capacitance of ions intercalation) (Fig. 1.11(D)).

In order to model the impedance of electrodes of supercapacitors, the influence of porous structure has to be taken into account for any mechanism of the interfacial charge storage. The impedance of the interface inside a pore is often modeled by the staircase-type equivalent circuit (Fig.1.12).

It was demonstrated that for the infinitely long cylindrical pore the interfacial impedance is described by the following equation [47]:

$$\vec{Z}_p = \sqrt{R|\vec{Z}|}e^{\frac{i\phi}{2}}. \qquad (1.22)$$

Here $R$ is an electrolyte resistance within the pore per unit length, $\vec{Z}$ is an impedance of a pore wall / electrolyte interface, and $\phi$ is a phase angle of $\vec{Z}$. (1.22) predicts that the interfacial impedance inside the pore has a phase angle equal to a half of an impedance of a flat electrode. For example, for the non-faradaic
<figure><img src="image_17.png" /><figcaption>Figure 1.12: The staircase equivalent circuit of the impedance model inside a single pore. The highlighted elements form the repeating pattern along the pore.</figcaption></figure>

impedance $\vec{Z}_{nf}$ the DeLevi model predicts a constant phase angle $\phi = 45^{\circ}$.

For the analysis of a pore with a finite length $l_p$ using the staircase model, it is convenient to define a length $\lambda$ of the penetration of the potential modulation into the pore [55]:

$$ \lambda = \frac{1}{2} \sqrt{\frac{\sigma d_p}{2\omega C_s}}. \qquad (1.23) $$

Here $\sigma$ is a conductivity of an electrolyte (in Ohm$^{-1}$.m$^{-1}$), $d_p$ is a pore diameter, and $C_s$ is a surface specific interfacial capacitance.

From (1.23) it can be concluded that $\lambda$ increases with a decrease in the modulation frequency $f = \omega/2\pi$. Thus, $\lambda > l_p$ for the frequencies below a certain threshold $f_0$. Under this condition, the potential modulation is uniform along the whole length of the pore, and the pore impedance is the same as the impedance of a flat electrode with the same surface area. In this case, one observes a change from the shift angle $\phi = 45^{\circ}$ at $f > f_0$ to $\phi = 90^{\circ}$ for $f < f_0$ in the Nyquist diagram (Fig.1.10(3)). This transition is often observed for the porous carbon electrodes (for example, in [10]). In general, the transition is much less sharp and occurs over a certain range of frequencies (Fig.1.10(4)) due to the presence of the pores with various diameters and shapes [55, 56].

This staircase model allows to correctly fit the experimental impedance spectra of the porous electrodes in the absence of faradaic processes and determine the
capacitance $C_{dl}$. Moreover, the average pore length $l_p$ can be estimated from the $f_0$ value, providing that the average diameter $d_p$ of the pores is known from the other methods.

In the above mentioned models it is assumed that individual pores are not interconnected. Therefore, the staircase circuit is applied independently to each pore. The supercapacitors electrode materials, for example the activated carbon, often have hierarchial pores: the larger pores branch into several smaller pores of next generation, which can branch further. The hierarchial porous structure makes the application of the staircase model more complicated, because the staircase circuits of pores from different generations are no longer independent.

The model of branching pores was proposed in [57] by introducing 2 generations of pores: the large (μm size) voids between carbon particles and the narrower (sub-μm size) internal pores. The larger pores were branching into $n$ smaller pores, and the length and diameter of smaller pores were scaled by $a_l$ and $a_d$ factors in comparison to the parent pores. Fitting the experimental spectra allowed to estimate the branching factor $n$ and scaling factors $a_l, a_d$. However, these values had to be considered as estimates only, because the porous structure is more complex.

The model also demonstrates that the pores branching is counter-productive for efficient utilization of the capacitance of porous electrodes. Indeed, the capacitance of the interface of the porous electrode was equal to the capacitance of a flat electrode $C_{dl}$ with the same surface area only if the lowest possible branching factor $n$ was assumed. This conclusion is an important guiding rule for material science in the optimization of the structure of electrode materials. However, the unambiguous analysis of the influence of various structural parameters is difficult because of their interdependence. In particular, in [57] the porosity of electrode material and the length scaling factor $a_l$ were fixed. An increase in the branching factor led to narrowing of branch pores which decreased the modulation penetration depth (1.23), and, thus, the efficiency of pore surface utilization.

It has also to be mentioned that the model in [57] assumed the serial connection of a parent pore and its branches: such as if the pores were branching only at their bottom points. This allowed to utilize the same polarization conditions for all pores of the same generation, thus simplifying the model. In reality, however, the branches may appear over the whole length of a pore (Fig.1.13).

In the following subsection the improved model of the impedance of a porous
<figure><img src="image_18.png" /><figcaption>Figure 1.13: The sketch of an electrode with hierarchial porous structure with 3 generations of pores.</figcaption></figure>

electrode is presented. This model utilizes staircase circuit and takes into ac-
count the presence of 3 generations of branching pores. The pores of 2<sup>nd</sup> and
3<sup>rd</sup> generations are formed by branching a parent pore. The branches are uni-
formly distributed over the length of the parent pore. The faradaic impedance
of pseudocapacitance is also included into the staircase model, allowing to model
the porous electrodes of pseudocapacitors. Thus, this model is further referred
to as "generalized staircase model". The goal is to predict the influence of the
structural parameters of electrode materials on their performance in supercapaci-
tors. Namely, the influence of the pores length and diameter, the porosity of the
material, and the branching factors is evaluated.

1.3.3 Generalized staircase model of impedance of porous electrode materials.

The sketch of the electrode material is depicted in Fig.1.13. The material is as-
sumed to have 3 hierarchial generations of pores. The pores of the 2<sup>nd</sup> and 3<sup>rd</sup>
generations are the branches of the pores of the parent generations and evenly dis-
tributed over the length of a parent pore. The pores of 3<sup>rd</sup> generation are terminal:
they have no branches.
Each generation of the pores has 2 characteristic parameters: a pore length $l_{p,i}$ and a pore diameter $d_{p,i}$, where $i$ is a generation index. The surface area of the individual pore of the $i^{th}$ generation is given as

$$A_i = \pi d_{p,i} l_{p,i} + \frac{\pi d_{p,i}^2}{4}. \qquad (1.24)$$

Neglecting the surface of the material outside the pores, the total surface area of the material is calculated as the surface area $A_p$ of all pores:

$$A_p = N_1(A_1 + \beta_{12}A_2 + \beta_{12}\beta_{23}A_3). \qquad (1.25)$$

In equation (1.25), $\beta_{12}$ and $\beta_{23}$ are the branching factors: $\beta_{12} = N_2/N_1$, $\beta_{23} = N_3/N_2$, where $N_1$, $N_2$, $N_3$ are the numbers of pores of the 1$^{st}$, 2$^{nd}$, and 3$^{rd}$ generations correspondingly. The factors $\beta_{12}$ and $\beta_{23}$ are the model parameters, whereas $N_1$ is found as

$$N_1 = \frac{m \cdot SSA}{A_1 + \beta_{12}A_2 + \beta_{12}\beta_{23}A_3}, \qquad (1.26)$$

where $m$ is a mass of the deposit (in g), SSA is a specific surface area of a the electrode material (in cm$^2$/g).

The impedance of the pores of each generation is modeled by the staircase circuit as described below.

### Model of an individual pore of the 3$^{rd}$ (terminal) generation.

An equivalent circuit utilized in the improved generalized staircase model of the impedance $\vec{Z}_{3g}$ of an individual pore of the 3$^{rd}$ generation (a pore with the smallest diameter and without branches) is depicted in Fig.1.14. The pore is divided into $K = 2^n$ identical segments. The value $n = 10$ is used for all calculations, as it was found that further increase in $n$ does not change the results of modeling.

For the pore with the diameter $d_p$ and the length $l_p$, the length of a segment is $\delta_l = l_p/K$, the surface area of the walls of the segment is $A_s = \pi d_p \delta_l = \pi d_p l_p / K$, and the surface of the bottom of the pore is $A_b = \pi d_p^2 / 4$.

A single segment has an interfacial impedance $\vec{Z}_s$. It consists of a component $C_{dl,s}$, which is the EDL capacitance of the walls of the segment. If the model includes also a faradaic impedance, then a faradaic impedance $\vec{Z}_{f,s}$ of the segment walls is added in parallel to $C_{dl,s}$. Therefore, in the presence of pseudocapacitance
<figure><img src="image_19.png" /><figcaption>Figure 1.14: The equivalent circuit applied in the generalized staircase model. The highlighted area corresponds to the repetitive pattern of the circuit. The inset depicts the model of the faradaic impedance related to the pseudocapacitance.</figcaption></figure>

the total impedance of the segment walls is given by the following equation:

$$ \frac{1}{\vec{Z}_s} = \frac{1}{i\omega C_{dl,s}} + \frac{1}{\vec{Z}_{f,s}}, \qquad (1.27) $$

$$ C_{dl,s} = C_s \cdot A_s = C_s \cdot \pi d_p \frac{l_p}{K}. \qquad (1.28) $$

The faradaic impedance $\vec{Z}_{f,s}$ is related to the pseudocapacitance and modeled as a serial combination of the charge transfer resistance $R_{ct,s}$ and the capacitance $C_{ps,s}$ (see the inset of Fig.1.14).

$$ \vec{Z}_{f,s} = R_{ct,s} + \frac{1}{i\omega C_{ps,s}}. \qquad (1.29) $$

Both $R_{ct,s}$ and $C_{ps,s}$ are calculated for the whole surface area $A_s$ of the segment walls:

$$ R_{ct,s} = \frac{RT}{nF i_0 A_s} = \frac{RTK}{\pi n F i_0 d_p l_p}, \qquad (1.30) $$

$$ C_{ps,s} = C_{ps} \cdot A_s = \pi C_{ps} \frac{d_p l_p}{K}. \qquad (1.31) $$

The final segment of the pore, beside the contribution from its walls ($\vec{Z}_s$), contains
also a contribution from the bottom ($\vec{Z}_{s,b}$). The parameters of the impedance $\vec{Z}_{s,b}$ of the bottom, namely, $C_{dl,b}$ and $\vec{Z}_{f,b}$ are calculated using the equations, same as (1.28), (1.29), (1.30), (1.31), but with the bottom surface area $A_b$ instead of the walls surface area $A_s$.

Thus, the impedance $\vec{Z}_K$ of the last segment of the pore is calculated according to the circuit in Fig.1.14 as follows:

$$ \vec{Z}_K = R_s + \frac{1}{\frac{1}{\vec{Z}_s} + \frac{1}{\vec{Z}_{s,b}}} \quad (1.32) $$

Here $R_s$ is the resistance of the electrolyte inside the segment of the pore:

$$ R_s = \frac{1}{\sigma} \frac{\delta_l}{\pi d_p^2} = \frac{l_p}{\pi \sigma d_p^2 K} \quad (1.33) $$

where $\sigma$ is the conductivity of an electrolyte.

The part of the pore containing last two segments ($K$ and $K-1$) has the impedance $\vec{Z}_{K-1}$:

$$ \vec{Z}_{K-1} = R_s + \frac{1}{\frac{1}{\vec{Z}_s} + \frac{1}{\vec{Z}_K}} \quad (1.34) $$

In general, the impedance of the part of the pore containing $i$ segments from the bottom of the pore (i.e. segments from $i$ to $K$) is calculated by the recursive equation, similar to (1.34):

$$ \vec{Z}_i = R_s + \frac{1}{\frac{1}{\vec{Z}_s} + \frac{1}{\vec{Z}_{i+1}}} \quad (1.35) $$

Thus the total impedance of a pore of the 3rd generation with $K$ segments is equal to an impedance of the pore with all segments from 1 to $K$: $\vec{Z}_{3g} = \vec{Z}_1$.

### Impedance of pores of the 1st and 2nd generations.

The impedance of the pores of the 1st and 2nd generations is calculated by the same generalized staircase model as for the pores of 3rd generation, except that the contribution of the impedance of the next generations are taken into account.

The interfacial impedance of each of the segments of the pores of the 1st and 2nd generations is modeled by the circuit depicted in Fig.1.15. For example, an individual pore of the 2nd generation is branched into $\beta_{23}$ pores of the 3rd generation.
ation. The total impedance of $β_{23}$ parallel pores with impedance $\vec{Z}_{3g}$ is equal to $\vec{Z}_{3g}/β_{23}$. It is assumed that the branch pores are uniformly distributed over the whole length of the parent pore. Thus, each segment of a parent pore contains an average additional impedance element $Z^*$ due to the impedance of pores of next generation. For the pores of the 2$^{nd}$ and the 1$^{st}$ generations this additional element is given by the following corresponding equations:

$$d\vec{Z}_{2g}^* = \frac{\vec{Z}_{3g}K}{\beta_{23}}, \qquad (1.36)$$

$$d\vec{Z}_{1g}^* = \frac{\vec{Z}_{2g}K}{\beta_{12}}, \qquad (1.37)$$

where $\vec{Z}_{3g}$ and $\vec{Z}_{2g}$ are the total impedances of an individual pore of the 3$^{rd}$ or 2$^{nd}$ generation correspondingly.

Thus, according to the circuit in Fig.1.15, the interfacial impedance $\vec{Z}_s$ of a segment of a pore of the 1$^{st}$ or 2$^{nd}$ generation, is calculated as

$$\frac{1}{\vec{Z}_s} = \frac{1}{i\omega C_{dl,s}} + \frac{1}{\vec{Z}_{f,s}} + \frac{1}{\vec{Z}^*}. \qquad (1.38)$$

Physically, this approach to account for a contribution of the impedance of branch pores is equivalent to adding a branch pore with impedance equal to $d\vec{Z}_{2g}$ or $d\vec{Z}_{1g}$ to each of K segments.

(1.38) differs from (1.27) for $\vec{Z}_s$ for the pores of 3$^{rd}$ terminal generation by the $1/\vec{Z}^*$ term. Besides this difference, the rest

of the calculation of the impedance of an individual pore is the same for the pores of all generations.

Finally, the total impedance of all pores of the material is calculated as the impedance of all $N_1$ pores of the 1$^{st}$ generation:

$$\vec{Z}_p = \frac{\vec{Z}_{1g}}{N_1}. \qquad (1.39)$$

<figure><img src="image_20.png" /><figcaption>Figure 1.15: The equivalent circuit modeling the interfacial impedance of a segment of an individual pore of the 1$^{st}$ and 2$^{nd}$ generations.</figcaption></figure>
## Results of modeling of impedance of porous electrode materials

The parameters of the materials of the electrode and electrolyte utilized in the calculations are listed in Table 1.4. The value of SSA = 230 m²/g is close to the one of the carbon Vulcan XC-72, often utilized as an electrode material. This carbon black material consists of the small primary carbon particles (ca. 30 nm) filled with the micropores with an average diameter $d < 2$ nm [58]. The primary particles form the agglomerates with the size of 100-300 nm, which include the pores with diameters of 2-20 nm. The agglomerates form the large (μm size) chain-like structures containing the macropores with the diameter $d > 50$ nm. Following this description, 3 types (generations) of pores of the model carbon material (see Fig.1.13) are defined in the generalized staircase model:

* micropores ($d_{p,3} = 0.7$ nm), representing the apex and terminal branches of the inner pores inside the primary particles;

* small mesopores ($d_{p,2} = 3$ nm), representing the major population of pores inside the primary particles;

* large mesopores ($d_{p,1} = 30$ nm), representing the opening mouth of the smaller mesopores and the zones of contacts between primary particles.

According to the classification of Eikerling [59], the diameters of the pores of the 1st and 2nd generations correspond to the secondary and primary pores of the carbon electrode materials in fuel cells. These pores are important for efficient accessibility of the electrode/electrolyte interface. In the generalized staircase model, the micropores (3rd generation) are also added, because their presence is known to be beneficial for the performance of the supercapacitors [60]. On the other hand, the presence of a large population of macropores ($d \approx 150$ nm)

<table><caption>Table 1.4: The parameters of materials used in calculations.</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>SSA</td><td>230 m²/g</td></tr><tr><td>σ</td><td>2 S.m⁻¹</td></tr><tr><td>C<sub>s</sub></td><td>5 μF/cm²</td></tr><tr><td>C<sub>ps</sub></td><td>300 μF/cm²</td></tr><tr><td>ρ<sub>c</sub></td><td>2 g/cm³</td></tr><tr><td>m</td><td>1 g</td></tr></tbody></table>
in Vulcan XC-72 is neglected for the simplicity of the model. These macropores make a relatively small contribution to the overall SSA of the carbon material and behave as a flat electrode under the conditions of the potential modulation utilized in the EIS measurements.

The length of the pores of the 2<sup>nd</sup> and 3<sup>rd</sup> generations are the variables of the model. The length of large mesopores (1<sup>st</sup> generation) is fixed to 0.1 µm, i.e. these pores are relatively short and represent the openings of pores within the agglomerates. The macropores branch into narrower and longer (3 µm) mesopores of the 2<sup>nd</sup> generation.

In order to evaluate the performance of a porous material as an electrode in supercapacitors, the measurable capacitance of the material is calculated as $-1/\omega \cdot \text{Im}(\vec{Z}_p)$ and the surface specific calculated capacitance $C_{s,c}$ as

$$C_{s,c} = -\frac{1}{A_s \omega \cdot \text{Im}(\vec{Z})} \qquad (1.40)$$

One of the main characteristics of the supercapacitor electrode material is the mass specific capacitance $C_g$:

$$C_g = C_{s,c} \cdot SSA. \qquad (1.41)$$

In the literature on supercapacitors, a high $C_g$ value is often considered as a criteria of an efficient electrode material. Indeed, an efficient electrode material must have a high $C_g$ value. However, it is not the only criterion. For example, a nm-thin film of an active storage material deposited on the current collector can demonstrate a high $C_g$ value. However, it is the impractical material due to the high relative contribution of masses of inactive components, such as the current collector and the separator. In reality, high loadings of active material (above 1 mg/cm²) are required. The modeling of such thick electrodes is difficult. Alternatively, for thick layers, volume specific characteristics should be analyzed [8], because they depend on both surface specific capacitance $C_s$ and porosity of the material. Thus, the model focuses on the analysis of volume specific capacitance $C_V$:

$$C_V = C_{s,c} \cdot SSA \cdot \rho, \qquad (1.42)$$
where $\rho$ is a calculated density of the material:

$$ \rho = \frac{m}{V_t} = \frac{m}{V_p + V_c} = \frac{m}{V_p + \frac{m}{\rho_c}}. \qquad (1.43) $$

Here $m$ is a mass of the material, $V_t$ is a total volume of the material, $V_p$ a total volume of the pores in the material, $V_c$ is a volume of the compact part of the material, $\rho_c$ is a compact density of the material (e.g. the density of a non-porous material; in the model $\rho_c = 2 \text{ g/cm}^3$, which is a typical value for the compact graphite materials). Parameters $m$ and $\rho_c$ are fixed model parameters (see Table 1.4), while $V_p$ is calculated by a similar approach to $A_p$ (eq.1.25):

$$ V_p = \pi \cdot N_1(d_1^2 l_1 + \beta_{12} d_2^2 l_2 + \beta_{12} \beta_{23} d_3^2 l_3). \qquad (1.44) $$

The impedance $\vec{Z}_p$ is calculated for the wide range of frequencies $f$: from 0.1 mHz to 100 kHz. It is important to define the range of $f$, which is the most relevant for the electrode materials of supercapacitors. As it was discussed in Subsection 1.2.4, the characteristics of electrochemical performance of the supercapacitor electrode materials should be measured under the operational conditions of supercapacitors, e.g. with the charging rates above 60C [8]. For a supercapacitor with 1V operational voltage, this charging rate corresponds to the potential sweep rate of ca. 16.7 mV/s or, for galvanostatic cycling, the charging current of 2 A/g (for a capacitor with $C_g=120 \text{ F/g}$). In the case of the EIS, the rate $v_E$ of modulation of the potential $\vec{E} = E_0 \sin(\omega t)$ is given by the equation

$$ v_E = \frac{dE}{dt} = E_0 \omega \cos(\omega t). \qquad (1.45) $$

According to (1.45), $v_E$ changes periodically with time. For the sake of comparison with other methods, one can use the root mean square rate $v_{rms}$

$$ v_{rms} = \frac{2\pi E_0 f}{\sqrt{2}}. \qquad (1.46) $$

Table 1.5 shows the correspondence between the modulation rate $v_{rms}$ and the modulation frequency $f$, calculated by (1.46). It can be concluded that the most relevant range of $f$ for supercapacitors is from ca. 0.1 Hz to ca. 100 Hz (or the range of $\log_{10} f$ is from -1 to 2).
<table><caption>Table 1.5: The values of $v_{rms}$ and the charging current densities $i_m$ as the estimations of the potential modulation rate for different modulation frequencies $f$, calculated by (1.46).</caption><thead><tr><th><i>v</i><sub>rms</sub>, mV/s</th><th><i>i</i><sub>m</sub>, A/g</th><th><i>f</i>, Hz</th><th>log<sub>10</sub>(<i>f</i>, Hz)</th></tr></thead><tbody><tr><td>0.1</td><td>0.01</td><td>0.0032</td><td>-2.49</td></tr><tr><td>0.2</td><td>0.02</td><td>0.0064</td><td>-2.19</td></tr><tr><td>0.5</td><td>0.05</td><td>0.016</td><td>-1.79</td></tr><tr><td>1.0</td><td>0.1</td><td>0.032</td><td>-1.49</td></tr><tr><td>2.0</td><td>0.2</td><td>0.064</td><td>-1.19</td></tr><tr><td>5.0</td><td>0.5</td><td>0.16</td><td>-0.79</td></tr><tr><td>10</td><td>1</td><td>0.32</td><td>-0.49</td></tr><tr><td>20</td><td>2</td><td>0.64</td><td>-0.19</td></tr><tr><td>50</td><td>5</td><td>1.61</td><td>0.21</td></tr><tr><td>100</td><td>10</td><td>3.21</td><td>0.51</td></tr><tr><td>200</td><td>20</td><td>6.43</td><td>0.81</td></tr><tr><td>500</td><td>50</td><td>16.1</td><td>1.21</td></tr></tbody></table>

Below the influence of various structural parameters of a porous system on its electrochemical properties, in particular, $C_V$, is analyzed. In order to decrease the number of varied parameters, the diameters of different generations of the pores $d_1, d_2, d_3$, and the length of the 1st generation of pores $l_1$ are fixed. Thus, the model is focused on the influence of $l_2$ and $l_3$, as well as the total pore length and the branching factors $\beta_{12}, \beta_{23}$.

### Simple model systems: influence of pore diameter, geometry, and branching

At first, the generalized staircase model was applied to the simple configurations of pores depicted in Fig.1.16. The results of the calculations with the model systems A, B, and C, containing 1 type of pores ($\beta_{12} = \beta_{23} = 0$), are shown in Fig.1.17. The pore length was $l = 3.2 \ \mu m$. The results are presented as the Nyquist plots of impedance ($-\text{Im}(\vec{Z})$ vs. $\text{Re}(\vec{Z})$), and the Bode plots of $C_V$.

The Nyquist plots demonstrate the expected transition from $\phi = 45^\circ$ at the frequencies higher than the threshold value $f_0$ to $\phi = 90^\circ$ at the frequencies $f < f_0$. The threshold $f_0$ decreases with the pore diameter (1.23). In the Bode plots one observes the transition from the constant $C_V$ at $f < f_0$ to the varying $C_V$, which decreases with the frequency $f > f_0$. As expected, the narrower pores have higher $C_V$ values due to the higher density of the material: $\rho = 1.513 \ g/cm^3$ for $d = 0.7$ nm, $\rho = 0.136 \ g/cm^3$ for $d = 30$ nm. The narrower pores also have a lower threshold frequency $f_0$ and a stronger $C_V$ loss with increase of $f$. Thus, at certain
<figure><img src="image_21.png" /><figcaption>Figure 1.16: The sketches of the simple pore model systems.</figcaption></figure>

$f > f_0$, for the pores with $d = 3$ nm the $C_V$ values are predicted to be higher than the $C_V$ values for the pores with $d = 30$ nm, Fig.1.17(B).

In order to validate the model, these results were compared with calculations for the model system with 3 generations of non-branching ("consecutive") pores ($\beta_{12} = \beta_{23} = 1$) with the same diameter $d_1 = d_2 = d_3$ (Fig.1.17). For the data in Fig.1.17, the lengths of the pores of the 1st, 2nd, and 3rd generations were $l_1 = 0.1$ µm, $l_2 = 3$ µm, $l_3 = 0.1$ µm. The total length of a single pore for the given model system was $l_p = l_1 + l_2 + l_3 = 3.2$ µm, which is the same as in the model with 1 generation of pores.

As expected, the results of the calculations for these 2 models coincide well with a slight negligible deviation observed in the Nyquis plots only for low frequencies. This slight deviation characterizes a small and acceptable inaccuracy of the model with 3 generations related to the uniform spreading of the impedance of branch pores over the length of a parent pore.

It was also verified that if all 3 generations of non-branching pores have the same diameter, then the results were the same for all models with the same total pore length, independently of the relative lengths of pores of each generations. Indeed, the results of calculations for the model system with $l_1 = 3$ µm, $l_2 = l_3 = 0.1$ µm coincide with the results for the model systems with $l_1 = l_3 = 0.1$ µm, $l_2 = 3$ µm, and $l_1 = l_2 = 0.1$ µm, $l_3 = 3$ µm.

The results for the model system D with non-branching pores, the diameters $d_1 = 30$ nm, $d_2 = 3$ nm, $d_3 = 0.7$ nm, the lengths $l_1 = l_3 = 0.1$ µm, $l_2 = 3$ µm, are
<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

Figure 1.17: The Nyquist plots (A) and the Bode plots (B) of the model systems. Dashed lines: the model systems with 1 generation of pores ($\beta_{12} = \beta_{23} = 0$) and various pore diameter $d$. Solid lines (except blue and cyan): the model systems with 3 generations of non-branching pores ($\beta_{12} = \beta_{23} = 1$) and the same pore diameter $d$. Blue lines: the model system with non-branching pores and decreasing pore diameters. Cyan line: the model system with branching factors $\beta_{12} = 2$, $\beta_{23} = 5$ and decreasing pore diameters.
shown in Fig.1.17 as blue lines. The Nyquist plots for this model system show a
more complex behavior, indicating the influence of all 3 generations of the pores.
The phase shift angle $\phi$ is larger than 45° even at relatively high frequencies, while
the transition to the behavior with $\phi = 90°$ is less sharp comparing to the system
with $d = 3$ nm (red lines). Such behavior of the Nyquist plots is similar to the one
predicted for the conical pores with decreasing diameter [55].

In comparison to the system B with straight pores, the system D with a wide
opening has a significantly lower density (0.841 and 0.372 g/cm³ correspondingly).
Thus, the system D shows the lower $C_V$ values if the frequencies $f$ are not too
high. The threshold frequency $f_0$ for this system is close to the model system B
with $d = 3$ nm (red line), despite of the narrower apex of the pores of the system
D ($d_3 = 0.7$ nm). A narrower apex is expected to result in a lower $f_0$, however,
this effect is weak and undetectable. On the other hand, due to a wider mouth
of the pore $d_1 = 30$ nm, the $C_V$ decrease in high frequency range is significantly
lower in comparison to the system B ($d = 3$ nm). Above a certain frequency, the
$C_V$ values of the system D with a wide opening are bigger than those of the system
B with the straight pores.

Finally, the results for the model system E with the same dimensions of 3
generations of pores as in system D, but with the branching factors $\beta_{12} = 2$ and
$\beta_{23} = 5$ are shown in Fig.1.17 (cyan lines). As a branching pore has a higher
surface area than a non-branching pore, the number of pores in the system E is
lower than in the system D. This results in a higher density (0.498 and 0.372 g/cm³
correspondingly), and, thus, higher $C_V$ values. Also, the value of the threshold
frequency $f_0$ of the system E falls in the interval between the values for the systems
with 1 generation of pores and the diameters $d = 3$ and $d = 30$ nm. This proves
that the modeled branching porous structure is more favorable for efficient surface
utilization than a non-branching structure at least in the considered frequency
range. The increased efficiency can be qualitatively explained by the fact that for
the same frequency $f$, and thus, the same penetration depth $\lambda$, the system with
$\beta_{12} = 2$ has ca. 2 times higher surface area than a system with non-branching
pores. However, for high frequencies, the efficiency of non-branching pores becomes
comparable with branching pores. This can be explained by a stronger contribution
of the wider mouths of pores to the total pore surface area in the non-branched
pores.

Summing up, the following preliminary conclusions can be made from the com-
parison of the models A-E.

* Straight and narrow (micro-)pores provide the highest interfacial capacitance at least at moderate potential modulation rates. For the applications requiring high power output (fast discharging rates), narrow mesopores might be more efficient.

* The presence of wide opening of the pores ("hierarchial" pores: wider pores transforming to the narrower ones) results in the smaller losses of capacitance with the potential modulation rate increase. However, the material density and the related $C_V$ values decrease. Thus, the wide opening of the pores might be not favorable for the applications requiring a high energy density.

* Comparing to a system with non-branching pores, the system with branching pores and the same pores dimensions has a similar dependence of capacitance on the modulation rate. However, pores branching results in a higher material density (lower porosity) for the same SSA and higher $C_V$ values. Therefore, the branching of pores appears to be favorable for the better material performance as supercapacitor electrodes.

In the next Subsection the influence of the structure of porous systems, namely, the pores diameter, length, and branching factors on the material performance is analyzed in details.

## Effect of pore "mouth"

First, the systems with the same total length of the non-branching pores $l_p = l_1 + l_2 + l_3 = 1.2~\mu\text{m}$ were modeled (see Fig.1.18). The system (F1) with the straight micropores with the diameter $d = 0.7~\text{nm}$ was compared to the systems (F2-F4) with wide pore openings, modeled as short (0.1 $\mu\text{m}$) segments with various diameters at the mouths of pores. The comparison of the results for the systems F1-F4 is given in Fig.1.19.

The Nyquist plots of the system with straight pores with $d = 0.7~\text{nm}$ (black lines in Fig.1.19) show the typical behavior described in the Subsection 1.3.3: a transition from $\phi = 45^\circ$ at $f > f_0$ to $\phi = 90^\circ$ at $f < f_0$. For the same threshold $f_0$, the Bode plots show the transition in $C_V$ behavior: from the independence of $f$ for $f < f_0$ to the decrease with increasing $f$ for $f > f_0$.
<figure><img src="image_24.png" /><figcaption>Figure 1.18: The sketches of the pore model systems F1-F4.</figcaption></figure>

Adding the short wide opening segments to the mouths of the pores shifts $f_0$ to slightly higher frequencies and decreases the loss of $C_V$ at higher $f$. These effects are stronger for wider openings. On the other hand, the presence of a wide opening increases a material porosity and decreases a material density: $\rho = 1.51$, 1.23 and 0.175 g/cm$^3$ for the systems F1 (no wider opening), F2 (opening segment with $d = 3$ nm) and F3 (opening segment with $d = 30$ nm) correspondingly. This results in a lower $C_V$ values at low and moderate frequencies. Thus, under the low and moderate potential modulation rates the pores without wide openings are more favorable for supercapacitors.

## Influence of pore branching

In order to evaluate the effect of pore branching on the capacitance of porous materials, the set of systems with pores of the same dimension ($l_1 = 0.1$ µm, $d_1=30$ nm, $l_2 = 3.0$ µm, $d_2=3$ nm, $l_3=0.1$ µm, $d_3=0.7$ nm) and varied branching factors was modeled. The first set (G) had fixed $\beta_{23} = 5$ and varied $\beta_{12}$ from 1 to 100, and the second set (H) had fixed $\beta_{12} = 5$ and varied $\beta_{23}$ from 1 to 100 (Fig.1.20). The Bode plots are shown in Fig. 1.21. For all model systems very close values of $f_0$ were obtained, close to one of the system B (pores with $l = 3.2$ µm, $d = 3$ nm, Fig.1.17, red lines). The increase in branching factors increases the loss of $C_V$ at high potential modulation rates. However, it also increases the density of the material. For example, for $\beta_{23} = 5$ the increase in $\beta_{12}$ from 1 to 10 and further up to 100 results in the increase in the material density from 0.375 to
<figure><img src="image_25.png" /><figcaption>Figure 1.19: The Nyquist plots (A) and the Bode plots (B) of the model systems with 3 generations of non-branching pores with the diameters $d_1 = 30$ nm, $d_2 = 3$ nm, $d_3 = 0.7$ nm. The relative lengths of the segments were varied, while the total pore length was kept constant $l_p = l_1 + l_2 + l_3 = 1.2$ µm.</figcaption></figure>
<figure><img src="image_26.png" /><figcaption>Figure 1.20: The sketches of the model system sets G and H.</figcaption></figure>

0.734 and further up to 0.840 g/cm³.

It may seem to be counter-intuitive that a system with stronger pore branching has a higher density. However, it can be rationalized as follows. Pore branching results in higher surface area of each individual pore, and thus, for the same SSA, a smaller amount of pores is required. This means the lower porosity and the higher density of the material. The effect of an increased material density is predominant in the wide range of $f$. Only at very high $f$ (above 10 kHz) the materials with a lower $\beta_{12}$ demonstrate a higher $C_V$. For the increase of $\beta_{23}$, the effect of stronger $C_V$ loss at higher frequencies is even less pronounced. This is explained by the fact that the contribution of terminal pores is smaller at the higher frequencies due to the short modulation penetration depth.

Thus, it can be generally concluded that pore branching is a favorable strategy to increase the capacitance of porous materials for supercapacitors with non-faradaic impedance, for example for the EDLC.

Below the model predictions for the systems with faradaic impedance are evaluated. These systems are relevant for the electrode materials for pseudocapacitors.

## Influence of faradaic impedance

The generalized staircase model was applied to calculate the impedance of porous electrodes with faradaic impedance involved in energy storage. The interfacial impedance in this case was modeled as the impedance of a parallel connection of the capacitive impedance $Z_{dl} = Z_{C=dl}$ and the faradaic impedance $Z_f = R_{ct} +$
<figure><img src="image_27.png" /><figcaption>Figure 1.21: The Bode plots for the systems G with varied $\beta_{12}$ (A) and the systems H with varied $\beta_{23}$ (B). The lengths of the pores are: $l_1 = 0.1$ µm, $l_2 = 3.0$ µm, $l_3 = 0.1$ µm.</figcaption></figure>
$Z_{C=C_{ps}}$ (Fig.1.14). The Nyquist plots clearly shows the influence of the faradaic impedance, especially at low frequencies, even for the relatively low values of $i_0$ (the exchange rate of electrochemical reaction), i.e. for the relatively high $R_{ct}$ (1.16). As the rate $i_0$ increases, the predominance of $\vec{Z}_f$ over $\vec{Z}_{dl}$ becomes pronounced also at the higher frequencies. However, the Bode plots show that the contribution of $C_{ps}$ becomes significant only for high $i_0$ and for low or moderate frequencies. For example, for the system with a very fast charge transfer rate ($i_0$ = 50 mA/cm$^2$) and high $C_{ps}$ = 300 $\mu$F/cm$^2$, the contribution of $C_{ps}$ becomes noticeable only at $f$ < 100 Hz, even though the $|\vec{Z}_{dl}| > |\vec{Z}_f|$ already at $f$ < 1000 Hz. Fig.1.23 compares the Nyquist and the Bode plots for the porous systems with the varied branching factor $\beta_{23}$, while $\beta_{12}$ = 2 was fixed. The Nyquist plots are practically identical, except slight deviations observable in the high frequency range (see inset of Fig.1.23(A)). It has to be reminded that data in the Nyquist plots, namely, $-\text{Im}(\vec{Z}_p)$ vs $\text{Re}(\vec{Z}_p)$, do not reflect the changes in material porosity. This shows that the influence of branching factor on the total impedance in the presence of the faradaic impedance is negligible. It can be explained by the fact that faradaic impedance becomes predominant in the low frequency range, where the penetration depth $\lambda$ is close or larger than the pore length $l_p$ and the effects of pore geometry are weak. In this case, one may expect that the main effect of branching factor increase on the capacitance of the materials with faradaic impedance is related to the changes in the material porosity and density. Indeed, the Bode plots in Fig.1.23 demonstrate a significant increase in the values of $C_V$ related to the increase of the material density from $\rho$ = 0.486 g/cm$^3$ for $\beta_{23}$ = 1 to $\rho$ = 0.663 g/cm$^3$ for $\beta_{23}$=100.

## 1.4 Conclusions

It was demonstrated that the generalized staircase model allows to model the impedance of the porous materials with relatively complex pore geometry. The pore geometry can include a system of hierarchial pores, e.g. pores with a wider diameter transforming into single narrower pores or branching into multiple narrower pores evenly distributed over a parent pore. Modeling of the interfacial impedance of an individual pore can be done by splitting a pore into small segments and using the recursive equations to calculate the impedance of a part of a pore by sequentially including all segments of the part. The generalized staircase
<figure><img src="image_28.png" /><figcaption>Figure 1.22: The Nyquist plots (A) and the Bode plots (B) for the model system with typical structural parameters: $d_1 = 30$ nm, $l_1 = 0.1$ µm, $d_2 = 3$ nm, $l_2 = 3$ µm, $d_3 = 0.7$ nm, $l_3 = 0.1$ µm, $\beta_{12} = 2$, $\beta_{23} = 5$, in the presence of the faradaic impedance.</figcaption></figure>
<figure><img src="image_29.png" /><figcaption>Figure 1.23: The Nyquist plots (A) and the Bode plots (B) for the model system with faradaic impedance ($C_{ps}$ = 300 µF/cm², $i_0$ = 50 mA/cm²) for the varied branching factor $\beta_{23}$ ($\beta_{12}$ = 2 was fixed).</figcaption></figure>
model allows to calculate the impedance of a single pore and the total impedance
of all pores for the material with both non-faradaic and faradaic processes.

The generalized staircase modal was developed to evaluate the influence of
structural parameters of the porous electrode materials on their electrochemical
performance in supercapacitors. it was demonstrated that the volume specific
capacitance $C_V$ serves as a informative parameter of the electrochemical perfor-
mance.

The variation of the structural parameters of the model porous systems causes
two groups of effects on the capacitance of materials.

The first group of effects is related to the limited depth of the penetration of
potential modulation. These effects are manifested as a decrease in capacitance at
modulation frequencies above the threshold value $f_0$. Both the magnitude of this
decrease and the value of $f_0$ depend on the structure of a porous system. In par-
ticular, a decrease in the pore diameter and an increase in the pore length result in
the stronger loss of $C_V$ and an increase of the value of $f_0$. Thus, the electrochemi-
cal performance of the material decreases. On the other hand, a wider opening of
pores (e.g. hierarchial pores) decreases the loss of $C_V$ at high frequencies.

The second group of effects is related to the change in material density caused
by the changes in the parameters of a porous system. A higher porosity results in
a lower density of the material, a lower volume specific capacitance and a lower
energy density. These effects are often counter-acting the effects of the first group.

The generalized staircase model allows to evaluate the influence of these two
groups of effects and detect the predominant effect for a chosen set of the struc-
tural parameters. It was observed that too wide opening of the pores results in a
strong density decrease and lower values of $C_V$. On the other hand, an increase in
pore branching results in the predominant effect of an increase in material density
and, thus, in higher values of $C_V$. The latter effect is the most important effect
for the capacitor electrode materials with faradaic interfacial reactions, i.e. pseu-
docapacitors. The faradaic impedance is predominant at lower rates of potential
modulation (low $f$), where the effects of the first group are weak.

The results of the modeling provide the guidelines for the focused design of
electrode materials with high performance in EDLC and pseudocapacitors.

In Chapters 2 and 3 the two main classes of materials for the electrodes of
supercapacitors, namely, carbon-based materials and transition metal oxides are
discussed in details.
Chapter 2

Carbon electrode materials

As mentioned above, the carbon-based materials are most often used for the elec-
trodes of supercapacitors. There are several advantages of using carbon materials
for the electrochemical energy storage.

* Carbon is an abundant element (12th most abundant element on Earth) and can be synthesized from various fossil or bio-sources.

* Carbon has the lowest electrical resistivity among all non-metal elements. For example, graphite resistivity in the plane-parallel direction is equal to 0.4 µΩ·m (in the plane-perpendicular direction is larger than 40 µΩ·m) [61]. This in plane resistivity is just a few times higher than the typical resistivity of transition metals (e.g. 0.096 µΩ·m for Fe) and ca. 10 times higher than that of coinage metals (e.g. 0.017 µΩ·m for Cu).

The conductivity of carbon is provided by the $\pi - \pi$ electron conjugation in the ordered $sp^2$ carbon sheets. Thus, the carbon conductivity is related to the ratio of $sp^2$ and $sp^3$ carbon and the size of the ordered $sp^2$ domains which form the graphene sheets. The conductivity of carbon materials spans from ca. $10^5 \Omega^{-1} \cdot cm^{-1}$ for the defect-free graphene sheet to ca. $10^{-15}$ for the diamond.

* Carbon is a light element: the density of graphite is 2.267 g/cm³, which is comparable to the density of Al (2.70 g/cm³)) and is lower than the densities of all transition metals. For carbon materials one has to distinguish between the density of carbon element (the above given value) and the apparent density of carbon materials. The apparent density is often much lower than the
density of carbon element due to the porosity or granular structure of carbon materials. For example, the typical densities of activated carbons are 0.6-0.8 g/cm³ [61].

* The structure of carbon materials can be strongly varied on the levels of its macro-, micro-, and nano-structures. In particular, carbon materials can be prepared in the form of powders, paper-like tissues, sheets, amorphous solids etc., i.e. with different macrostructures. Furthermore, carbon materials exist in the form of nanoparticles, nanofibers, nanotubes, atomically thin layers, nanosheets etc., i.e. in different nanostructures. Such variability of carbon materials provides possibilities for optimization of their structure according to the demands of a particular application. For example, in the EDLC, the charge is stored at the electrode/electrolyte interface. Thus, a high SSA, which is accessible for an electrolyte, is required for the best performance of carbon as EDLC electrode. The SSA of carbon materials can be as high as 3000 m²/g. However, one has to take into account that the surface specific charge capacitance has a complex dependence on the SSA, which will be discussed in more details below.

Carbon can be found in various allotropies: sp³ (diamond structure), sp² (graphene sheet or graphite basal plane), distorted sp² (fullerene structure), sp¹ (carbyne structure) [62]. Carbon easily forms the oligo- and polymeric chains giving rise to the vast variability of existing carbon structures: both natural (diamond, graphite, coal, tars, etc.), and engineered. The author suggests to classify these various structures according to their architectures as 0D, 1D, 2D and 3D materials. Below the examples of the characteristic structures of are described with particular attention to the materials relevant for the EESD applications.

## 2.1 0D-carbon

Carbon nano-onions (CNO) or onion-like carbon (OLC) [63] are the examples of the 0D-carbon. CNO can be obtained by the high temperature annealing (1500-2000°C) of nanodiamonds (produced by the detonation synthesis) under inert atmosphere or in vacuum. CNO consists of concentric carbon spheres with the diameters less than 10 nm. The typical SSA of CNO is less than 500 m²/g. The high temperature annealing results in the graphitization of carbon. As a result,
CNO materials have a high conductivity, a low defect density, and a relatively low surface specific capacitance (≈ 4 μF/cm²) and mass specific capacitance (≈ 2 F/g). Considering the low specific capacitances and the high production costs, CNO represent a little practical interest for the energy storage applications. However, it is an interesting model electrode material with a well controlled and regular porosity.

## 2.2 1D-carbon

Carbon nanotubes (CNT) and nanofibers (CNF) are the examples of the 1D-carbon. This is one of the most widely studied type of carbon materials for supercapacitor electrodes. There are numerous methods of synthesis of CNT/CNF, including the chemical vapor deposition via the decomposition of hydrocarbon precursors [7]. This method offers a preparation of ordered (vertically aligned) nanotubes electrode with a relatively easy scale-up to mass production.

The main advantage of using 1D-carbon materials as electrodes is their controlled porosity (with predominant presence of mesopores), especially in the case of vertically aligned nanotubes or nanofibers. There are numerous studies demonstrating the possibility of the fine control on the geometry and the composition of carbon nanotubes. This makes CNT a very suitable model material to study the influence of various structural parameters on the performance in supercapacitor electrodes.

On the other hand, the disadvantage of CNT is a relatively low SSA (less than 250 m²/g) of the as-synthesized materials, resulting in a low mass specific capacitance. However, the mass specific capacitance can be significantly increased by the chemical post-treatment, due to the increase in SSA and carbon surface functionalization. In particular, the introduction of the O- and N-containing surface groups allows to improve the accessibility of pores for aqueous electrolytes and increases the contribution of pseudocapacitance due to the redox transformations related to these groups [64]. The optimization of geometry and composition of CNT allows to increase the SSA up to 500 m²/g resulting in the mass specific capacitance equal to 203 F/g.

Carbon nanotubes and nanofibers are often utilized in combination with metal oxides and/or conductive polymers as the components of the composite electrodes for supercapacitors [65]. The reason is the regular controllable structure of carbon
<figure><img src="image_30.png" /><figcaption>Figure 2.1: The Lerf-Klinowski chemical model of GO produced by the chemical oxidation of graphite [68].</figcaption></figure>

nanotubes, namely, their geometry, porosity, surface and bulk composition. As mentioned above, the modification in chemical composition allows to increase the intrinsic charge capacitance of carbon nanotubes due to the pseudocapacitance reactions. The introduction of heteroatoms (N, S) also allows to create the adsorption sites for metal precursors, thus, facilitating the synthesis of metal oxide / carbon composites.

## 2.3 2D-carbon

Graphene is the 2D carbon material which consists of the separated (exfoliated) sheets of sp²-carbon. This material has a high mechanical stability: the Young modulus of a defect-free single graphene sheet is equal to 1.0 TPa and break stress is equal to 130 GPa [66]. Graphene has a high theoretical SSA equal to 2630 m²/g [67]. The single sheet graphene is characterized by a relatively low surface resistance 400 Ohm/sq, which is decreased to 30 Ohm/sq for the film of 4 graphene sheets with thickness of ca. 2 nm [68]. The bulk conductivity of such graphene-type film is estimated to be ca. $1.6 \cdot 10^5$ Ohm⁻¹·cm⁻¹.

The high electronic conductivity and mechanical stability make graphene a very attractive electrode materials for EDLC. However, there are two main challenges related to the application of graphene as an electrode material: (i) the tendency of re-stacking of graphene sheets during the deposition on the electrode and (ii) the cost of scaling-up of the synthesis of graphene above laboratory quantities.

The methods for the graphene synthesis are typically divided into two groups: the top-down methods (e.g. mechanical exfoliation, chemical and sonochemical
exfoliation) and the bottom-up methods (chemical vapor deposition (CVD)). The CVD and the mechanical exfoliation are known to produce high quality graphene films [67], however, its application is limited to non-porous substrates. Among the top-down methods, the mechanical exfoliation is the hardest method to scale up. Alternatively, the chemical exfoliation route can be easily scaled up to mass production. Below two main strategies of the chemical exfoliation methods are considered: chemical (Hummer method) and sonochemical exfoliation.

### 2.3.1 Chemical exfoliation of graphite

The chemical exfoliation consists of the chemical oxidation of graphite by the Hummer's original [69] or improved synthesis [70], producing the graphene oxide (GO). This step is followed up by the reduction of GO, either chemical or thermal, to the reduced graphene oxide (rGO) [5, 71]. The chemical oxidation of $sp^2$ carbon introduces the oxygen-containing functional groups (Fig.2.1) and results in the formation of the graphite oxide. The polar functional groups decrease the interaction between the graphene sheets. A hydrophilic nature of the oxygen containing groups facilitates the exfoliation in aqueous solutions. Eventually, this results in the exfoliation of graphite, often assisted by the controlled sonication of the solution. Thus, the stand-alone GO sheets suspension is formed in the solvent. The difficulty of this step of the synthesis is to tune the oxidation rate: a stronger oxidation results in a more efficient exfoliation of graphite, as well as in shorter flakes of GO with more structural defects [72].

The exfoliation efficiency is manifested by the change in the XRD of the material. The graphite shows a narrow and strong peak at $2\theta = 26.5^{\circ}$, corresponding to the (002) plane and the sheets separation distance equal to 3.35-3.37 Å [73]. The graphite oxidation results in the shift of this peak to $2\theta = 10^{\circ}$, indicating a wider separation of the graphene sheets. The chemical oxidation also induces the distortions and structural defects in the 2D-ordered graphene sheets.

These transformations are often detected by the Raman spectroscopy. The spectra of undistorted graphene contain the characteristic peaks: the G peak at ca. 1580 $cm^{-1}$ related to the scattering of $E_{2g}$ phonons [73] and the 2D peak at ca. 2700 $cm^{-1}$ due to second-order boundary phonons. The relative intensity of the 2D peak is sensitive to the number of layers in the graphene stack (Fig. 2.2). Indeed, a systematic study of the Raman spectra of the graphene sheets with different
thicknesses on the atomically flat SiO$_{2}$/Si substrate demonstrated the empirical linear dependence (2.1) of $I_{G}/I_{2D}$ ratio on the number of graphene layers $n$ [74]:

$$ \frac{I_{G}}{I_{2D}} = 0.14 + \frac{n}{10}. \qquad (2.1) $$

The position and intensity of the 2D Raman band of graphene depends also on the density of strains and defects in a graphene sheet [75]. Thus, the equation (2.1) should be understood only as a rough estimate, which is useful for the comparison of the thicknesses of the graphene materials with comparable structures.

The appearance of structural defects is manifested by the intensive D peak in the Raman spectra at ca. $1320 \text{ cm}^{-1}$ related to the first-order boundary phonons [66, 77]. The ratio of the intensities of the D- and G-peaks, $I_{D}/I_{G}$, is often used to characterize the density of structural defects in graphene materials. For example, in the well crystallized graphite, $I_{D}/I_{G}$ lies in the range of 0.05-0.35, depending on the origin of the graphite [73]. $I_{D}/I_{G}$ ratio increases up to 3.0-3.5 as an average distance between the structural defects $L_{D}$ decreases down to $\approx 4$ nm [68].

The distance $L_{D}$ can be approximated as an average size of the ordered sp$^{2}$ domain. For disordered activated carbons with a high density of structural defects, the typical $I_{D}/I_{G}$ ratio is much higher: 4.8-5.6 [77]. The formation of GO from the graphite results in a significant decrease in the conductivity due to the distortion of the $\pi-\pi$ conjugation. The surface resistance of GO is reported to be as high as $10^{12} \text{ Ohm/sq}$. Thus, the reduction of GO is required. The reduction can be done either thermally at temperature above 750°C, or chemically by NaBH$_{4}$, N$_{2}$H$_{4}$, or electrochemically [68]. The reduction of the functional groups is necessary to re-establish a predominant population of the conductive sp$^{2}$ domains. On the other hand, the recovery of these domains increases the hydrophobicity of rGO and can trigger the re-stacking of graphene sheets. Thus, the optimal reduction ratio has to be found experimentally.

The stacking degree and the concentration of defects can be controlled by the

<figure><img src="image_31.png" /><figcaption>Figure 2.2: The dependence of Raman spectra of the few layer graphene stacks on the number of layers [76].</figcaption></figure>
conditions of graphite oxidation and GO reduction [6]. According to the XPS data, the reduction of GO to rGO results in an increase in the atomic C/O ratio from 4:1-2:1 (typical for GO) to ca. 12:1 [68], which indicates that the reduction is not complete. Also, the typical $I_D/I_G$ ratio for rGO lies in the range 0.9-1.5, which indicates the presence of structural defects in the resulting material.

From the viewpoint of the energy storage applications, the advantage of GO/rGO route is that the partially reduced oxygen-containing functional carbon groups are electrochemically reversible and add the charge storage capacitance due to the pseudocapacitance phenomena [67]. The pseudocapacitance phenomena is discussed in details in the Section (1.2.3). The challenge of the GO/rGO route is to find the optimal conditions of each of its steps. The typical SSA and the mass specific capacitance of rGO electrode materials are reported in the ranges of 400-700 $m^2/g$ and 150-260 F/g correspondingly.

### 2.3.2 Sonochemical exfoliation of graphite

The sonochemical exfoliation of graphite allows to avoid the formation of high density of structural defects, because it relies on the presence of an exfoliating chemical agent in the sonicated liquid dispersion of graphite instead of the formation of oxygen functional groups. At the sufficiently powerful sonication, the exfoliation of graphite occurs in organic solvents, such as DMF and NMP [66] even in the absence of exfoliation agents. However, the exfoliation yield is relatively low [72]. In the last decade, the non-toxic perfluorinated organic solvents were shown to be efficient in the graphite exfoliation [76].

The exfoliating agents added to a solvent assist the exfoliation and can also act as a surfactant, stabilizing the graphene sheets and avoiding their re-stacking [67]. The variety of exfoliation agents allows to form stable aqueous and organic dispersions of partially stacked graphene sheets, most commonly with few (less than 10) layers stacked. Therefore, the resulted materials are commonly called as the few layer graphene (FLG) [76]. Mostly the expensive and toxic organic solvents are used, making scaling up of the process more difficult [72]. However, the utilisation of the bio-compatible oligopeptide exfoliating agents is reported [78], which offers the possibility for preparation of a stable aqueous dispersion of the FLG materials.

A comparison between the chemical or electrochemical oxidation/reduction
(GO/rGO) routine and the sonochemical exfoliation (FLG) demonstrated [79] that the latter has a generally lower yield and requires a longer time for the formation of comparable amount of the material. The exfoliation during the chemical or electrochemical oxidation of graphite is due to the chemical processes occurring in the bulk of carbon, such as oxidation, formation of oxygen-containing groups and, more importantly for the exfoliation, formation of gas ($CO_2$ and $O_2$) bubbles. The exfoliation during the sonochemical treatment is due to the formation of cavity bubbles located at the very vicinity of the solid/liquid interface. This results in a lower yield but a smaller damage in the graphene sheets structure.

### 2.3.3 Graphene-like carbon in supercapacitors

The stand alone graphene-like materials (rGO or FLG) are rarely applied in the supercapacitors due to the re-stacking phenomenon or the limited surface accessibility in the presence of surfactants [5]. In order to avoid the re-stacking, the spacers are often introduced between the graphene sheets: for example, the carbon nanotubes [80, 81] or the metal oxide nanoparticles [82, 83]. The latter belongs to the class of the composite metal oxide / carbon materials, which will be considered in details in Chapter 3.2.

The synthesis of graphene-based electrode materials by using structuring compounds (*template synthesis*) is another common and promising approach. Polymers are most commonly applied as the structuring elements [67]. The materials prepared by the template synthesis have the high SSA and conductivity of the graphene based materials and the controlled optimized porous structure of templates. Such structural design of the composite graphene/polymer electrode materials results in the outstanding capacitance at high electrode loading. For example, the stand-alone graphene/PANI foam with the loading equal to 0.90-0.96 mg/cm² demonstrated the capacitance equal to 790 F/g at the charging current $I_g = 1$ A/g [84]. The reported capacitance is higher than theoretical capacitance for the stand alone graphene (ca. 550 F/g), indicating the contribution of the polymer. It explains a ca. 20% loss of capacitance of the composite during 1 A/g cycling in 1M $H_2SO_4$, while the capacitance retention of the stand alone carbon electrodes is close to 100% under the same conditions.

Alternatively, the inorganic templates can be applied to control the structure of a synthesized electrode material. For example, MgO or $Mg(OH)_2$ are reported
<figure><img src="image_32.png" /><figcaption>Figure 2.3: The typical SEM images of carbon material prepared by the KOH impregnation / etching technique in the absence (a-c) and in the presence of the templating NaCl agent (d-f) [10].</figcaption></figure>

as template agents [85, 86]. The materials synthesized with these templates show the high specific capacitance up to 450 F/g at 0.5 A/g in 6M KOH. Contrary to the polymer/graphene composites, the retention of capacitance up to 134% was detected after $10^4$ cycles at 200 mV/s, i.e. a noticeable increase in capacitance upon cycling was observed, which often occurs for carbon electrodes.

### 2.3.4 Case study: Graphene-like carbon from sugar/salt mixture

In this subsection the graphene-type carbon materials synthesized and tested for the supercapacitors in ICPEES with the participation of the author are discussed [10].

The graphene-like carbon materials with the ordered 2D-carbon structure were synthesized by the high temperature KOH etching of the refined sugar in the presence of NaCl as an inorganic template, which is soluble in water [10]. These materials are further referred to as salt-to-sugar or the *StS* materials. Thermal KOH etching is a conventional technique to prepare the carbon materials with high SSA. An impregnation of the carbon source by KOH is followed by its thermal activation, resulting in the development of a complex porous 3D carbon structure with a high SSA up to the theoretical limit $2630 \text{ m}^2/\text{g}$ (Fig. 2.3(a-c)). The technique of thermal chemical etching is considered in details in Section 2.4.
In contrast to the conventional etching technique, the impregnation of carbon source (sugar), was done in the presence of NaCl as a templating agent. The thermal etching results in the carbon material with predominantly disordered graphene 2D-sheets, providing a high accessibility of carbon surface for an electrolyte (Fig. 2.3(d-f)). The properties of the $StS_{x/y}$ carbons depended on the mass ratio $x/y$ of salt to sugar. The optimal ratio was found to be $x/y = 1/1$. The graphene-type structure of the carbon sheets is evident from the presence of 2D Raman peak at $2400 \text{ cm}^{-1}$ [74]. The intensity ratio $I_G/I_{2D} \approx 5.3$ shows that the observed planes contain few tens of single graphene sheets. This fact is in agreement with a relatively small SSA of the $StS$, 630 - 890 m$^2$/g, comparing to the SSA of activated carbons prepared by the conventional KOH etching technique.

Despite a lower SSA, the $StS_{1/1}$ carbon demonstrated an outstanding specific capacitance $C_g = 293 \text{ F/g}$ in 0.5M $H_2SO_4$ at 1 A/g with a very moderate decrease to 234 F/g at a higher charging rate of 50 A/g, (Fig.2.4(A)). Moreover, the capacitance retention close to 100% was observed in more than $12 \cdot 10^3$ cycles after the 11.5% capacitance increase in first 1000 cycles at 5 A/g (Fig.2.4(B)). An estimate of the surface specific capacitance of the $StS_{1/1}$ carbon (SSA=780 m$^2$/g) demonstrated a high value of $C_s \approx 38 \mu\text{F/cm}^2$, i.e. few times higher than the one expected for the double-layer capacitance of the $sp^2$ carbon surface. This indicates the contribution of the surface- and sub-surface pseudocapacitance phenomenon to the charge storage in the $StS$ carbon. For example, the H intercalation into the graphitic layers can be responsible for the pseudocapacitance phenomenon. Indeed, the contribution of pseudocapacitance is manifested by the presence of a distinct pair of the CV peaks, which are centered around 0.5 V RHE (Fig.2.4(A)). These peaks are similar to the CV peaks for carbon electrodes submitted to the strong chemical oxidation, (Fig.1.7) [29]. It is important to emphasize that as the charging rate increases from 1 A/g to 50 A/g, $C_s$ decreases only slightly from 38 to 30 $\mu\text{F/cm}^2$, indicating a fast rate of pseudocapacitance processes within $StS$.

More details on the mechanism of the interfacial phenomena can be obtained from the impedance spectra (Fig. 2.5). The Nyquist plots demonstrate the typical behavior for a porous material in the capacitor. Namely, the transition from low $\phi < 45^\circ$ at high $f$ to $\phi \approx 90^\circ$ at $f < 100 \text{ Hz}$ is observed. The low frequency part of the Nyquist plots can be modeled by the CPE impedance with $a \approx 0.89$, which is a typical value for the carbon electrodes. In the high frequency range ($10^5 - 10^2 \text{ Hz}$), one clearly observes the phase angle $\phi$ decreasing to the values close to or
<figure><img src="image_33.png" /><figcaption>Figure 2.4: (A) The CV curves of the $StS_{x/y}$ carbon electrodes in 0.5M $H_2SO_4$ at 20 mV/s. (B) The dependence of the specific capacitance of the $StS_{1/1}$ electrode on the number of galvanostatic cycles applied at different charging rates [11].</figcaption></figure>

even less than 45°. The low phase angle is related to the influence of porosity of $StS$ material, as discussed in details in the Section 1.3.

The above discussed results allows to relate the strong performance of the $StS_{1/1}$ electrodes to their structural properties. The presence of the graphitic layers appears to be important for the electrolyte ions intercalation. The presence of the oxidized carbon surface groups seems to be preferable for the strong interaction of the electrolyte species with the carbon surface. The optimal combination of these two factors is responsible for an outstanding electrochemical performance of the $StS$ carbon

materials. In order to support these conclusions, a more detailed study of the mechanism of pseudocapacitance in the $StS$ carbon materials in various electrolytes is needed (see Chapter 4).

<figure><img src="image_34.png" /><figcaption>Figure 2.5: The Nyquist impedance plots of the $StS_{1/1}$ carbon electrode at the open circuit potential in 0.5M $H_2SO_4$ [11].</figcaption></figure>
## 2.4 3D-carbon

The formation of the EDL is the main mechanism of charge storage in carbon electrodes of the EDLC. The charge capacitance of an electrode materials of the EDLC is determined by its accessible interfacial surface area. This parameter depends on (i) the SSA of carbon materials and (ii) the accessibility of carbon surface area for an electrolyte. According to (1.41), the mass specific capacitance $C_g$ is linearly proportional to the SSA of carbon: the higher is SSA, the higher is $C_g$. This trend has been confirmed by the analysis of experimental data [28]. However, the experimental dependence of $C_g$ on SSA is not strictly linear due to a decrease of $C_s$ with the SSA increase. There are 2 origins of the dependence of $C_s$ on SSA.

* In carbon materials, a higher SSA is provided by a higher volume contribution of micropores: the surface-to-volume ratio of pores increases with the decrease of their diameter $d$. However, as $d$ decreases to the values below the doubled size of the solvated ions (e.g. below ca. 1 nm), the surface specific capacitance of carbon decreases down to $5 \ \mu\text{F/cm}^2$, because the diffusion of ions in these pores becomes more difficult [60]. However, a further decrease in pores size (below 1 nm) is expected to result in a significant increase in the surface specific capacitance (up to $15 \ \mu\text{F/cm}^2$) due to the distortion of ions solvation shell [60]. Another origin of the increase of surface specific capacitance is the increase in surface quantum capacitance, related to the formation and screening of the image charge in the conductor induced by the adsorbed ion [28]. Nevertheless, the experiments reported in the literature demonstrate the decrease of $C_s \approx 15 \ \mu\text{F/cm}^2$ for SSA < 1000 m$^2$/g to $C_s \approx 5 \ \mu\text{F/cm}^2$ for SSA > 2500 m$^2$/g.

* It has been discussed in details in Section 1.3 that the depth of the penetration of potential modulation inside the pores decreases for a higher modulation rate and a narrower pore diameter $d$ (see 1.23). Thus, a material with the narrower pores have a stronger loss of interfacial capacitances $C_g$ and $C_V$ at higher charging rates. However, narrower pores also provide a lower porosity of the material with the same SSA, resulting in a higher material density, and, thus, a higher $C_V$. As it was discussed in Section 1.3, the effect of porosity is predominant in the range of potential modulation rates which is
typical for supercapacitors, both EDLCs and pseudocapacitors. Thus, the micropores are favorable for the materials with high $C_g$ and $C_V$.

The activated carbons (AC) form the class of 3D carbon materials with the highest practically achievable SSA, larger than 3000 m²/g [62]. The other advantage of the activated carbons is relatively straightforward synthesis routines, which can be classified in general as (i) the physical and (ii) the chemical carbon activation methods [87].

The physical activation occurs due to the pyrolysis of a carbon source in the inert atmosphere at high temperature (800 °C or more). The resulting structure depends on the duration and temperature of the pyrolysis. The optimal temperature which provides the maximal carbon surface area, depends on the source of carbon. For example, a detailed study of the activation of the pine-derived biomass [88] demonstrated that the highest BET surface area was obtained after the pyrolysis at 600°-800°C, while further increase in the temperature resulted in a sharp decrease of the surface area. In general, 4 typical temperature domains of carbonisation were proposed [88].

* The initial structure of source was predominantly kept below 350°C, providing the predominant crystalline cellulose structure and a relatively high content of hydrogen, oxygen and other volatile elements.

* The initial degradation started above 350°C, resulting in the increase of the content of the $sp^2$ carbon, and a slight growth of the BET surface area and porosity.

* The formation of carbon nanocrystalline structure occurred at ≈ 500°C; resulting in the further increase in the $sp^2$ carbon content. In this temperature range, the maximums of the BET surface area and the volume of pores were obtained.

* At the temperature above ≈800°C the graphitic structure developed, the maximum of $sp^2$ carbon content was reached, while the BET surface area decreased.

The first step of the chemical activation is the impregnation of carbon source with an etching agent, most typical of which are NaOH/KOH, $H_3PO_4$, $H_2SO_4$,
$K_2CO_3$, $ZnCl_2$. The next step is the etching, which is performed at a high tem-
perature but below the temperatures typical for the physical activation. Then
the traces of etching agents are washed out from resulting material after its cool-
ing. The chemical activation is preferred due to a lower activation temperature
and shorter time. The porous structure of the final material is controlled by the
duration of activation and the amount of etching agent.

Various sources of carbon can be used to prepare the activated carbon by the
chemical activation method. From the sustainable development point of view, the
utilization of natural, abundant and cheap carbon sources, such as peat, wood, or
coconut shells is specially interesting for the mass production of carbon materials.
In particular, the utilization of natural carbon-rich sources is a viable strategy for
the biomass valorization [89]. The chemical activation of a bio-source precursor
eliminates most of the elements other than carbon and creates the porous carbon
structure with a well developed surface area. The structure of the final product, i.e.
the activated carbon, depends on the activation parameters: namely, nature and
concentration of an activating agent, temperature and duration of the activation
step.

**2.4.1 Case study: activated carbon from ripe fig fruit**

In this subsection the 3D activated carbon materials, which were synthesized from
a bio-sourced precursor in the ICPEES with the participation of the author, are
discussed [11].

The utilization of bio-wastes as precursors for the porous carbon electrodes for the EESD is an attractive topic of research from the sustainable development point of view. The electrodes for the EDLCs and the Li-S batteries have been prepared from the banana peels by chemical activation [90, 87]. A fig fruit is another natural precursor for the carbon materials. It can be used as a model of various bio-wastes due to the high content of carbonhydrate lignocellulose fibers.

A highly porous carbon was prepared by the KOH activation of the natural ripe fig fruit at the temperatures above 700°C [10]. Similarly to the thermal activation, the chemical activation at a higher temperature results in a more graphitized structure. It is manifested by the appearance of the (002) XRD peak and a significantly lower *I*<sub>D</sub>*I*<sub>G</sub> ratio of Raman peaks observed for the AC prepared at 900°C in comparison to the AC prepared at 700°C (Fig.2.6).
<figure><img src="image_35.png" /><figcaption>Figure 2.6: (A) The Raman spectra and (B) the XRD of the bio-sourced carbon synthesized by the KOH thermal activation at 700°C and 900°C [10].</figcaption></figure>

<figure><img src="image_36.png" /><figcaption>Figure 2.7: (A-D) The SEM images of the bio-sourced carbon synthesized by the KOH thermal activation at 700°C and 900°C [10].</figcaption></figure>
<figure><img src="image_37.png" /><figcaption>Figure 2.8: The pores distribution as measured by the BET [10].</figcaption></figure>

Contrary to the physical activation, an increase of the annealing temperature of the chemical activation above 700°C results in a higher BET: 1003 m²/g at 700°C and 2337 m²/g at 900°C [10]. These high SSA values reflect a well developed porous structure (Fig.2.7). The higher SSA of carbon obtained at 900°C is related to strong contribution of micropores and small mesopores with $d < 4$ nm (Fig.2.8). The SEM images demonstrate that the pores of the AC sample prepared at

900° have the hierarchical branching structure: the wider pores are branching in several narrower pores. As concluded in Section 1.3, such structure provides a compact material with a well developed surface and is favorable for the materials with a high specific capacitance.

The electrochemical properties of the activated carbon obtained by the KOH thermochemical activation of a fig fruit are summarized in Fig.2.9. This AC material demonstrates an outstanding charge capacitance above 300 F/g in 0.5M H₂SO₄ at the charging rates below 2 A/g. Using the BET value of SSA, the surface specific capacitance was estimated as $C_s = 13$ µF/g, which is a value expected for carbon/electrolyte interface. It allows to conclude that:

* the EDL charging is the main mechanism of charge storage for the studied AC material,

* the utilization of an entire surface of the studied AC material is close to 100% because of the well developed open porous structure.

The increase in charging rate up to 50 A/g resulted in the decrease of the mass specific capacitance to 170 F/g. This might be related to the pronounced dependence of the penetration depth on the rate of potential modulation in the case of micropores. The effect of porosity was clearly detected by the impedance measurements. The EIS spectra of the activated carbon electrode, freshly prepared as well as after $10^4$ cycles are shown in Fig. 2.9(D). The freshly prepared activated carbon
<figure><img src="image_38.png" /><figcaption>Figure 2.9: The electrochemical properties of the bio-sourced activated carbon in 0.5 M H₂SO₄. (A) CV curves measured at 20 mV/s. (B) The dependence of the mass specific capacitance on the charging rate. (C) The evolution of the mass specific capacitance with the galvanostatic cycling at 2 A/g charging rate. (D) The impedance spectra measured at open circuit voltage at 5 mV amplitude [11].</figcaption></figure>

demonstrated a change in phase angle at the frequency decrease. A constant phase angle behavior, corresponding to $a=0.70$, was reached at $f<100$ Hz.

After 10000 charging cycles, the impedance behavior was changed. The effect of the dependence of the modulation penetration depth on the frequency was detected at high frequency ($f > 1$ kHz) only. The CPE behavior with $a=0.88$ was observed at the intermediate and low frequencies. The value of $a$ close to 0.9 corresponds to a non-homogeneous flat carbon surface. This indicates that the AC electrodes behave as flat carbon electrodes at these frequencies. 10000 charging cycles also resulted in a significant increase in the capacitance (Fig. 2.9(C)), which can be explained by an increase in the accessible surface area.

These phenomena can be interpreted as changes in the pore structure and the surface chemistry during the charge/discharge cycles, which result in the improvement of pore accessibility for an electrolyte. One may suggest that charging cycles result in the formation of more polar carbon surface groups, making the interface more hydrophilic, and, thus, making the pores more accessible. The presence of the polar surface groups also increases the interaction of electrolyte species with
the carbon surface and results in the faster surface redox transitions.

It can be seen that the impedance spectroscopy can provide valuable insights
into the influence of porous structure of a material on its electrochemical prop-
erties. In order to better understand these effects, a model of the interfacial
impedance for porous electrode materials in supercapacitors is needed. Such gen-
eralized staircase model and the results of its application for porous electrode
systems are described in the Section 1.3.

**2.5 Conclusions**

The carbon materials are among the materials most commonly used for the elec-
trodes of supercapacitors. The reasons are their high conductivity, electrochemical
stability, abundance and non-toxicity. The main mechanism of charge storage in
carbon electrodes is the formation of the EDL. However, the specific capacitance
of the carbon electrodes in the EDLC is limited by relatively low surface specific
capacitance of the carbon/electrolyte interface, which spans from 13.5 µF/cm² for
a single graphene sheet to 5 µF/cm² for the stacked sheets of the graphitized car-
bon [28]. A highest theoretical SSA of the exfoliated separate graphene sheets is
equal to 2680 m²/g, even though the SSA above 3000 m²/g was reported for the
activated carbons [62]. Thus, the highest specific capacitance of the purely EDL
electrodes cannot be higher than ca. 360 F/g.

The porosity of the carbon is an important factor determining its total charge capacitance. The micropores with $d < 1$ nm provide a high SSA even for relatively compact materials. Thus, these materials have high mass and volume specific capacitancies $C_g$ and $C_V$. Branching of wider pores into the narrower ones also leads to the high utilization of pores surface area and the high capacitance of electrode materials. Indeed, the AC electrodes, prepared from a fig fruit, demonstrated the open and well developed porous structure, the high SSA (2337 m²/g), the high mass specific capacitance (above 300 F/g) and the high retention above 100% after 10000 charging cycles at 2 A/g.

The contribution of pseudocapacitance to the charge capacitance of a carbon electrode is relatively small. The oxidation of carbon surface results in the generation of oxygen-containing functional groups responsible for the surface-bounded pseudocapacitance reaction [29], providing ca. 10 µF/cm². This contribution is comparable to the EDL surface specific capacitance.
The electrolyte ions intercalation into the carbon allows to significantly increase its charge capacitance. For example, the highest reported specific capacitance of the CH900-200 carbon (1194 F/g) is related to the intercalation of $H^+$ into the strongly swelled carbon structure [38]. However, the process of the $H^+$ intercalation is claimed to be very slow: the apparent diffusion coefficient of $H^+$, which is intercalated from an aqueous electrolyte, is several orders of magnitude smaller than diffusion coefficient of $Li^+$ cations intercalated from a non-aqueous electrolyte [91]. The slowness of the $H^+$ intercalation limits a practical interest in this swelled carbon structures.

Contrary to the CH900-200 carbon, the high specific capacitance of StS carbon materials, which is also related to the pseudocapacitance phenomenon, decreases only slightly at higher charging rates (up to 50 A/g). A combination of several factors is claimed to be responsible for this outstanding electrolchemical behavior:

* open porous structure with the surface of carbon easily accessible for an electrolyte,

* presence of the oxidized carbon surface groups enhancing an interaction of electrolyte species with the carbon surface,

* formation of the densely packed graphitic layers with the estimated thickness of few tens of graphene sheets, providing the medium for electrolyte ion intercalation.

Doping of the carbon materials by the heteroatoms (N,B) allows to increase their charge capacitance. In particular, N-doping enhances the interaction of solvent molecules with the carbon surface and, thus, enhances the pseudocapacitance phenomenon. Simultaneous tuning of the carbon structure and the amount of doping element allows to reach a relatively high specific capacitance even at relatively fast charging rates: above 350 F/g at 10 A/g charging rate [92].
# Chapter 3

## Transition Metal Oxide (TMO) electrode materials

### 3.1 General aspects

The metal oxides are the second largest class of materials utilized in EESD, in particular, in pseudocapacitors [1]. The oxides of transition and some post-transition metals are of particular interest due to the existence of at least 2 stable oxidation states for these metals (Fig. 3.1).

As several stable oxidation states for transition metal cations exist, the reversible redox transformation of these cations occurs upon the modification of the electrode potential. This results in an additional interfacial charge storage due to the phenomenon of pseudocapacitance, as discussed in Section 1.2.3. If this reaction is accompanied by an insertion of an ion into the sub-surface interfacial area, it provides a mechanism of propagation of the redox transition reaction into the bulk of oxide material according to (3.1). This equation is the same as the equation (1.10) in Section 1.2.3). As discussed in Chapter 1.2.4, the intercalation of an electrolyte cation $Ct^+$ ($H^+$ or $Na^+$, $K^+$, ...) results in the pseudocapacitance, providing that the process is sufficiently fast to be quasi-reversible in the electrochemical sense [8]:

$$ MO_n + \delta Ct^+ + \delta e^- \rightleftharpoons Ct_\delta MO_n \quad (3.1) $$

The efficiency of the charge storage by the intercalation mechanism depends on the maximal amount of cations that can be intercalated into the crystal cell of the
<figure><img src="image_39.png" /><figcaption>Figure 3.1: The oxidation states and the electron configurations in the TMO of the 4th (3d), 5th (4d), and 6th (5d) periods. B indicates a well-characterized binary oxide, T – a ternary or a more complex oxide. (B) and (T) indicate doubtful or poorly characterized or mixed valence systems [93].</figcaption></figure>
TMO. Ideally, all transition metal cations of the oxide (including the bulk of the oxide phase) can undergo the redox transition, meaning that $\delta = 1$ in (3.1). In this case, the maximal theoretical specific charge storage $Q_C$ (in C/g) for a given oxide can be estimated by using Faraday's law:

$$Q_C = \frac{x \cdot F}{M}, \qquad (3.2)$$

where $x$ is the stoichiometric factor of the metal in the oxide formula $M_xO_y$, $M$ is the molecular weight of metal oxide (in g/mol).

In the literature on the batteries, the value of charge storage capacitance $Q_C$ is traditionally given in A.h/kg (or mA.h/g) units. On the other hand, in the literature on the supercapacitors, the specific capacitance $C_g$ is the most common characteristic of charge storage capacitance (along with the energy density, see Section 1) and traditionally given in F/g units. $C_g$ can be related to $Q_C$ by the simple relation:

$$C_g = \frac{Q_C}{\Delta V}, \qquad (3.3)$$

where $\Delta V$ is the operational voltage of the supercapacitor, providing that $Q_C$ is expressed in C/g. Thus, $Q_C$ is expressed in C/g units further in the text. The transformation between the units is straightforward: C/g = 3.6 A.h/kg.

For $RuO_2$ ($M = 133$ g/mol, $x = 1$), $Q_C = 725.4$ C/g according to (3.2). This value is in a good agreement with the values of $Q_C = 614$ C/g, measured experimentally for the amorphous $RuO_2$ [44]. This agreement indicates that the value of the redox transformation degree $\delta \approx 0.85$. Thus, the most of Ru cations in the bulk of the oxide is involved into the charge storage due to a relatively high conductivity (both electronic and ionic) of $RuO_2$.

The (3.2) demonstrates that the highest mass-specific charge storage is expected for the oxides of the light metals with low $M$. This is why the oxides of the transition metals of the 4th period (Ti, V, Cr, Mn, Fe, Co, Ni) attract more interest, while the oxides of the transition metals of the 5th (Zr, Nb, Mo) and 6th (Hf, Ta, W) periods are much less studied in relation to the supercapacitors. According to Fig.3.1, the oxides of 4th period also have a stronger variability of stable phases with different oxidation states, which is beneficial for pseudocapacitance. However, these factors cannot be generalized: a good example is $RuO_2$, which is a metal oxide belonging to the 5th period. Despite having a relatively
<figure><img src="image_40.png" /><figcaption>Figure 3.2: The diagram of the abundance of elements in the continental crust [94].</figcaption></figure>

high molecular weight, typical for the 5th period metals, RuO$_{2}$ possesses a high electronic and ionic conductivity. This is why RuO$_{2}$ is the most studied oxide for supercapacitors.

High capacity of the ion intercalation (a large $\delta$) and a high conductivity are important criteria of the efficiency of metal oxides in supercapacitors. From the sustainable development point of view, the other important criteria of the efficiency of electrode materials for supercapacitors, are their abundance, affordable price, non-toxicity and environmental compatibility. Fig.3.2 illustrates the relative abundance of the elements in the continental crust of the Earth [94]. The transition metals of 4th period are apparently more abundant than the heavier metals. The Fe and Ti are the most abundant among the metals of the 4th period. A non-uniform distribution of the metals all over the world has to be taken into account as well. Co, for example, is a relatively abundant element, but more than 50% of Co is located in the mines of the Democratic Republic of Congo. Such non-uniform distribution is critical for the sustainable development of the technologies relying on Co.

The crystalline structure of many binary TMO consists of MO$_{6}$ octahedra arranged in a regular way by sharing corners, sides, or faces. It creates the crystal sites and channels for the cations transport and the intercalation without losing a
structural integrity of the oxide crystal structure. This phenomenon is utilized, for example, in the Li-ion batteries, where the cathode (for example, a lithium-metal mixed oxide) is emitting $Li^+$ cations and the anode (for example, the carbon) is hosting them by the intercalation mechanism during the charging [95, 45, 96]:

$$ \begin{gathered} \mathrm{LiMO}_n \rightarrow \mathrm{Li}_{1-x}\mathrm{MO}_n + x\mathrm{Li}^+ + xe^- : \text{ cathode} \\ x\mathrm{Li}^+ + \mathrm{C} + xe^- \rightarrow \mathrm{CLi}_x : \text{ anode} \\ \mathrm{LiMO}_n + \mathrm{C} = \mathrm{Li}_{1-x}\mathrm{MO}_n + \mathrm{CLi}_x : \text{ total} \end{gathered} \quad (3.4) $$

As it has been discussed in Section 1.2.4, the reaction (3.1) has to occur under a quasi-reversible conditions in supercapacitors. It means that the equilibrium concentration of the intercalated cation $Ct^+$ in the oxide at the interface, as defined by the Frumkin isotherm, must be attained. This concentration depends on the interfacial potential and the concentration of $Ct^+$ in an electrolyte, and is not influenced by the distribution of cations in the bulk of oxide (see 1.12). In the other words, the conditions of the cation transport in the oxide phase must not influence the intercalation ratio $\delta$, requiring a high ionic conductivity of the TMO and a relatively short diffusion path of cations. Thus, the pseudocapacitive electrode materials are often prepared as thin oxide films on a conductive substrate or as the composite materials of the nanostructured oxides (nanoparticles, nanofibers) mixed with a conductive component.

In [95], the criteria of an ideal electrode material for the Li-ion batteries were formulated. These criteria can be adapted for the electrode materials for super-capacitors, taking into account similarities and differences in the performances of batteries and supercapacitors.

* *Capacitance.* The oxide electrode materials ($MO_n$) should be light and able to intercalate a significant amount of the electrolyte cation $Ct^+$ in order to have a high specific charge capacitance.

* *Electrochemical stability.* The $MO_n$ should be stable in a wide potential range, insoluble and inert toward an electrolyte in order to provide a high voltage available for the supercapacitor. The high voltage $V$ and the capacitance $C_g$ provide a high energy density (1.3).

* *Conductivity.* The electrode materials should have a high electronic conduc-
tivity. The oxide MO$_n$ should also have a high ionic conductivity for the fast and efficient cation intercalation. A fast charge transfer within the electrode material is crucial for a high power density of supercapacitors.

* *Structural integrity.* The MO$_n$ oxide should demonstrate none or minimal modification of its crystalline structure for a possible widest range of the intercalation efficiency $\delta$ ($0 < \delta < 1$) for a high cyclability retention.

* *Availability.* From the scaling up point of view, the electrode materials should be abundant, inexpensive and non-toxic.

The capacitance and the availability criteria were discussed above. The conductivity, the electrochemical stability and the structural integrity of the TMO are discussed below.

## 3.2 Conductivity of TMO

The sufficient electronic conductivity is necessary to ensure the uniform distribution of the potential over the electrode material deposited on the current collector. In contrast to the carbon materials, used in the EDLC electrodes, most of the metal oxides are either semiconductors or insulators with a low conductivity at the ambient conditions ($10^{-3} - 10^{-17} \text{ S} \cdot \text{cm}^{-1}$). The conductivity increases exponentially with the increase of the temperature [97].

However, there exist the oxides which have metallic conductivity, which decreases with the temperature increase ($10^2 - 10^6 \text{ S} \cdot \text{cm}^{-1}$): TiO, CrO$_2$, NbO, RuO$_2$, ReO$_3$. There is also a numerous class of the metal oxides, which have a low conductivity at low temperatures, but demonstrate the transition to the metal conductivity at the certain temperature: Ti$_2$O$_3$, Ti$_3$O$_5$, Fe$_2$O$_4$, NbO$_2$, several vanadium oxides.

The electronic conductivity of the oxide material depends on its properties, and it is difficult to find general correlations between various parameters. The influence of various parameters of the oxide chemical and crystalline structures on its conductivity is discussed below.

The intrinsic electronic conductivity of the metal oxides depends on the position of the *d*-electron band relative to the Fermi level and the shape of the *d*-band. The conductivity depends on the *d*-band width: the narrowing of the band due to a
larger lattice distance results in a lower conductivity [97]). The conductivity is also influenced by the hybridization of the *d*-band with *s*- and *p*-orbitals: a stronger hybridization results in a lower conductivity [98]. These dependencies can be explained by the band theory combined with the Mott concepts of the dependence of band narrowing on the lattice parameters, the band overlap model and the antiferromagnetism effects.

The combined theory correctly predicts that the oxides $M_2O_3$ with 1,3,5,7 and 9 *d*-electrons (the corrundum crystalline structure) are the insulators ($Ti_2O_3-3d^1$, $Cr_2O_3-3d^3$, $Fe_2O_3-3d^5$, and $Ni_2O_3-3d^7$), and $V_2O_3-3d^2$ has the metallic conductivity [97]. The monoxides TiO, VO, MnO, FeO, CoO, NiO have the same crystalline structure (monoclinic NaCl), but the systematically varying lattice constant: from 4.09 Å for VO to 4.43 Å for MnO. In this set of oxides, TiO has the metallic conductivity, VO has the transition from the semiconducting behavior at low temperature to the metallic conductivity above 126 K, and the other oxides are the insulators [98].

The electronic conductivity of a material depends on its crystallinity. In general, a polycrystalline material has a lower conductivity than single-crystals due to the grain boundaries resistance [97]. It can also be expected that amorphous materials have a lower conductivity than the crystalline oxides due to a higher concentration of the grain boundaries. However, this is not always the case: the crystalline $V_2O_5$ is an insulator, while the amorphous $V_2O_5$ is a semi-conductor with conductivity $\sigma = 6.5 \cdot 10^{-6} \text{ S.cm}^{-1}$ (as explained by the small polaron nearest-neighbor hopping model [99]).

The electronic band structure and, thus, the conductivity of the oxide materials, can be tuned by altering its stoichiometry, for example, by the creation of oxygen deficiencies or the controlled doping [100]. One of the most impressive examples is the formation of Magnéli phases. Magnéli phases are the stable oxygen-deficient phases, which are formed by inducing regular stacking faults into the rutile-type oxides $MO_2$. These phases have a general formula $M_nO_{2n-1}$ [101]. Fig.3.3 demonstrates how the intrinsic conductivity of the titania oxides changes with its stoichiometry: $TiO_2$ anathase is a typical semiconductor, while $Ti_4O_7$ has the conductivity $1.5 \cdot 10^3 \text{ S.cm}^{-1}$, which is comparable to the graphite.

The doping of the oxide materials by the cation insertion is another commonly used approach to increase the conductivity [100]. For example, the doping of the electrochemically deposited $MnO_2$ by Au up to 9.9 at.% results in inducing the
<figure><img src="image_41.png" /><figcaption>Figure 3.3: The dependence of the intrinsic electronic conductivity of Magnélli oxide Ti<sub>n</sub>O<sub>2n-1</sub> on its stoichiometry [101].</figcaption></figure>

intermediate electronic states close to the conductance band and the conductivity
increase [102]. This results in the increase of C<sub>g</sub> of the 9.9 at.% Au doped MnO<sub>2</sub> to
626 F/g comparing to 380 F/g for the non-doped MnO<sub>2</sub>. The further increase in
the Au doping level results in the segregation of Au into the nanoparticles under
the potential cycling and in the decrease in specific capacitance. The cycling of
the potential of the optimally doped Au–MnO<sub>2</sub> oxide results in the capacitance
increase due to more homogeneous redistribution of Au in the oxide. This obser-
vation highlights an importance of controlled and uniform distribution of a dopant
in the oxide for the efficient improvement of the electrochemical performance.

The conductivity of the oxides with the same stoichiometry may significantly
vary depending on its crystalline phase. For example, MnO₂ can be synthesized
with various crystallographic structures: α-MnO₂, β-MnO₂, γ-MnO₂, δ-MnO₂,
ramsdellite, delithiated spinel λ-MnO₂, todoroskite MnO₂ [95] (see Table 3.1). The
difference in the structure is related to the way of arranging of MnO₆ octahedra
into a regular structure and the formation of molecular tunnels. The conductivity
of these oxides (mixed with 20%wt. of carbon) was found to vary from 10⁻⁶ for
<table><caption>Table 3.1: The description of some of the crystalline phases of MnO<sub>2</sub> [95, 103]</caption><thead><tr><td>Phase</td><td>Crystallographic structure</td><td>Relative tunnel size</td><td>Comments</td></tr></thead><tbody><tr><td>α-MnO<sub>2</sub></td><td>cryptomelane</td><td>(2x2) + (1x1)</td><td>regular combination of 2x2 (4.6x4.6 Å) and 1x1 (1.9x1.9 Å) tunnels; stabilized by cations or water</td></tr><tr><td>β-MnO<sub>2</sub></td><td>pyrolusite</td><td>(1x1)</td><td>compactly arranged MnO<sub>6</sub> octahedra with (1.9x1.9) Å tunnels</td></tr><tr><td>γ-MnO<sub>2</sub></td><td></td><td>(2x1) + (1x1)</td><td>combination of ramsdellite 2x1 (1.9x2.3 Å) and β-MnO<sub>2</sub> structure; ratio depends on synthesis conditions</td></tr><tr><td>δ-MnO<sub>2</sub></td><td>birnessite</td><td></td><td>layers of β-MnO<sub>2</sub> with 7.0 Å interlayer distance</td></tr><tr><td>λ-MnO<sub>2</sub></td><td>spinel</td><td></td><td>formed by delithiation of LiMn<sub>2</sub>O<sub>4</sub> spinel; narrow tunnels</td></tr><tr><td></td><td>ramsdellite</td><td>2x1</td><td>regular arrangement with (2x1) tunnels; γ-MnO<sub>2</sub> with nearly no β-MnO<sub>2</sub></td></tr></tbody></table>

the pyrolusite (β-MnO<sub>2</sub>) to 2.8 · 10<sup>-3</sup> S.cm<sup>-1</sup> for the spinel (λ-MnO<sub>2</sub>) or 3 · 10<sup>-3</sup> S.cm<sup>-1</sup> for the cryptomelane (α-MnO<sub>2</sub>) [103]. It is important to note that various phases may contain various amounts of water. For example, β-MnO<sub>2</sub> contains no water after the synthesis, while α-MnO<sub>2</sub> contains significant amount of water. The water in α-MnO<sub>2</sub> stabilizes the wide tunnels structure and can enhance the charge transfer rate [95]. The difference in the electrochemical performance of various phases of MnO<sub>2</sub> will be discussed below.

The sufficient intrinsic electronic conductivity (i.e. the low resistivity) is necessary for the electrode materials for supercapacitors due to the requirement of a high specific power (see 1.4). However, even for MnO<sub>2</sub> with a relatively high conductivity, this requirement is fulfilled only for the thin film electrodes, where the values of pseudocapacitance are close to the theoretical value [104, 105]. The
increase of electrode loading above ca. 10 µg/cm² results in the significant decrease of the specific capacitance. This is related to the hindered charge transfer at interparticles contacts, which have much lower effective cross-sections comparing to the bulk volume of the particles. A typical loading of carbon electrodes in the commercial supercapacitors is, however, 1-10 mg/cm², i.e. 2-3 orders of magnitude larger than the above mentioned limit.

This problem is typically addressed by adding a conductive carbon powder (15-20%$_{wt.}$ of C) to the oxide in order to form a thicker electrode. However, the mechanical mixing of the components often results in the non-uniform distribution of the oxide particles and the existence of interparticles contacts even at the carbon content as high as 66%$_{wt.}$ [13]. A more promising approach is the synthesis allowing the formation of the metal oxide and the carbon composites with the uniform mixing of the components directly during its formation. The example is the solid state combustion (SSC) synthesis, which is discussed in details in Section 3.3.3. The 34%$_{wt.}$ Mn₃O₄/C composite, prepared by the SSC method, was studied by the author [13]. The material demonstrated the capacitance which was several times higher comparing to the mechanical mixture of the same components. The difference in capacitance values became stronger at the faster potential sweep rate, i.e. at the higher power of charging/discharging (Fig.3.4).

As discussed above, the sufficient electronic conductivity is necessary for the uniform polarisation of the electrode material and its efficient utilisation in supercapacitors. The ionic conductivity is another factor, which is often claimed to limit material utilization in supercapacitors. For example, a study of different crystalline phases of MnO₂ [103] demonstrated that their specific capacitance was correlated with SSA and, more strongly, with the ionic conductivity. In particular, very close specific capacitances of the spinel and the birnessite δ - MnO₂ were detected despite the fact that the electronic conductivity of the spinel was 3 orders higher. The corresponding ionic conductivities were much closer (0.015 and 0.017 S.cm⁻¹), allowing the authors of [103] to conclude that the ionic conductivity was a main factor influencing the specific capacitance of the oxide materials.

One has to emphasize that measuring the ionic conductivity of the TMO under the ambient conditions is more difficult than measuring the electronic conductivity. In particular, separating ionic and electronic conductivities of the TMO is not simple. The ionic conductivity is often measured as the difference of total conductivity of the material filled with an electrolyte and the conductivity of the
<figure><img src="image_42.png" /><figcaption>Figure 3.4: The TEM images of Mn₃O₄ synthesized by the SSC method (A) without carbon, (B) with 66%wt. of carbon. (C) The dependence of specific capacitance Cg on the potential sweep rate for the 34% Mn₃O₄/C electrode [13].</figcaption></figure>

dry material (which is assumed to have the electronic conductivity only) [103]. However, in this method the conductivity of an electrolyte in the interparticles volume is either neglected or approximated by a simple geometric volume. Thus, one may anticipate a relatively strong incertitude in the reported values of the ionic conductivities.

Thus, it is important to evaluate the theoretical requirements to the ionic conductivity for the TMO materials for supercapacitors. As mentioned in Section 1.2.4, the ionic transport in the bulk of the oxide phase should not influence the interfacial equilibrium concentration of intercalating ions. The intercalation follows the interfacial redox reaction 3.1, resulting in the gradient of Ct⁺ concentration from the interface to the bulk of the oxide particle. Assuming the uniform distribution of the electrical potential within the oxide phase (due to sufficient electronic conductivity), the charging current I is related to the flow Jₐt⁺ of Ct⁺ ions from the interface to the bulk of oxide according to the first Fick's law <fnref n="1" />:

$$I = -FAJ_{\text{Ct}^{+}} = -FAD_{\text{Ct}^{+}}\frac{\partial C_{x}}{\partial x}, \qquad (3.5)$$

<footnote n="1">¹For the simplicity of evaluation, the flat diffusion model is used, neglecting the curvature of the electrode/electrolyte interface</footnote>
<figure><img src="image_43.png" /><figcaption>Figure 3.5: The illustration of the $C_t^+$ cation intercalation into the oxide particle.</figcaption></figure>

where A is the surface area of the electrode/electrolyte interface, $C_x$ is the concentration of $C_t^+$ inside the oxide phase at distance $x$ from the interface (Fig.3.5).

Using the second Fick's law and the boundary conditions $C_{x=0} = C_0$, $C_{x=\infty} = 0$, the equation (3.5) can be integrated into the following equation for the charging current:

$$I = A F C_0 \frac{D_{C_t^+}}{\delta_L}, \qquad (3.6)$$

where $\delta_L$ is the apparent thickness of concentration gradient layer.

For the supercapacitors materials, the requirement for the concentration of $C_t^+$ is $C_x \approx C_0$ for any $x$ inside the particle. This means that $\delta_L \gg d$, where $d$ is the thickness of the oxide phase. For example, $d$ can be the diameter of the oxide particle.

For the efficient intercalation material, $\delta \approx 1$, and the value of $C_0$ can be approximated as the concentration of metal cations of the oxide. For example, for $MnO_2$, $C_0 \approx \rho/M$, where $\rho$ is the density of the oxide material, and $M$ is its molecular weight. Thus, for $MnO_2$, $C_0 \approx 0.058 \text{ mol/cm}^3$.

The value of $\delta_L$ depends on the charging rate $I$ (the faster the charging, the shorter is $\delta_L$) and on the solid state diffusion coefficient $D_{C_t^+}$ (the bigger $D_{C_t^+}$, the longer is $\delta_L$). The diffusion coefficient can be related to ionic conductivity $\sigma_i$
by using the Smolukhowski-Einstein equation:

$$ \sigma_i = \frac{F^2 C_{\text{Ct}^+} D_{\text{Ct}^+}}{RT}. \qquad (3.7) $$

For $\sigma_i \approx 0.02 \text{ Ohm}^{-1}\text{cm}^{-1}$ reported in [103], $D_{\text{Ct}^+} \approx 10^{-7} \text{ cm}^2/\text{s}$. From the practical point of view, it is useful to relate the charging rate $I$ (in A) with the specific charging current density $i_m$ (in A/g) by considering the geometry of a spherical particle with the diameter $d$:

$$ i_m = \frac{6IA}{\rho d}. \qquad (3.8) $$

Combining the equations (3.7) and (3.8), the equation for $\delta_L$ dependence on $i_m$ is obtained:

$$ \delta_L = \frac{6FC_{\text{Ct}^+} D_{\text{Ct}^+}}{i_m \rho d}. \qquad (3.9) $$

For $i_m = 100 \text{ A/g}$ (the fast charging rate for the supercapacitor electrodes) and $d = 100 \text{ nm}$ (large particles), (3.9) gives the estimate $\delta_L \approx 0.6 \text{ cm}$, which apparently satisfies the condition $\delta_L \gg d$. The values of $D_{\text{Ct}^+}$ estimated from the values of $\sigma_i$ reported in [103] are few orders of magnitude higher than those reported for the $\text{Li}^+$ solid state diffusion [39]. This fact points to the above mentioned incertitude of the measurements of $\sigma_i$ for the TMO materials. However, even for the values of $D_{\text{Ct}^+}$, which are 3 orders of magnitude lower, the estimated value of $\delta_L$ is 6 $\mu\text{m}$ and, thus, the condition $\delta_L \gg d$ still holds.

The above discussion leads to the assumption that for the nanostructured oxides (for example, in the form of large nanoparticles) the ion mass transport is not a limiting factor even for relatively high charging rates. However, this requires that the electronic conductivity of the oxide phase is sufficiently high to assume the uniform distribution of the potential inside the oxide phase.

The electrochemical stability of the TMO as another important criterion of efficiency of the oxide materials as supercapacitor electrodes is considered in the next section.
## 3.3 Electrochemical stability and structural integrity of TMO

The other important parameter determining the performance of electrode materials in supercapacitors is the capacitance retention during charge/discharge cycling. The pseudocapacitance phenomena involves the redox reaction which can be not completely reversible due to the change of the metal oxide structure or the formation of an insulating or soluble oxide phase. In the electrochemistry, the thermodynamic stability of a material in aqueous solutions is conveniently analyzed by the E-pH diagrams, which are thoroughly presented in Pourbaix atlas [106] for the most of the metals. The Pourbaix diagrams for some of the metal/metal oxide systems are presented in Fig.3.6. Generally, there are 3 domains of stability of metal/metal oxide system in contact with an electrolyte.

*   **Immunity (blue domains in Fig.3.6).** Under these conditions, a metal in contact with an electrolyte is thermodynamically stable and not corroding. The charge can be accumulated on the metal/electrode interface, similar to the EDLC. The charge storage in the bulk of the metal is possible by the absorption of electrolyte components, for example, by the absorption of hydrogen in Pd and the formation of Pd hydrides [107]. However, this type of processes provides either lower energy density or weaker cyclability comparing to the pseudocapacitance in the metal oxides.

*   **Corrosion (yellow domains in Fig.3.6).** The thermodynamically stable form of the metal under these conditions is water soluble: it can be either the metal cation $M^{z+}$ (in general, at acidic pH) or the metal anion $MO_y^{z-}$ or $H_xMO_y^{z-}$ (in alkaline solutions). The diagram does not show the kinetics of the corrosion, thus, the corrosion might be kinetically hindered. For example, the chromium (III) oxide $Cr_2O_3$ is thermodynamically unstable at acidic pH (Fig.3.6(C)). However, the kinetics of its dissolution in the most of acids is so slow that the layers of $Cr_2O_3$ are used as the corrosion protection [108]. Similarly, Al oxide $Al_2O_3$ is stable in the acids due to the kinetically hindered dissolution despite being thermodynamically unstable. Nevertheless, in general, the lack of thermodynamic stability in the corrosion domains does not allow the sufficient cyclability of a metal oxide for the application in supercapacitors.

*   **Passivation (green domain in Fig.3.6).** Under these conditions, the given
<figure><img src="image_44.png" /><figcaption>Figure 3.6: The Pourbaix diagrams for different metals (the sketch is adapted from [106]).</figcaption></figure>