Theoretical and Experimental
Studies of Electrode
and Electrolyte Processes in
Industrial Electrosynthesis
Rasmus K. B. Karlsson
Doctoral Thesis, 2015
KTH Royal Institute of Technology
Applied Electrochemistry
Department of Chemical Engineering and Technology
SE-100 44 Stockholm, Sweden
TRITA CHE Report 2015:66
ISSN 1654-1081
ISBN 978-91-7595-781-4
Akademisk avhandling som med tillstånd av KTH i Stockholm framlägges till
offentlig granskning för avläggande av teknisk doktorsexamen 18 december kl.
10.00 i sal F3, KTH, Lindstedtsvägen 26, Stockholm.
Abstract
Heterogeneous electrocatalysis is the usage of solid materials to decrease the amount
of energy needed to produce chemicals using electricity. It is of core importance
for modern life, as it enables production of chemicals, such as chlorine gas and
sodium chlorate, needed for e.g. materials and pharmaceuticals production. Furthermore, as the need to make a transition to usage of renewable energy sources
is growing, the importance for electrocatalysis used for electrolytic production of
clean fuels, such as hydrogen, is rising. In this thesis, work aimed at understanding
and improving electrocatalysts used for these purposes is presented.
A main part of the work has been focused on the selectivity between chlorine gas,
or sodium chlorate formation, and parasitic oxygen evolution. An activation of
anode surface Ti cations by nearby Ru cations is suggested as a reason for the high
chlorine selectivity of the “dimensionally stable anode” (DSA), the standard anode
used in industrial chlorine and sodium chlorate production. Furthermore, theoretical methods have been used to screen for dopants that can be used to improve
the activity and selectivity of DSA, and several promising candidates have been
found. Moreover, the connection between the rate of chlorate formation and the
rate of parasitic oxygen evolution, as well as the possible catalytic effects of electrolyte contaminants on parasitic oxygen evolution in the chlorate process, have
been studied experimentally.
Additionally, the properties of a Co-doped DSA have been studied, and it is found
that the doping makes the electrode more active for hydrogen evolution. Finally,
the hydrogen evolution reaction on both RuO2 and the noble-metal-free MoS2
electrocatalyst material has been studied using a combination of experimental and
theoretically calculated X-ray photoelectron chemical shifts. In this way, insight
into structural changes accompanying hydrogen evolution on these materials is
obtained.
Keywords: Electrocatalysis, metallic oxides, ruthenium dioxide, titanium dioxide,
DSA, doping, selectivity, ab initio modeling, density functional theory
i
Sammanfattning
Heterogen elektrokatalys innebär användningen av fasta material för att minska
energimängden som krävs för produktion av kemikalier med hjälp av elektricitet.
Heterogen elektrokatalys har en central roll i det moderna samhället, eftersom det
möjliggör produktionen av kemikalier såsom klorgas och natriumklorat, som i sin
tur används för produktion av t ex konstruktionsmaterial och läkemedel. Vikten av
användning av elektrokatalys för produktion av förnybara bränslen, såsom vätgas,
växer dessutom i takt med att en övergång till användning av förnybar energi blir
allt nödvändigare. I denna avhandling presenteras arbete som utförts för att förstå
och förbättra sådana elektrokatalysatorer.
En stor del av arbetet har varit fokuserat på selektiviteten mellan klorgas och biprodukten syrgas i klor-alkali och kloratprocesserna. Inom ramen för detta arbete har
teoretisk modellering av det dominerande anodmaterialet i dessa industriella processer, den så kallade “dimensionsstabila anoden” (DSA), använts för att föreslå
en fundamental anledning till att detta material är speciellt klorselektivt. Vi föreslår att klorselektiviteten kan förklaras av en laddningsöverföring från ruteniumkatjoner i materialet till titankatjonerna i anodytan, vilket aktiverar titankatjonerna.
Baserat på en bred studie av ett stort antal andra dopämnen föreslår vi dessutom
vilka dopämnen som är bäst lämpade för produktion av aktiva och klorselektiva
anoder. Med hjälp av experimentella studier föreslår vi dessutom en koppling mellan kloratbildning och oönskad syrgasbildning i kloratprocessen, och vidare har
en bred studie av tänkbara elektrolytföroreningar utförts för att öka förståelsen för
syrgasbildningen i denna process.
Två studier relaterade till elektrokemisk vätgasproduktion har också gjorts. En experimentell studie av Co-dopad DSA har utförts, och detta elektrodmaterial visade
sig vara mer aktivt för vätgasutveckling än en standard-DSA. Vidare har en kombination av experimentell och teoretisk röntgenfotoelektronspektroskopi använts för
att öka förståelsen för strukturella förändringar som sker i RuO2 och i det ädelmetallfria elektrodmaterialet MoS2 under vätgasutveckling.
Nyckelord: Elektrokatalys, metalloxider, ruteniumdioxid, titandioxid, DSA, dopning, selektivitet, ab initio-modellering, täthetsfunktionalteori
ii
Publications
This thesis is based on the following publications:
1. Christine Hummelgård, Rasmus K. B. Karlsson, Joakim Bäckström, Seikh
M. H. Rahman, Ann Cornell, Sten Eriksson, and Håkan Olin. Physical and
electrochemical properties of cobalt doped (Ti, Ru)O2 electrode coatings.
Mater. Sci. Eng., B, 178:1515–1522, 2013.
2. Hernan G. Sanchez Casalongue, Jesse D. Benck, Charlie Tsai, Rasmus K.
B. Karlsson, Sarp Kaya, May Ling Ng, Lars G. M. Pettersson, Frank AbildPedersen, Jens K. Nørskov, Hirohito Ogasawara, Thomas F. Jaramillo, and
Anders Nilsson. Operando characterization of an amorphous molybdenum
sulfide nanoparticle catalyst during the hydrogen evolution reaction. J. Phys.
Chem. C, 118(50):29252–29259, 2014.
3. Rasmus K. B. Karlsson, Heine A. Hansen, Thomas Bligaard, Ann Cornell,
and Lars G. M. Pettersson. Ti atoms in Ru0.3Ti0.7O2 mixed oxides form active and selective sites for electrochemical chlorine evolution. Electrochim.
Acta, 146:733–740, 2014.
4. Staffan Sandin, Rasmus K. B. Karlsson, and Ann Cornell. Catalyzed and
uncatalyzed decomposition of hypochlorite in dilute solutions. Ind. Eng.
Chem. Res., 54(15):3767–3774, 2015.
5. Rasmus K. B. Karlsson and Ann Cornell. Selectivity between oxygen and
chlorine evolution in the chlor-alkali and chlorate processes - a comprehensive review. Submitted to Chem. Rev., 2015.
6. Rasmus K. B. Karlsson, Ann Cornell, and Lars G. M. Pettersson. The electrocatalytic properties of doped TiO2. Electrochim. Acta, 180:514-527,
2015.
7. Rasmus K. B. Karlsson, Ann Cornell, Richard A. Catlow, Alexey A. Sokol,
Scott M. Woodley, and Lars G. M. Pettersson. An improved force field for
structures of mixed RuO2-TiO2 oxides. Manuscript in preparation.
iii
iv
8. Rasmus K. B. Karlsson, Ann Cornell, and Lars G. M. Pettersson. Structural
changes in RuO2 during electrochemical hydrogen evolution. Manuscript in
preparation.
Author contributions
• Paper 1:
– I prepared the electrodes together with Joakim Bäckström, and performed all electrochemical measurements, as well as all data analysis
of those measurements. I also contributed to the overall design and
planning of the study, and wrote parts of the paper.
• Paper 2:
– I performed all XPS calculations and wrote parts of the paper.
• Paper 3:
– I performed all calculations and wrote most of the paper.
• Paper 4:
– I contributed to the initial design and continuous planning of the study.
However, all experiments were performed by Staffan Sandin. I also
wrote parts of the paper.
• Paper 5:
– I wrote most of the review paper.
• Paper 6:
– I performed all calculations and wrote most of the paper.
• Paper 7:
– I performed all calculations and wrote most of the paper.
• Paper 8:
– I performed all calculations and wrote most of the paper.
Contents
1
2
Introduction
1.1 State of the art electrodes . . . . . . . . . . . . . . . . . . . . .
1.1.1 Anodes for chlor-alkali and sodium chlorate production
1.1.2 Cathodes for HER . . . . . . . . . . . . . . . . . . . .
1.2 Selectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Process conditions . . . . . . . . . . . . . . . . . . . .
1.2.2 Electrolyte contamination . . . . . . . . . . . . . . . .
1.2.3 Electrode structure and composition . . . . . . . . . . .
1.3 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
Computational methods
2.1 Theoretical descriptions of matter . . . . . . . . . . . . . . . . .
2.1.1 Force fields . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Quantum mechanics . . . . . . . . . . . . . . . . . . . .
2.1.3 Density functional theory . . . . . . . . . . . . . . . . . .
2.1.4 Approximate self-interaction correction with DFT+U . . .
2.1.5 Quantum mechanics for periodic systems . . . . . . . . .
2.1.6 The frozen-core approximation . . . . . . . . . . . . . . .
2.1.7 Describing the wave function: Plane waves, linear combination of atomic orbitals and real-space grids . . . . . . .
2.2 The theory of X-ray photoelectron spectroscopy . . . . . . . . . .
2.3 Theoretical electrochemistry . . . . . . . . . . . . . . . . . . . .
2.3.1 The computational hydrogen electrode method . . . . . .
2.4 Electrocatalysis from theory . . . . . . . . . . . . . . . . . . . .
2.4.1 Brønsted-Evans-Polanyi relations . . . . . . . . . . . . .
v
1
5
5
7
8
8
9
10
12
15
15
15
18
20
26
26
29
30
31
32
33
37
37
vi
3
CONTENTS
Sabatier analysis . . . . . . . . . . . . . . . . . . . . . .
38
2.4.3
Scaling relations . . . . . . . . . . . . . . . . . . . . . .
39
Experimental methods
43
3.1
Preparation of mixed oxide coatings using spin-coating . . . . . .
43
3.2
Electrochemical measurements . . . . . . . . . . . . . . . . . . .
44
3.2.1
Cyclic voltammetry . . . . . . . . . . . . . . . . . . . . .
44
3.2.2
Galvanostatic polarization curve measurements . . . . . .
44
Combined determination of gas-phase and liquid-phase compositions during hypochlorite decomposition using mass spectrometry
and ion chromatography . . . . . . . . . . . . . . . . . . . . . .
46
3.3
4
2.4.2
Results and discussion
47
4.1
Co-doped Ru-Ti dioxide electrocatalysts . . . . . . . . . . . . . .
47
4.2
Selectivity between Cl2 and O2 . . . . . . . . . . . . . . . . . . .
48
4.2.1
4.2.2
5
The uncatalyzed and catalyzed decomposition of hypochlorite species . . . . . . . . . . . . . . . . . . . . . . . . .
48
Theoretical studies of the connection between oxide composition and ClER and OER activity and selectivity . . . .
53
4.3
Semiclassical modeling of rutile oxides . . . . . . . . . . . . . .
66
4.4
XPS modeled using DFT . . . . . . . . . . . . . . . . . . . . . .
69
4.4.1
The active site for HER on MoS2 . . . . . . . . . . . . .
70
4.4.2
Structural changes in RuO2 during HER . . . . . . . . . .
73
Conclusions and recommendations
79
5.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.2
Recommendations for future work . . . . . . . . . . . . . . . . .
80
6
Bibliography
85
7
Acknowledgments
99
List of
symbols and
abbreviations
E
Energy or electrode potential
e
The charge of an electron
E0
The ground-state energy
EB
Electron binding energy in
XPS
Exc [ρ (r)] The exchange-correlation
functional (the sum of the two
correction terms ∆T [ρ (r)] and
∆Vee [ρ (r)])
F[ρ(r)] A functional of the electronic
density
H
The atomic Hamiltonian
kB
The Boltzmann constant
He
The electronic Hamiltonian
∆Gr
The reaction free energy
hKS
i
The one-electron Kohn-Sham
Hamiltonian
εxc
The per-particle exchangecorrelation energy density of
a system
I
Electrical current
j
Current density, current per
area
me
The mass of the electron
mk
The mass of atomic nucleus k
k
The k-vector
ri
The position of a particle,
(x, y, z)
Ψ
The wave function
N
The total number of electrons
in a system
ψi (r)
A one-particle Kohn-Sham orbital
q∗
Electrochemically active surface area
ρ(r)
The total electronic density at
position r
qi
The charge of an ion i
τ (r)
The gradient in the kinetic energy density
rab
The distance between particles
a and b
S
Entropy
ψ̃n (r) A pseudo wave function
p̃ai
b
T
A projector function in the
PAW method
t
Temperature
Tafel slope
Kinetic energy of electron i
Ti
vii
Time
viii
CONTENTS
U
Electrode potential
Ueq
The reversible potential for a
certain reaction
uik (r) Unit-cell periodicity vector
v(r)
The external potential
VN
The nuclear-nuclear repulsion
energy
Vee
The electron-electron interaction energy, also known as the
Hartree energy
ve f f [ρ (r)] The effective potential (the
sum of electron-nuclei and
classical charge density interactions and the exchangecorrelation potential)
CUS
Coordinatively
unsaturated
site, e.g. on the rutile (110)
surface
CV
Cyclic voltammetry
DFT
Density functional theory
DFTB Density-functional
binding
tight-
DSA
Dimensionally stable anode
DSC
Differential scanning calorimetry
EXX
Evaluation of the exchange energy exactly using the HartreeFock method on Kohn-Sham
orbitals
FF
Force field
GA
Genetic algorithm
vext
The external potential (the
electron-nuclei interaction)
vH
The Hartree potential, the functional derivative of the classical
charge density interaction
GGA
Generalized gradient approximation
GTO
Gaussian-type orbital
vxc
The functional derivative of Exc
HER
Hydrogen evolution reaction
Z
Atomic number
HF
Hartree-Fock
AFM
Atomic force microscopy
IC
Ion chromatography
AOM
Angular overlap model
IR
BEP
Brønsted-Evans-Polanyi
The change in potential across
an electrolytic cell due to electrolyte resistance (I × R)
BZ
Brillouin zone
KS
Kohn-Sham
CC
Coupled-cluster
CHE
Computational hydrogen electrode
CI
Configuration interaction
ClER
Chlorine evolution reaction
LCAO Linear combination of atomic
orbitals
LDA
Local density approximation in
DFT
ML
Monolayer
CONTENTS
ix
MP
Møller-Plesset
MS
Mass spectrometry
OER
Oxygen evolution reaction
ORR
Oxygen reduction reaction
PAW
Projector-augmented wave
PEM
Polymer electrolyte membrane
PGM
Platinum-group metal (Ru, Rh,
Pd, Os, Ir or Pt)
PW
Plane wave
QMC
Quantum Monte Carlo
RPA
Random phase approximation
SE
Schrödinger equation
SEM
Scanning electron microscopy
SHE
Standard hydrogen electrode
TEM
Transmission
croscopy
TST
Transition-state theory
XPS
X-ray photoelectron
troscopy
XRD
X-ray diffraction
ZPE
Zero-point energy
electron
mi-
spec-
x
CONTENTS
Chapter 1
Introduction
Modern life would be impossible without heterogeneous catalysis. Heterogeneous
catalysts are used to reduce the energy required for production of the fertilizers
necessary to supply humanity with food, to make cleaner fuel for transportation,
and to make emissions less harmful. The heterogeneous catalysts that are used
today are applied mainly in thermal processes, where a solid catalyst is used to
decrease reaction barriers for chemical reactions. However, heterogeneous electrocatalysts are also increasingly important. Electrocatalysts have long been used
in chlor-alkali and sodium chlorate production, two inorganic electrosynthesis processes that have long histories[1]. Electrosynthesis is the usage of electrical energy
to produce chemicals. Electrocatalysts are also used in fuel cells, which might be
increasingly important in future transportation. The focus of this thesis is on the
understanding and improvement of electrocatalysts applied for electrosynthesis,
specifically on electrocatalysts used for the anodic processes of chlorine gas and
chlorate production, and for the cathodic process of hydrogen evolution.
Chlorine is a fundamental building block in modern chemical industry[2]. The
world-wide production capacity exceeds 50 million metric tons[3, 4]. Chlorine gas
is used in production of most (more than 85%) pharmaceuticals, and is involved
in the production of a large part of all other modern chemicals and materials[2, 4].
A few important products that require chlorine gas are shown in Table 1.1. Furthermore, chlorine, and its oxide sodium chlorate, have long been used as oxidants
for e.g. bleaching fabrics or pulp and paper. Today, bleaching of pulp and paper
products is usually performed using chlorine dioxide (ClO2) which is produced using sodium chlorate[5]. This application drives the production of sodium chlorate,
which exceeds three million metric tons[6].
Both chlorine gas and sodium chlorate are produced through electrolysis of sodium
chloride solutions. A main difference between the processes is the usage of some
sort of divider, in chlorine production, between the anode, where chlorides are
1
2
CHAPTER 1. INTRODUCTION
Table 1.1: Some important uses for chlorine gas[4].
Product
Polyvinyl chloride (PVC) and
several other plastics
TiO2
Highly pure Si for solar cells and electronics
Highly pure HCl
Pharmaceuticals
Chlorine use
Raw material
In precursor TiClx
In precursor SiCl4
As reactant
Final product, reactants or intermediates
oxidized to form chlorine gas according to
2Cl− → Cl2 + 2e−
(1.1)
and the cathode, where hydrogen gas is evolved according to
2H2 O + 2e− → H2 + 2OH− .
(1.2)
The anode used in both processes is the so-called Dimensionally Stable Anode
(DSA), with is Ti coated with a mixed rutile oxide of ca 30% RuO2 (possibly
together with other dopants) and 70% TiO2. If a divider is used, the pH at the
anode side is kept low and chlorine gas is liberated. Without a divider the pH in
the electrolyte can reach close to neutral values, allowing sodium chlorate to form
in the following reactions, starting with hydrolysis of formed chlorine
Cl2 + H2 O H+ + HOCl + Cl− ,
(1.3)
HOCl + H2 O OCl− + H3 O+ ,
(1.4)
followed by chemical formation of chlorate,
+
−
2HOCl + OCl− → ClO−
3 + 2H + 2Cl .
(1.5)
The detailed mechanism of reaction 1.5 has still not been determined[1]. To maximize the chemical formation of sodium chlorate, chlorate plants have low-volume
electrolysis cells connected to larger reactors where the chemical conversion can
be maximized. In the divided cell, highly concentrated sodium hydroxide (alkali)
can be extracted from the cathode side, explaining why this process is known as
the chlor-alkali process. The overall reaction occurring in the divided (chlor-alkali)
cell is
2NaCl + 2H2 O → Cl2 + H2 + 2NaOH,
(1.6)
3
Table 1.2: Typical conditions for chlor-alkali production in membrane processes
[3, 4, 8].
Cell voltage / V
Current density / kA m−2
Temperature / ◦ C
NaCl concentration in the anolyte / g dm−3
Anolyte pH
NaOH concentration in the catholyte / wt-%
2.4-2.7
1.5-7
90
200
2-4
32
Table 1.3: Typical operating conditions in sodium chlorate production [5, 9, 10].
Cell voltage / V
Current density / kA m−2
Temperature / ◦ C
NaCl concentration / g dm−3
NaClO3 concentration / g dm−3
NaOCl concentration / g dm−3
Na2Cr2O7 concentration / g dm−3
Electrolyte pH
2.9-3.7
1.5-4
65-90
70-150
450-650
1-5
1-6
5.5-7
whereas in undivided (sodium chlorate) cells the overall reaction is
NaCl + 3 H2O
NaClO3 + 3 H2
(1.7)
The type of “divider” used separates the different chlor-alkali processes into three
subtypes: the mercury cell process, the diaphragm process and the membrane process. Only the last type is currently acceptable from the environmental and economic points of view[7], and the other processes are to be discontinued in the
European Union. Due to the high yearly production rates of both chemicals, with
chlor-alkali production alone consuming an estimated 150 TWh of electricity per
year[4], it is clear that high selectivity and energy efficiency in both processes
is required.[1] Typical operating conditions for the membrane chlor-alkali and
sodium chlorate processes are found in Tables 1.2 and 1.3, respectively.[3–5, 8–10]
A challenge in both processes is parasitic oxygen evolution. The possible pathways
for this oxygen evolution include direct anodic water oxidation, competing with
reaction 1.1,
2H2 O → O2 + 4H+ + 4e− ,
(1.8)
direct anodic chlorate formation
−
+
−
6ClO− + 3H2 O → 2ClO−
3 + 4Cl + 6H + 1.5O2 + 6e ,
(1.9)
4
CHAPTER 1. INTRODUCTION
as well as decomposition of hypochlorites (HOCl or OCl–) according to
2HOCl → 2HCl + O2 .
(1.10)
The decomposition of hypochlorites might be due both to bulk electrolyte and electrode surface reactions. The reduction in current efficiency due to these parasitic
reactions ranges from 1-5%. While measures such as usage of selective electrodes
(see the following section) and anolyte acidification can reduce the oxygen side
reaction to enable a current efficiency of close to 99% in the chlor-alkali process,
this is not possible in sodium chlorate production, where 5% losses in current
efficiency are common. Research also indicates that the oxygen evolution sidereaction not only lowers the current efficiency of the processes, but that it is also
directly connected to the rate of deactivation of the DSA.[1]
The cathode reaction in both chlor-alkali and sodium chlorate production is hydrogen evolution, reaction 1.2. The interest in this reaction has been rising in recent
years, as it is a way of producing hydrogen for use as fuel (or in production of other
fuels, e.g. by Fischer-Tropsch synthesis) in a renewable fashion[11]. In both industrial electrosynthesis of chlor-alkali and sodium chlorate, and in water electrolysis
for production of hydrogen, the hydrogen evolution takes place in a strongly alkaline solution, which places limitations on the cathode material. Although early
industrial electrosynthetic hydrogen production was carried out already in 1927,
electrosynthetic hydrogen makes up only a very small part (about 5%) of the total
hydrogen produced today. Production of hydrogen based on fossil hydrocarbons
is the dominant production method, as it requires less energy. The theoretical energy consumption required for production using methane is 41 MJ/kmol H2 while
the theoretical energy consumption for electrolytic hydrogen is about six times
higher at 242 MJ/kmol H2[12]. Today, industrial electrolytic cells for chlor-alkali
and water electrolysis often use steel; Ni or Ni alloys; or RuO2 deposited on Ni
as hydrogen evolution reaction (HER) cathodes[3, 5, 12, 13]. However, due to
the more demanding conditions, mainly steel or Ti cathodes are used in chlorate
production[5]. Ni-based cathodes are not acceptable in chlorate production as Ni is
a well-known catalyst for decomposition of HOCl[1]. As interest in electrochemical hydrogen production has risen, so has the research into “activated cathodes”,
which are cathodes with an activity exceeding that of Ni[13] .
A primary goal of the present thesis has been to further the understanding for how
the electrode composition determines the selectivity, activity and stability in these
processes. To this end, the first comprehensive literature review of the selectivity
issue in industrial chlor-alkali and sodium chlorate production has been carried
out as a part of the present thesis project[1]. As the review discusses the present
knowledge regarding the effect of anode structure and composition, electrolyte
contamination, and process parameters on the selectivity in great detail, the reader
is advised to read the Discussion section of that review for a thorough discussion
of these aspects. A brief overview will still be provided in Section 1.2 of this
1.1. STATE OF THE ART ELECTRODES
5
chapter. Based on the findings in the review, the selectivity issue in the chlorate
and chlor-alkali processes has been investigated in a broader sense, by performing also studies of how electrolyte contamination affects the decomposition of
hypochlorite[14]. Still, a main focus has been put on the influence of electrode
structure and composition[15–20], and I will therefore summarize the present status of anodes in chlor-alkali and sodium chlorate production and of activated cathodes for HER in Section 1.1. I have made use of first principles modeling to gain
a further understanding of the connection between composition, activity and selectivity, and to clarify the connection between experimental X-ray photoelectron
spectroscopic (XPS) results and the actual electrode structure. As the application
of such theoretical methods to problems in heterogeneous (electro)catalysis and
materials science is recent, Chapter 2 contains a thorough overview of these methods, and their application in the study of solid materials, (electro)catalysis and
X-ray photoelectron spectroscopy. Experimental methods employed in this thesis
are reviewed in Chapter 3. The results are presented and discussed in Chapter 4,
and some conclusions and an outlook for future work is presented in Chapter 5.
1.1
The state of the art of electrodes used for the inorganic electrosynthesis of chlorine, sodium chlorate and hydrogen
1.1.1
Anodes for chlor-alkali and sodium chlorate production
As has already been mentioned, DSA are used for the oxidation of Cl– in both
processes. Dimensionally stable anodes were invented by Henri Beer in the 19501960s[21]. Previously, graphite anodes had been applied, but these electrodes were
consumed during use, requiring frequent adjustment of the anode position. Beer
realized that since titanium forms a highly corrosion-resistant TiO2 oxide layer
when used as an anode, it could be used as a stable anode support material in
chlor-alkali electrolysis[22]. The current DSA coating developed in several steps.
First, Beer suggested that deposition of noble metals onto titanium should serve to
activate the anode. If a part of the active coating would be removed from the surface, the anode would not be destroyed since corrosion-resistant TiO2 would form.
Beer tried depositing noble metals, such as platinum, on the surface of titanium,
to activate the electrode, and patented the procedure[23]. However, the stability of
this coating was found to be insufficient during industrial testing[21, 22, 24]. As a
next step, the deposition method was changed from electrodeposition to a method
consisting of painting a noble metal salt solution onto titanium. The painted titanium was then heated in air at temperatures between 400 − 500 ◦ C, converting
the salt into what was thought to be metallic form. Different coating solutions,
6
CHAPTER 1. INTRODUCTION
containing e.g. platinum, ruthenium, iridium and rhodium were in this way applied on titanium. This change took place in the early 1960s. One coating which
was frequently used was a mixture of 70% platinum and 30% iridium. This metal
combination had actually been known to be stable in these processes already since
the early 1900s[25]. Except for coatings containing platinum, the heating procedure that was used to prepare such coated titanium anodes actually converted the
noble metal into metal oxide form. Electrodes with this coating were preferable
to the old graphite electrodes and to platinum coatings, but the anodes were still
not stable enough in the mercury cell process. Furthermore, the prices of iridium and platinum were also high in comparison with other noble metals, such as
ruthenium[21, 26]. By the middle of the 1960s, Beer proposed that a coating of
ruthenium oxide could be preferable to the Pt/Ir coating. It was eventually found
that mixing a platinum-group metal (PGM) oxide with a film-forming metal oxide
could create a mixed oxide with improved stability. The mixture that was most
promising was that of ruthenium oxide and titanium oxide[21].
The first metal oxide patent, for a coating of 50% ruthenium oxide and 50% titanium oxide was filed by Beer in 1965 (“Beer 1”)[27]. Anodes coated with this
mixture were claimed to be more durable in mercury cell processes and to have a
stronger adhesion of the coating layer to the titanium. Two years later, a second
patent was filed (“Beer 2”)[28]. It introduced that the coating should be made up
of 30% RuO2 and 70% TiO2 . This combination had several advantageous properties. It gave a high activity and was well suited for use in chlor-alkali and chlorate
production. The usage of 70% titanium dioxide in the coating reduced the price
of the coating. Furthermore, the mixed oxide of ruthenium oxide and titanium oxide showed a high stability, enabling operation at higher temperatures and current
densities compared with those possible when using graphite electrodes. Titanium
oxide has a low electric conductivity but, in the mixed oxide, RuO2 increases the
electric conductivity of the coating. Another important advantage, especially of
the Beer 2 combination, is that the selectivity for chlorine evolution over oxygen
formation is increased when the ruthenium amount in the coating is decreased[29–
35]. Combined with the self-healing properties of the supporting titanium material,
these properties made the “Beer 2” DSA a great step forward in anode technology
for the chlor-alkali and chlorate industries.[1, 21, 22, 26, 36]
The “Beer 2” DSA is considered the standard industrial DSA composition. However, electrode manufacturers usually make further modifications of the electrode,
although the details of these modifications are usually kept secret. For example, coatings containing a reduced amount of ruthenium oxide replaced with tin
oxide have been developed to improve the stability of the coating further. In
other coatings, some ruthenium is replaced with iridium oxide, also to improve
stability.[1, 26]
Nevertheless, the standard 30% RuO2 - 70% TiO2 coating has been studied in detail, and it is now known that it is made up of small electrode particles of rutile
1.1. STATE OF THE ART ELECTRODES
7
structure, with sizes ranging from 10 nm to 30 nm[37–39]. The particles likely
consist of a solid solution of RuO2 and TiO2, meaning that Ti and Ru cations are
mixed on the atomic scale[40–42]. On the micrometer scale, the coating has a
structure similar to that of dry mud, with large cracks separating more dense areas
(the “cracked mud” or “mud crack” structure)[38, 39, 43]. These cracks have been
found to facilitate chlorine bubble detachment, resulting in improved activity[44].
The nanoscale structure of the particles, specifically regarding whether any preferred arrangement of Ru and Ti cations exists in the surface layer of these particles, is still unknown.
1.1.2
Cathodes for HER
Today, steel or Ti cathodes are used for HER in chlorate cells[5], while Ni or
RuO2-coated cathodes are used in chlor-alkali cells[3, 12]. Ni cathodes are also
used in industrially-sized water electrolysis plants. In both chlor-alkali and water
electrolysis, the cathode is exposed to highly concentrated caustic solutions at a
high temperature, as this accelerates the kinetics of the process and maximizes the
electrical conductivity of the electrolyte[12, 13]. Under such conditions, Ni combines an acceptable activity with a sufficient stability. Nevertheless, Ni electrodes
exhibit overpotentials of several 100 mV. This overpotential can be reduced significantly by measures that increase the surface area of the electrode (e.g. in so-called
Raney Ni). This does not change the per-site activity of the electrode, but improves
the activity by exposing a larger number of active sites[13]. Furthermore, RuO2
deposited on Ni combines acceptable stability with a lower overpotential than that
of smooth Ni, making it an alternative to Raney Ni[3].
The metal with the highest activity for HER is Pt[45, 46], but its limited stability in
alkaline conditions and high cost prevents is use in industrial cells. However, an alternative process for water electrolysis is based on cells using polymer electrolyte
membranes (PEM) similar to those used in fuel cells[12, 47]. In this case, the
cathode material can be graphite, usually with Pt deposited on the graphite. Here
it is also possible to apply other activated cathode materials, e.g. sulphides, oxides
or alloys of several elements (see Trasatti [13]), that have insufficient stability in
alkaline solutions. MoS2 deposited on graphite is one such material. While it has
been known that a variety of sulphides, including MoS2, are active for HER[13],
an increased interest in this material probably originated in the theoretical study of
Hinnemann et al. [48], which indicated that edge sites in MoS2 should exhibit a
similar electrocatalytic activity as Pt. This might enable electrolytic hydrogen production at lower overpotentials than those of Ni without use of noble metals. The
correlation between the activity of MoS2 and the edge length was experimentally
validated by Jaramillo et al. [49]. The details of the mechanism of HER on this
material is discussed in one part of the present thesis[20]. However, MoS2 has of
yet not been studied in a PEM electrolyzer, and electrolysis at industrially relevant
8
CHAPTER 1. INTRODUCTION
current densities (several kA m−2 ) is yet to be realized[47].
As mentioned above, RuO2 deposited on Ni is used in both alkaline water electrolysis and chlor-alkali production[3, 13]. However, RuO2 is not thermodynamically
stable versus reduction under normal HER conditions. This has been associated
with the decomposition of the cathode, especially under shutdown conditions[50].
Indeed, Trasatti [13] points out that degradation during shutdown is actually a main
factor limiting the stability of cathodes. Still, the degree to which RuO2 is reduced
is debated in literature, with some even suggesting a bulk conversion of the oxide
into metallic form[51]. However, other studies have come to the conclusion that
the reduction that occurs is due to incorporation of H into the coating, and that
formation of metallic Ru does not occur[52, 53]. This topic has been studied in
the present thesis, by comparing calculated XPS shifts with experimental ones of
Näslund et al. [51].
1.2
The present understanding of the selectivity in
chlor-alkali and sodium chlorate production
The factors that control the selectivity between parasitic oxygen evolution and the
desired chlorine gas, or sodium chlorate, production, can essentially be divided
into three groups: effects resulting from process parameters (including temperature and concentrations of the main intermediates and reactants), effects resulting
from contaminants in the electrolyte (such as those possibly released from the
electrodes or construction materials) and finally effects of the electrode structure
and composition. In this section, only the more well-understood effects will be
presented, and a detailed discussion is provided in [1].
1.2.1
Process conditions
Overall, the effects of process conditions are likely the most well-studied, in comparison with electrolyte impurity and electrode composition effects. The main
anodic reaction is chloride oxidation, reaction 1.1. Oxygen evolution by water
oxidation, reaction 1.8, or hypochlorite oxidation under chlorate conditions, are
likely important anodic side reactions. Electrochemical chlorate and perchlorate
formation will be disregarded for the moment, as these can be controlled by keeping the hypochlorite or chlorate concentration at the electrode low[9].
Water oxidation according to reaction 1.8 is clearly pH dependent, and this reaction
can be controlled by keeping the pH low. Furthermore, HOCl formation from
Cl2 is prevented if the pH is kept low. This is the basis for anolyte acidification,
which is practiced in chlor-alkali membrane cells[54]. Still, oxygen does form
in industrial cells, although there is some dispute regarding which reaction (or
1.2. SELECTIVITY
9
reactions) is the main source. On the other hand, for industrial chlorate production
keeping the pH low to control the rate of oxygen formation is not possible as an
optimal pH value between pH 6-7 exists for the homogeneous chlorate formation
reaction 1.5[30, 55–58].
Furthermore, the chlorine evolution reaction is concentration dependent. Increasing the chloride concentration is one of the main ways (together with pH control,
and as will be discussed, current density increase) that the selectivity for the desired products can be maximized[31, 35, 57–65]. This is due partly to improved
thermodynamics, but perhaps mainly due to increased mass transfer rates of chloride to the surface as well as competitive adsorption of chlorides on the electrode
surface[62, 66].
For modern DSA, the Tafel slope is about 40 mV/decade for chlorine evolution[67],
while the Tafel slope for OER has been found to be about 60 mV/dec in acidic solutions and ca 120 mV/dec in neutral solutions[68]. With increasing current density, the overpotential for OER should thus increase more rapidly than the overpotential for ClER. A high current density should result in a high selectivity for
chlorine evolution, as is indeed noted in practice[35, 43, 55, 57, 61, 62, 66, 69–72].
However, the picture seems to be more complicated in chlorate production, since
it has been found that the optimal chlorate efficiency is found close to the so-called
critical potential, close to E = 1.4 V vs SHE[73].
The effect of temperature is complex. Increasing the cell temperature increases the
rate of both main and parasitic reactions, and it has been connected to increased
oxygen selectivity under chlorate conditions[57, 69, 71, 72].
When it comes to concentrations of intermediates, increased hypochlorite concentrations have been found to yield increased oxygen selectivity[57, 60, 64, 69,
71, 74–77], likely due to decomposition of hypochlorite, while increasing concentrations of chlorate have been found to yield decreased oxygen selectivity[10,
57, 58, 69]. Both of these species should be avoided in chlor-alkali production,
and in chlorate production the hypochlorite concentration is kept relatively low
(at 1 g/dm3 to 5 g/dm3 [5, 9, 10]) while the chlorate concentration is kept high
(450 g/dm3 to 650 g/dm3 [5, 9, 10]). However, the concentration of chlorate is
kept high primarily to allow crystallization of NaClO3.
1.2.2
Electrolyte contamination
The effects of electrolyte contamination, e.g. by metals, is less well-studied. Furthermore, most contaminants have been studied either in pure electrolytes (e.g.
including only hypochlorite species and chloride) or otherwise under conditions
(e.g. temperature and concentrations) that depart from those used industrially.
The main effects that have been studied revolve around competitive adsorption on
the electrode, with phosphates[58, 78, 79], sulphates[79, 80] and nitrates[78] all
10
CHAPTER 1. INTRODUCTION
having been found to result in increased rates of parasitic oxygen evolution and
effects on the decomposition rate of hypochlorite species. F– has been found to
result in decreased selectivity for oxygen[58, 78, 81], probably due to competitive
adsorption. There is likely no reason to consider F– addition to industrial cells (in
fact, it has been claimed that F– might lead to electrode deactivation[82]), since the
same effect most probably is achieved by increasing the chloride concentration. As
hypochlorite concentrations should be kept low in chlor-alkali production, oxygen
evolution due to decomposition of hypochlorite is a less severe problem in that
process. The situation is more challenging in chlorate production. Co[83–91],
Cu[83, 84, 90, 92, 93], Ni[83, 87–90, 92] and Ir[88, 94, 95] are all metals that
have been found to catalyze hypochlorite decomposition, and might thus lead to
oxygen evolution in both processes.
1.2.3
Electrode structure and composition
The effects of electrode structure and composition can essentially be divided into
two parts, as pointed out by Trasatti [36]: effects due to changes in electrode active
surface area (the exposed number of active sites) and actual catalytic effects. The
first aspect is most likely the one that is most well understood. Increasing the active
surface area, though necessary for achieving a high electrode activity, is connected
with an increased selectivity for parasitic oxygen formation[43]. Similar effects
have been noted when decreasing coating particle sizes[96]. The effect is fundamentally the same as that of increased current density, as an increased active
surface area results in a decreased local (per-site) current density. It is likely that
control of the surface area is one aspect that electrode manufacturers make use of
already today. Doping a pure oxide often results in increased surface area[97, 98],
but not necessarily in clear changes in actual catalytic activity (such as e.g. changes
in Tafel slope[13]).
Nevertheless, exceptions do exist. One such case is DSA itself, where a reduction
in the Ru content of the mixed oxide down to about 20 to 30 mol − % has been
found to result in increased selectivity for chlorine evolution[29–35]. While this is
likely partially an effect of the surface area also decreasing when reducing the Ru
content[97], measurements of exchange current densities and Tafel slopes for OER
on RuO2-TiO2 electrodes indicate that the Tafel slope increases and the exchange
current density decreases as the Ru content is lowered, even at temperatures of
80 ◦C[99], both in alkaline[99, 100] and acidic[32, 39, 100] solutions. While some
studies note increased Tafel slopes only below 30% Ru[99, 100], others find a more
gradual increase in Tafel slopes when reducing the Ru content in the range 80 %
to 20 % Ru[39]. Changes in Tafel slope reflect true electronic effects on catalysis, rather than surface area effects, while the exchange current density might be
affected by changes in true surface area[13]. In the case of ClER, Tafel slopes
have generally been found to be constant down to 20% Ru[100], in some cases
1.2. SELECTIVITY
11
even to less than 20% Ru at high temperature (90 ◦C)[101]. This could indicate
that the selectivity of the Ru-Ti mixed oxide, which can be likened to Ru-doped
TiO2 at lower Ru concentrations, is altered due to an electronic effect, related to
charge transfer between the Ru and Ti components. XPS measurements indeed
indicate that charge transfer occurs[102, 103]. Furthermore, an increase in surface
area does not seem to be enough by itself to account for the maintained activity of
ClER (and OER) as the Ru percentage in the coating is reduced, as e.g. Burrows
et al. [100] found a difference in overpotential between 80% and 20% Ru of 5 mV,
while the change in overpotential according to Tafel kinetics for such a decrease in
active site numbers (if Ru is the only active site) should be more than 20 mV (see
Karlsson et al. [16]). Nevertheless, as there is a lack of experimental methods to
fully account for changes in surface area when determining the activity of electrocatalysts, it is challenging to fully decouple true catalytic effects from effects of
surface area[104].
The same problem exists in studies of other mixed oxide combinations. Several
studies exist of selectivities of ternary Ru-Ti oxides (i.e. where an additional
dopant has been added to the coating), or of RuO2 combined with other transition metals, and there are many more studies that do not examine the selectivity
specifically. For example, doping RuO2 with SnO2 has been found to result in an
improved chlorine evolution selectivity[43, 77, 105, 106], and this has again been
coupled with changes in Tafel slopes[43] indicating an electronic effect.
During the past decade, a deeper understanding of the connection between electronic effects and selectivity has been gained from studies using density functional
theory. When it comes to the electrocatalysis of oxygen and chlorine evolution on
oxide surfaces, the studies of Rossmeisl et al. [107] (focusing on the OER) and
Hansen et al. [108] (focusing on ClER and OER) are possibly the most important
theoretical contributions since the early 1970’s. The workers found likely reaction mechanisms for both ClER and OER on rutile oxide surfaces through first
principles calculations. Furthermore, scaling relations (linear relations between
adsorption energies of different adsorbates, e.g. intermediates in reactions[109],
see Section 2.4.3) were identified that enabled the activity for both ClER and OER
to be coupled to one single descriptor, the adsorption energy ∆E(Oc ) of O on the
rutile coordinatively unsaturated site (CUS). In this way, the relative activities of
different oxides could be accounted for in a single theoretical framework. The
present thesis will use these results as a basis for a consideration of the selectivity
of mixed oxides. The results of Rossmeisl et al. [107] and Hansen et al. [108] were
later reconsidered by Exner et al.[110–113], essentially confirming their results. A
question that still remains regards the effect of solvation[107, 110, 114], a factor
that is challenging to account for in first principles studies. While both Rossmeisl
et al. [107] and Siahrostami and Vojvodic [114] found, by modeling water explicitly, that the hydration itself should have a minor effect on activity trends for OER,
Exner et al. [110] found large effects when using an implicit water model.
12
CHAPTER 1. INTRODUCTION
An additional complication when it comes to the selectivity between oxygen evolution and chlorine evolution in chlor-alkali production is that electrodes with decreased oxygen selectivity have been found to instead yield increased chlorate production rates[8, 31, 56, 73, 115]. Whether this is purely a pH effect (a decreased
rate of reaction 1.8 resulting in a higher pH at the anode) or due to direct formation
of chlorate at the anode is so far not known.
The electrocatalytic properties of electrodes have mostly been studied using electrochemical measurements of OER and ClER activities, sometimes with direct
measurements of selectivity. However, the possibility that electrodes might catalyze chemical decomposition of hypochlorite in the chlorate process has not been
studied in much detail. Kuhn and Mortimer [116] suggested this possibility when
using unpolarized RuO2 electrodes, while Kokoulina and Bunakova [63] and Kotowski and Busse [117] did not find that unpolarized RuO2-TiO2 electrodes catalyzed the decomposition.
To summarize, while the connection between the active surface area of the electrode and the selectivity for e.g. ClER is well understood, the connection between
the electronic structure of the electrode material and the activity, and selectivity, is
lacking.
1.3
Aims
As has been made clear in the preceding sections, there are still significant gaps
in the understanding of both oxygen-forming reactions in chlor-alkali and sodium
chlorate production, and of the hydrogen evolution reaction on cathodes for water
electrolysis. The aim of this thesis has been to further the understanding of these
technical aspects of electrocatalysis for industrial electrosynthesis. The stability of
electrodes has also been examined, primarily by XPS simulation to strengthen the
connection between electrode structure and deactivation processes. In this work,
both experimental and theoretical methods have been applied. Some key issues
that have been examined include:
• Why do mixed RuO2-TiO2 coatings exhibit increased chlorine evolution selectivity in comparison with pure RuO2?
• How are the electrochemical properties of DSAs and of TiO2 altered by
doping with other elements?
• Which materials catalyze decomposition of hypochlorite?
• Which is the mechanism of HER on Mo sulphide cathodes?
• What structural changes occur during HER on RuO2?
1.3. AIMS
13
• Can simple force fields based on Buckingham potentials be used to simulate
the structure and energetics of TiO2-RuO2 mixed oxides?
14
CHAPTER 1. INTRODUCTION
Chapter 2
Computational methods
2.1
Theoretical descriptions of matter
In the present thesis, both first principles (using density functional theory) and
semiclassical modeling (using force fields) of atoms, molecules and matter have
been carried out. In the following sections, both types of modeling will be described in some detail, with a focus specifically on the fundamentals of first principles modeling using density functional theory. I thereby set the stage for the
modeling results that are presented in the current thesis, which include an application of force fields to study mixed rutile RuO2-TiO2, and of density functional
theory to further the understanding for the HER on RuO2 and MoS2 and for the
electrocatalytic properties of doped TiO2.
2.1.1
Force fields
2.1.1.1
Fundamentals of force fields
Force fields (FFs) based on interatomic potentials have one main advantage over
first-principles methods in their significantly lower computational cost. They therefore see a main application in the description of e.g. proteins and other macromolecules. However, there are also several force fields for the description of inorganic materials, including oxides. In the present thesis, the FFs that have been
used are based on the combination of Coulombic interactions and Buckingham
interatomic potentials with a shell model on the oxygen anions. The Coulombic
interactions describe the electrostatic contribution to the interaction, and are given
by
15
16
CHAPTER 2. COMPUTATIONAL METHODS
UiCoulomb
=
j
qi q j
,
4πε0 ri j
(2.1)
where the i and j indices correspond to two different ions, qi is the charge of ion
i, ε0 is the vacuum permittivity and ri j is the distance between ions i and j. The
Coulombic interactions are pairwise interactions between all ions in the system,
and to evaluate this efficiently in three dimensions a special Ewald summation is
applied[118]. The dipolar polarizability of the O 2p electrons is described by a
shell model, in which each oxygen is modeled as being composed of a positive
core (with, in this case, a charge of +0.513) and a negative shell (with a charge
of −2.513 in the present work). The distance between the shell and the core is
determined by an additional potential, which is simply a classical spring with a
certain spring constant (k = 20.53 eV/2 ). An additional short-ranged potential is
then added to describe bonding between ions. Several forms of this interaction
can be used, but in the present thesis the Buckingham interatomic potential, which
describes the interaction between O shells and Ru and Ti cores or other O shells,
has been used. It has the following form
ri j
C6
Buckingham
− 6,
(2.2)
Ui j
= A exp −
ρ
ri j
with three adjustable parameters A, ρ and C6 . The first term in the expression can
be seen as representing the repulsion between electronic densities at close range,
with A giving the strength of the repulsion and ρ how far it reaches, while the
second term is the attractive component (with C6 giving the strength of the attraction). The Buckingham potentials for O-O interactions and oxygen shell model
parameters in the present thesis are the same as in the work of Bush et al. [119].
A part of the present thesis has been focused on finding improved Buckingham
parameters for Ru-O and Ti-O interactions. Buckingham potentials and Coulomb
interactions are radial, and thus best describe spherical ions. This is not a suitable description for transition metals with incompletely filled d-shell, in which the
non-spherical d-orbitals play an important role. A recent improvement is the introduction of an additional interatomic potential based on the angular overlap model
(AOM)[120, 121]. This model, which is related to ligand field theory, allows for
the simulation of the angular distribution of d orbitals. It adds two new parameters,
ALF and ρLF , which determine the energies of the d orbitals (εd ) according to
εd = ALF S2 ,
(2.3)
and an additional parameter for the spin state (high or low spin) of the system.
In equation 2.3, S is the overlap between a transition metal d orbital and an O2−
ligand. Thus, ALF is a scaling coefficient that describes the relationship between
2.1. THEORETICAL DESCRIPTIONS OF MATTER
17
the overlap and the d orbital energy. The ρLF parameter is used to describe the
exponential distance dependence of the overlap
S = exp (−r/ρLF ) .
(2.4)
The AOM is described in more detail in Woodley et al. [121].
2.1.1.2
Using genetic algorithms to find the globally optimal FF
While FFs can be used to successfully model structures and properties of materials, a chief difficulty exists in the parametrization of the FF. Typically, this process
requires chemical insight and several attempts before yielding values that achieve
the desired accuracy. This is the case even when using partially automatic methods, such as those where the potential parameters are varied to minimize the force
on, e.g., the experimental structure that is used as the “target structure” for the
parameters. While such deterministic methods facilitate the fitting of interatomic
potentials, they still require that the initial parameters are well chosen. Furthermore, if a successful FF is not found, with this method it is not possible to make
certain that the failure is because there simply is no set of FF parameters that would
be successful, or if it is due to a failure on the part of the investigator in exploring
the complete parameter space.
In such a situation, to make sure that the complete parameter space has been
searched, a global optimization method is needed. One such optimization method
is optimization by use of a suitably designed genetic algorithm (GA)[122]. Genetic
algorithms are similar to the Monte Carlo method[123] or the method of simulated
annealing[124, 125], both of which invoke thermodynamic arguments for exploring a parameter space in a way that is more efficient than a random search. As the
name implies, GAs are inspired by natural evolution. The GA works in the following way. First, a number of chromosomes, e.g. in the present case an array of FF
parameters, is generated randomly. These chromosomes are then evaluated using
a fitness function, which ranks the quality of all chromosomes, e.g. based on their
ability to model a certain structure accurately. The fittest chromosomes are carried
over to the next generation. However, a crossover operator is usually also applied.
This crossover operator combines the chromosomes with the highest fitnesses in
the population to generate a new chromosome that possibly can exceed the quality
of the “parent” chromosomes. Furthermore, a mutation operator is usually also
applied, which e.g. changes one value in the chromosome of a certain individual
randomly. This process is then repeated for a certain number of generations, and
the chromosome with the highest fitness after the conclusion of this process is then
taken as the globally optimal solution.
Some challenges in the application of this method are immediately obvious. First
off, there is no guarantee that the search has converged to the actual global opti-
18
CHAPTER 2. COMPUTATIONAL METHODS
mum in a certain search space. There are ways to check whether the solution has
converged close to the optimum, e.g. by repeating a certain search for more generations, or by repeating it with another random seed to see if the best chromosome
is similar to the one previously found. The second problem is that there are no
good rules for how to choose the values for the parameters in the GA, e.g. what
mutation rate (how many percent of the individual chromosomes in a certain generation should be mutated) or what crossover rate to use (i.e. how many of the best
individuals should be used for crossover to generate new chromosomes). Which
parameters to use is problem-dependent. Studies exist of which GA parameters are
optimal for a certain class of problems[126, 127], but when being applied to a new
problem, several attempts with different GA parameter values are necessary. In
fact, there are even methods which attempt to optimize the GA parameters while
the GA is being used to solve a main problem, which indicates the difficulty in
deciding parameters[128]. Finally, as in any global optimization method, for the
solution to be found quickly the evaluation of the fitness function must require a
short time. This is the case when applying GAs for optimization of FF parameters,
as one structural relaxation using a force field based on Coulomb interactions and
two-body potentials such as those of the Buckingham type usually requires just a
few seconds on a single modern processor core.
Despite these problems, GAs have been used with success for many optimization
problems, including the determination of low energy crystal structures[129] and
fitting of parameters for force fields[130]. It is used in the present thesis to attempt
to find a globally optimal force field, to describe both structural and energetic
properties of DSA, based on Buckingham potentials and the AOM model.
2.1.2
Quantum mechanics
After the discussion of semi-classical modeling of molecules and matter in the last
section, we will now continue the discussion by focusing on quantum mechanics.
This theory, which was developed to describe the behavior of small particles, such
as atoms, electrons and photons, had to also account for the wave-like characteristics of matter. This was achieved through the description of a particle using a wave
function, Ψ. The wave function contains all information that can be known about a
system, such as that of electrons around atoms. For example, for the one-electron
wave function, the square (or complex conjugate |Ψ|2 = Ψ∗ Ψ, for complex wave
functions) of the wave function at a certain point is the probability of finding the
electron around that point, and the integral of the square of the wave function over
all space is unity (the probability of finding the electron somewhere in space is
1). If matter is described as a wave, then there should be a similar kind of wave
equation for matter as for waves in classical physics. The equation that was found
to describe the properties of matter is the Schrödinger equation (SE),
2.1. THEORETICAL DESCRIPTIONS OF MATTER
HΨ = EΨ,
19
(2.5)
wherein the Hamiltonian (H) operates on the wave function to provide the energy E of the wave function. H mainly appears in two forms. The first one,
yielding the time-independent Schrödinger equation, describes the properties of
systems with no external electric or magnetic fields, and systems for which relativistic effects (which become important for the heavier elements) can be disregarded . The second one, which is capable of describing such systems, is the timedependent Schrödinger equation. In the present thesis, only the time-independent
Schrödinger equation is considered.[131, 132]
The form of the Hamiltonian for the time-independent SE is the following[132]:
H = −∑
i
h̄2 2
e2 Zk
e2 Zk Zl
e2
h̄2 2
∇i − ∑
∇k − ∑ ∑
+∑
+∑
,
2me
i k rik
i< j ri j
k 2mk
k<l rkl
(2.6)
wherein indices i and j correspond to electrons, indices k and l correspond to nuclei, h̄ is h/2π and h is Planck’s constant, me is the mass of the electron, mk is
the mass of nucleus k, e is the charge of an electron, Z is an atomic number and
rab is the distance between two particles with indices a and b. The Hamiltonian
can thus be understood as being composed of five components: the kinetic energy
of the electrons, the kinetic energy of the nuclei, the electron-nuclei attraction, the
electron-electron repulsion and the nuclei-nuclei repulsion. The Schrödinger equation (or indeed, any complete equation for the interaction between three or more
particles[133]) is analytically unsolvable for more than two interacting particles.
Even a simple system such as a single water molecule is a thirteen-particle system
(three nuclei and 10 electrons). It is quite clear that even the approximate solution
of the Schrödinger equation for any realistic atomic, molecular or solid system is
a daunting task.
A first approximation that is essentially always made is that the wave function can
be separated into two components, one describing the energy of the electrons at
fixed positions of the nuclei and one describing the energy of the nuclei. This
assumption, known as the Born-Oppenheimer approximation, allows the motion
of electrons to be decoupled from the motion of the nuclei. This assumption holds
in many cases, but fails e.g. at avoided crossings. Within the Born-Oppenheimer
approximation, the contribution to the total energy of a system can be calculated
separately for the electronic system and the nuclei. The electronic Hamiltonian
then becomes
He = − ∑
i
h̄2 2
e2 Zk
e2
∇i − ∑ ∑
+∑ ,
2me
i k rik
i< j ri j
(2.7)
20
CHAPTER 2. COMPUTATIONAL METHODS
and the electronic SE becomes
(He +VN ) Ψe = Ee Ψe ,
(2.8)
where VN is the nuclear-nuclear repulsion energy, which is constant for a certain
atomic geometry. This is a significant decrease in complexity, but further approximations are needed to be able to simulate atoms, molecules, and solids.
Not only is the evaluation of even the electronic Hamiltonian a challenging problem, the exact description of the wave functions Ψ themselves is also not obvious. The SE is only able to provide the energy for a certain wave function, but
says nothing about how the correct wave function for a certain system might be
found. However, it can be proven that any arbitrary trial wave function that is an
eigenfunction of the electronic Hamiltonian will, through the Hamiltonian, always
be associated with (or, more precisely, be the eigenfunction of) an energy that is
higher than or equal to the ground state energy E0 . This implies that one can rank
the suitability of different trial wave functions by how low the associated energy is.
This is the variational principle. Designing and selecting mathematical descriptions of the wave function with high enough variational freedom is still a topic of
research in quantum chemistry and physics.
The further approximations applied to enable the solution of the SE is what separates different levels of theory within quantum mechanics. Furthermore, the methods can be separated into molecular orbital based methods and density functional
methods. The first group includes methods such as Hartree-Fock (HF), MøllerPlesset perturbation theory (MP), Coupled Cluster (CC) and Configuration Interaction (CI), ordered by increasing level of accuracy (CC and CI can achieve
similar accuracies) and cost. Density functional methods achieve a balance between low computational cost and accuracy, which motivates their wide use for
study of molecules, solids and heterogeneous reactions, the interaction between
the molecules and solids.
2.1.3
Density functional theory
In the electronic Hamiltonian, equation 2.8, the energy of a system is dependent
on the unique positions ri = (x, y, z) of every electron in the system1 . In turn, the
wave function is, through the SE, a function of the interaction between every electron and nucleus in the system. The strength of density functional theory is the
recognition that the properties of a chemical system does not need to be described
as a function of every electronic position. Instead, the total electronic density ρ (r)
(the integrated number of electrons at every point, a function of the electronic positions) can be used directly. The total electronic density is a unique description
1 Where
not explicitly stated, the discussion in this section is based on [132, 134–141].
2.1. THEORETICAL DESCRIPTIONS OF MATTER
21
of a given system. This is intuitively understandable. The integral of the electron density is the total number of electrons. Chemical bonds are found in areas
between the nuclei where the electronic density is higher than that far from the
nuclei. The shape of the electronic density indicates the character of the bonds
and the electronic structure of the atoms. Cusps in the electronic density indicate
the positions of the nuclei. The derivative of the electronic density at a cusp indicates the atomic number of the nucleus. However, while the idea is simple to
understand, the mathematical implementation is much more challenging.
2.1.3.1
The Hohenberg-Kohn theorems
Density functional theory rests upon two theorems of Hohenberg and Kohn [137].
The theorems prove that there is a unique functional (a function of a function)
F [ρ (r)] such that
ˆ
E0 ≡
v (r) ρ (r) dr + F [ρ (r)] ,
(2.9)
where ρ (r) is the ground state electron density and v (r) is an external potential
(the potential of the interaction between nuclei and electrons). The functional
F [ρ (r)] is thus universal and independent of v (r).[137]
The second Hohenberg-Kohn theorem is a variational theorem. Since the first
theorem proves the unique relation between a certain electronic density and an
external potential, the density also determines an energy. Thus, upon varying the
density, an energy that is higher than or equal to the true ground state energy is
obtained. Hence, the variational principle of quantum mechanics is also applicable
if the electronic density is the fundamental quantity.
2.1.3.2
The Kohn-Sham equations
The next key step in the development of density functional theory (DFT) was presented in the paper by Kohn and Sham [138] in 1965[132, 134, 135, 138]. Kohn
and Sham realized that the task of solving the SE within the context of density
functional theory could be made much simpler if one started by describing (i.e.
by using the Hamiltonian for such a system) a system of non-interacting electrons
with the same ground state electronic density as a real system of interacting electrons. The advantage now is that several of the terms in the Hamiltonian for a
non-interacting system are known exactly, and can be described using expressions
from classical physics. The exact energy functional can then be written as
E [ρ (r)] = Tni [ρ (r)] +Vne [ρ (r)] +Vee [ρ (r)] + ∆T [ρ (r)] + ∆Vee [ρ (r)] (2.10)
22
CHAPTER 2. COMPUTATIONAL METHODS
where the first three terms are the kinetic energy of the non-interacting electrons
(indicated by index ni), the nuclear-electron interaction
nuclei ˆ
Vne [ρ (r)] =
∑
k
Zk
ρ (r) dr,
|r − rk |
(2.11)
and the electron-electron repulsion (or rather, self-repulsion of the charge distribution) in a classical picture
ˆ ˆ
ρ (r) ρ (r0 ) 0
1
Vee [ρ (r)] =
dr dr,
(2.12)
2
|r − r0 |
respectively. Vee [ρ (r)] is also known as the Hartree energy. Now, to be able to
evaluate the first term, the kinetic energy of the non-interacting electrons (Tni ),
Kohn and Sham introduced a set of one-particle orbitals for the non-interacting
electrons. These one-particle orbitals are known as Kohn-Sham orbitals, and have
the property that
N
N
i
i
ρ (r) = ∑ ψi (r)∗ ψi (r) = ∑ |ψi (r) |2
(2.13)
Then one can write the one-electron kinetic energy as
N
Tni [ρ (r)] = ∑
i
ˆ
1
ψi (r)∗ − ∇2 ψi (r) dr.
2
(2.14)
The last two terms of equation 2.10 are corrections to describe real interacting
electrons. The term ∆T [ρ (r)] is the correction to the kinetic energy due to the
interaction between electrons. The term ∆Vee [ρ (r)] includes all other quantum
mechanical corrections to the energy due to electron-electron repulsion. These
corrections include the energetic corrections due to
• quantum mechanical exchange interaction (interactions due to the Pauli exclusion principle, stating that no two electrons can have exactly the same
quantum numbers)
• quantum mechanical electronic correlation (that the movement and position
of one electron is correlated with the movement and positions of all other
electrons)
• the correction for the self-interaction energy (the classical self-repulsion energy as written in equation 2.12 is non-zero even for systems of just one
electron, i.e. the electron-electron repulsion is overestimated)
2.1. THEORETICAL DESCRIPTIONS OF MATTER
23
The exchange interaction correction can be calculated exactly, as is done in HartreeFock theory. Furthermore, the self-interaction energy is completely corrected for
in Hartree-Fock theory. On the other hand, apart from a part of the same spin
correlation, most of the correlation is disregarded at the Hartree-Fock level.
To continue, the two correction terms ∆T [ρ (r)] and ∆Vee [ρ (r)] are brought together into one, the so-called exchange-correlation functional Exc [ρ (r)]. Now,
if the Kohn-Sham orbitals are introduced into equation 2.10, the energy can be
evaluated using the following expression
N
E [ρ (r)] = ∑
i
ˆ
ˆ
1 2
ψi (r) − ∇ ψi (r) dr − ρ (r)
2
∗
ˆ
nuclei
∑
k
Zk
|r − rk |
ˆ
ρ (r0 ) 0
1
dr dr + Exc [ρ (r)]
ρ (r)
2 |r − r0 |
!
dr+
(2.15)
The equations can be expressed as a set of one-electron equations
hKS
i ψi (r) = εi ψi (r) , i = 1, 2, ..., N
(2.16)
where, εi are the KS orbital energy eigenvalues and
1 2 nuclei Zk
hKS
=
−
∇ − ∑
+
i
2 i
k |ri − rk |
ˆ
ρ (r0 )
dr0 + vxc
|ri − r0 |
(2.17)
and vxc is the functional derivative of Exc :
vxc =
δ Exc
.
δ ρ (r)
(2.18)
The last three terms of equation 2.17 are often collected into a term known as the
effective potential, ve f f [ρ (r)], as these terms are direct functionals of the electronic density, while the kinetic energy requires the Kohn-Sham orbitals. Furthermore,
ˆ
ρ (r0 ) 0
dr
|r − r0 |
(2.19)
is known as the Hartree potential, vH (the functional derivative of equation 2.12 )
and
nuclei
∑
k
Zk
|ri − rk |
(2.20)
24
CHAPTER 2. COMPUTATIONAL METHODS
is known as the external potential vext .
The set of all one-electron eigenvalue equations is known as the Kohn-Sham equations. The set of KS equations (equation 2.16) is solved to yield the energies and
wave functions (after deciding on some suitable description of the wave functions)
of each one-electron orbital. Then, the ground state density can be calculated using
equation 2.13, before calculating the total energy for the interacting system using
equation 2.15.
This line of reasoning is the basis for the vast majority of all applications of DFT
in electronic structure studies, and is called Kohn-Sham density functional theory.
Alternative density functional theories not following the KS scheme exist, such
as reduced density matrix functional theory[134], but these methods are not in
general use yet.
The KS equations represent a significant simplification, as a large part of the
ground state energy is captured using classical expressions, while still yielding
a formally exact connection between the ground state density and the ground state
energy, since all corrections are captured in Exc . The equations can be extended to
take electronic spin into account [132], but this is not treated in further detail here.
2.1.3.3
The solution of the Kohn-Sham equations in practice
The Kohn-Sham equations are evaluated iteratively. The iterative process starts
with a trial electronic density ρ (r). This density can be obtained e.g. starting from
linear combination of atomic orbitals of all atoms in the system, as is done in the
DFT code GPAW (the DFT code that has been used to obtain all results presented
in the current thesis)[142]. The next step is to calculate the part of the total energy
due to the effective potential Ve f f [ρ (r)] (the last three terms of equation 2.17).
Then, the KS equations are solved to yield the new KS orbitals, which in turn
(using equation 2.13) yield a new electronic density. The process is then repeated
until self-consistency in the electronic density is achieved.
2.1.3.4
Exchange-correlation functionals
Hohenberg and Kohn [137] proved that a universal functional F [ρ (r)] exists, but
the proof says nothing about how this functional might be constructed or how it
might look. The KS equations show how a part of the expression can be constructed exactly, and placed all corrections into the exchange correlation functional Exc . Since 1965, a large number of different approximate Exc expressions
have been designed. This is the reason why DFT is approximate in practice, even
though it is formally exact. One of the greatest challenges involved in DFT calculations is the choice of a proper Exc for the type of problem studied. A number
2.1. THEORETICAL DESCRIPTIONS OF MATTER
25
of papers exist where the accuracy of different functionals is benchmarked, giving
some direction in this task[143–147].
Exchange-correlation functionals are generally categorized based on their dependence on ρ (r). The simplest group, the local density approximation (LDA) functionals,consists of the functionals that depend only on the local electronic density
at every point. These functionals can be described using the general expression
ˆ
LDA
Exc
=
εxc (ρ (r)) ρ (r) dr
(2.21)
where εxc is the exchange-correlation “energy density” per particle for a system.
In practice, most LDA functionals use εxc of the homogeneous electron gas (a
theoretical model system where electrons interact with a uniformly spread positive
charge yielding a completely homogeneous electronic density).
The next level in Exc sophistication, and computational cost, is the group of functionals depending on both the electronic density and the gradient in electronic density. These functionals are known as generalized gradient approximation (GGA)
functionals. Some of the more well-known GGA functionals include the functional
of Perdew et al. [148] (“PBE”, widely used in physics), “BLYP” (the Becke exchange functional [149] combined with the Lee-Yang-Parr correlation functional,
widely used in chemistry) and “RPBE” (revised PBE, the Exc functional of Hammer et al. [150]). RPBE is of special interest for the study of heterogeneous catalysis as this functional was designed to yield accurate chemisorption energies, although at the cost of a worse description of the properties of solids. It achieves
an accuracy in chemisorption energies of between 0.2 to 0.3 eV[150, 151]. For
this reason, combined with the relatively low computational cost of DFT calculations at the GGA level, RPBE has seen wide use in the study of heterogeneous
(electro)catalysis[108, 152–154].
The next logical step in complexity after the GGA is the inclusion of not only
the local electronic density and its gradient (first derivative), but also the second
derivative of the electronic density (or, alternatively, the gradient in the kinetic1
2
energy density τ (r) = ∑occ
i 2 |∇ψi (r) | ). These functionals are called meta-GGA
functionals. One example is “TPSS” of Tao et al. [155]. These functionals yield,
in general, a slightly improved accuracy in comparison with GGA functionals.
Next is usually the combination of a correlation functional with a fraction a of
exact exchange from Hartree-Fock theory:
DFT
+ aExHF
Exc = (1 − a) Exc
(2.22)
These functionals are known as “hybrid functionals”. The most popular DFT functional, B3LYP [156, 157], is a hybrid functional. While a clear improvement in
accuracy is often found when using hybrid functionals, the improvement is obtained at a significantly increased computational cost due to the evaluation of
26
CHAPTER 2. COMPUTATIONAL METHODS
HF exchange. As mentioned above in the description of the universal Exc , the
exchange-correlation functional should also correct for the self-interaction error
inherent in using the classical expression for the electron density self-repulsion
(equation 2.12). This is not done in most LDA, GGA or meta-GGA functionals, but the inclusion of HF exchange in hybrid functionals enables at least partial
self-interaction error correction[158].
Even further increases in accuracy with more complex functionals, such as functionals based on the random phase approximation[159–161], are possible. This
level of treatment is experimental and quite computationally demanding, and is
thus not in general use.
2.1.4
Approximate self-interaction correction with DFT+U
It is well known that both LDA and GGA functionals predict too small band gaps
for many materials with partially filled d− or f −orbitals, predicting that such
materials should be metallic conductors rather than insulators. Included in this
group of materials are several transition metal monoxides, e.g. NiO[162]. The
reason for this deficiency is that common LDA and GGA functionals do not fully
correct for the self-interaction energy, and thus favor an unphysical delocalization
of the electron density[158]. An approximate way of correcting for this error is the
DFT+U method, which is also known as the Hubbard-U method, where a U value
is chosen to alter the occupation of a certain set of orbitals, e.g. the d orbitals. In
GPAW, when applying the DFT+U method, the total energy is evaluated as[142,
162]
U
(2.23)
EDFT+U = EDFT + ∑ Tr (ρ a − ρ a ρ a ) ,
a 2
where U is the chosen Hubbard-U value, Tr is the trace operator (the sum of the
main diagonal elements of a matrix) and ρ a is the atomic orbital occupation matrix
(e.g. the d-orbital density matrix). As discussed by Himmetoglu et al. [158], the
application of such an energy expression results in a potential that favors integer
occupation (localization) of electrons in e.g. the d orbitals. The DFT+U method
is often applied when modeling metal oxides[163–165].
2.1.5
Quantum mechanics for periodic systems
Heterogeneous catalysis concerns reactions on the surface of a solid catalyst. Two
different kinds of methods are usually used for modeling surfaces: cluster-based
methods and periodic methods.
2.1. THEORETICAL DESCRIPTIONS OF MATTER
2.1.5.1
27
Cluster-based methods
In cluster-based methods, the surface is modeled as a bunch of atoms (a cluster) of
large enough size to capture the properties relevant for the study. By using a cluster
of atoms in vacuum as the model, wave-function-based methods can easily be
applied. This allows for using methods such as coupled cluster, which give results
approaching chemical accuracy (errors of the order of kB T at room temperature
i.e. ca 0.025 eV or about 1 kcal/mol), albeit at significant cost. However, edge
effects due to the finite size of the cluster may affect the results if not corrected
for[166]. Furthermore, for adsorption energy calculations, the state of the cluster
might have to be adjusted (e.g. by saturating bonds) to yield an adsorption energy
that is similar to that of a proper periodic surface[167]. For other properties, such
as the band structure and properties depending on it, convergence towards a proper
description of a solid by using larger and larger clusters is very slow[168].
2.1.5.2
Periodic methods
Periodic descriptions of a material or a surface allow the properties of a macroscopic solid to be reproduced using a small unit cell that is repeated infinitely far
in the surface-parallel dimensions. With such a model, properties of the periodic
solid, such as the complete band structure, are captured correctly at the level of
theory chosen. It is possible to make such a model by using Bloch’s theorem and
Bloch functions[135, 169–171]
ψik (r) = uik (r) exp (ik · r) ,
(2.24)
where i is a certain band index, uik (r) are functions describing the periodicity of
the system and exp (ik · r) is a plane wave. Bloch functions are the one-electron
solutions of the Schrödinger equation in a periodic potential, such as that set up by
atomic nuclei in a periodic solid. They are thus useful as eigenfunctions in periodic
KS DFT. Each component of the k-vector is associated with a wave function, but
unique wave functions are only found for values in the interval of −π/a ≤ k ≤ π/a
where a is the periodicity (lattice constant) of the cell in a certain direction of
the reciprocal unit cell. Values of k in this range are said to describe the “first
Brillouin zone (BZ)”, which spans reciprocal space. In a linear combination of
atomic orbitals picture, each value of the k-vector describes different combinations
of atomic basis functions. These combinations cover the range from completely
bonding to completely antibonding[170]. By integrating over the first BZ, properties of a periodic system can be determined. One example is the electronic charge
density[171]:
ˆ
1
fik |ψik (r) |2 dk,
(2.25)
ρ (r) =
VBZ ∑
BZ
i
28
CHAPTER 2. COMPUTATIONAL METHODS
where VBZ is the volume of the first BZ and fik is the number of electrons in state
ik (the occupation number, 0 ≤ fik ≤ 1). The sum is evaluated over i bands, and
the integral over the first BZ is an integral over all k-points. Another example is
the kinetic energy, which in a periodic KS scheme can be evaluated as[135]
ˆ ˆ
Ekin,s [{ψ}] = ∑
i
k
1
∗
ψik
(r) − ∇2 ψik (r) drdk
2
r
(2.26)
The range of k-points in the first BZ is of course continuous, but an integral over
an infinite number of k-points (not to mention the additional sum over all i bands)
is not practical to evaluate. However, one can make use of the fact that wave
functions ψik at k-points spaced only a short distance apart in the first BZ are very
similar. Then, the integral over all k-points can be approximated as a weighted sum
over a number of k-points. Using more k-points gives an improved approximation
of the integral. There are several methods that attempt to find ways of selecting
(or sampling) the k-points efficiently, to reduce the number of k-points needed.
One method is that of Monkhorst and Pack [172]. The number of k-points needed
also depends on the volume of the first BZ. As the BZ spans reciprocal space,
an increased real space volume of a unit cell will yield a smaller BZ. Therefore,
fewer k-points will be needed to yield the same spacing between points in reciprocal space. A doubling of the real unit cell size will thus mean that half as many
k-points need to be sampled for maintained accuracy of description. However, the
number of k-points that needs to be sampled can also be reduced by using symmetries in the BZ. From a computational cost point of view, it is more advantageous to
use more k-points rather than a larger unit cell, since the cost scales linearly with
k-points but (usually) more than linearly with increasing the number of atoms.
Surfaces are commonly treated in a periodic picture. The surface is then modeled
as a slab of e.g. four layers of atoms which is separated by vacuum from its
own image in the neighboring cell. Usually e.g. the bottom layers of the slab
are fixed to the bulk geometry, and the number of layers of atoms in the slab is
high enough to reach convergence in the property of interest. The slabs must be
separated by a large enough vacuum distance so that they do not interact. The
smallest distance that can be used is found by performing several calculations with
larger and larger slab distances and seeing at which distance the property of study
(e.g. an adsorption energy) no longer changes. The distance between the slabs is
then said to be converged.
Treatments based on the Bloch theorem can also be applied to study defects (such
as vacancies or interstitial atoms) in a material or on a surface. In a similar way
as when using periodic slabs to model surfaces, it is important that the unit cell is
chosen to be large enough that the defect in a unit cell is located far away from its
own repeated image so that the defects do not interact, if that is the situation to be
modeled.
2.1. THEORETICAL DESCRIPTIONS OF MATTER
2.1.6
29
The frozen-core approximation
Chemistry is in general associated with interactions between valence orbitals in
atoms. Therefore, for many purposes, calculations can be made less computationally demanding if the electrons in the core regions of the atoms are frozen to the
configurations assumed in the free atom. When this is done, the full system is
simulated within a frozen-core approximation. This can be achieved in many different ways, for example by using pseudopotentials of different types (e.g., normconserving[173] or ultrasoft[174]), or by using projector-augmented wave (PAW)
setups. These approximations differ in the accuracy that can be achieved, with
PAW setups in general achieving a higher accuracy than e.g., pseudopotentials[175].
However, what is common in all practical implementations of the frozen-core approximation is that only the orbitals that are unimportant for the property under
study should be kept frozen. The choice of which electrons to keep in the frozen
core must be made based on careful benchmarking versus chemical and physical
properties.
The PAW description will now be explained in more detail, as it is used for the
frozen core in GPAW. Although the PAW method can be extended beyond a frozencore description[176], it is only used as such in GPAW. A key advantage with the
PAW description is that smooth functions (called pseudo wave functions, ψ̃n (r))
can be used in regions outside the atomic cores, while the orbitals in the atomic
core are frozen and described by a pre-generated atomic setup, consisting of core
orbitals φia,core . However, in contrast to pseudopotential descriptions, the allelectron wave function (through ψn (r)) for the system is still available, and can be
reconstructed using a linear transformation
ψn (r) = T̃ ψ̃n (r)
(2.27)
where the transformation operator T̃ is given as
T̃ = 1 + ∑ ∑ |φia i − φ˜ia h p̃ai |
a
(2.28)
i
where φia (r) are atom-centered all-electron wave functions, φ̃ia (r) are the smooth
partial wave functions corresponding to that all-electron wave function, and p̃ai
are projector functions characterizing the transformation from the all-electron to
the smooth description. In this notation, n is a certain band index, a is a certain
atom index and i is given by the n, l and m quantum numbers. The atomic core
all-electron wave functions φia (r) are calculated for the isolated atom, using a
certain radial cutoff distance rca (defining the atomic augmentation sphere). The
valence states are described by smooth functions (partial waves, φ˜ia ), that must
agree with the all-electron (wave) functions at all distances larger than rca . The
projector functions are also smooth, and one is needed for each valence state partial
30
CHAPTER 2. COMPUTATIONAL METHODS
wave. These parameters are determined in an atomic calculation where the atomic
setup is generated. The quality of these setups is important for the accuracy of the
final result.[177, 178]
2.1.7
Describing the wave function: Plane waves, linear combination of atomic orbitals and real-space grids
To describe the wave function in an efficient way, several different mathematical
forms can be applied. In many DFT codes used for modeling of heterogeneous
catalysis, the basis set (the combination of mathematical functions used to represent the wave function) is the combination of plane waves (PWs)
χk (r) = eik·r
(2.29)
with different kinetic energies, given by
1
(2.30)
E = k2 ,
2
when using atomic units. The basis set is limited based on a cutoff, which is the
plane wave with the highest energy that is included in the set. Higher energies are
associated with more rapid oscillations in the basis function. PWs are thus suitable
for describing periodic systems where the electronic density varies slowly, such as
in a metallic solid material. However, the atomic core regions of a system are
challenging to describe using PW basis sets, as these regions are characterized
by localized and rapidly oscillating orbitals, requiring a large range of k-values.
Furthermore, the singularity in the electronic density at the position of the nucleus
is also hard to describe. For these reasons, PWs are used together with a frozen
core described by a pseudopotential or a PAW.[134]
Another common basis set is that of linear combination of atomic orbitals (LCAO).
This type of basis is very common in quantum chemistry. In this type of basis set,
the functions are chosen to be able to describe atomic orbitals accurately, with each
one-electron orbital being given as a sum of a number of atomic orbital functions
with different weights ai
N
ψ = ∑ ai φi .
(2.31)
i=1
These orbitals are well suited to describe localized states, and fewer terms can
be used to accurately represent atoms or molecules than if e.g, PWs are used.
Nevertheless, if diffuse atomic orbital functions are used, delocalized states can
also be described in a LCAO basis set. A common type of function used in LCAO
2
basis set is the Gaussian, i. e., an expression including a e−ζ r factor. Such basis
sets are known as Gaussian-Type Orbital (GTO) basis sets. It is also possible to
2.2. THE THEORY OF X-RAY PHOTOELECTRON SPECTROSCOPY
31
use numerical basis sets, where the individual basis functions are represented by
numerical values on a radial grid rather than as an analytical function. The LCAO
mode in GPAW makes use of numerical localized pseudo-atomic orbitals.[132,
134, 179]
An alternative to using a specific basis set to describe e.g. the wave functions is the
usage of a three-dimensional real-space grid, where physical quantities such as e.g.
the electronic density or the KS wave functions are represented by values at individual grid points[142, 180]. The accuracy in the description is then dependent on
the distance between grid points in the system. Furthermore, it is simple to parallelize a calculation, as e.g. the calculation of a derivative in the electronic density
is a local operation between neighboring grid points, allowing a straightforward
division of a whole system into smaller parts handled on different processors.
GPAW[142] allows calculations to be carried out using either PW or LCAO basis sets, or using real-space grids, in all three cases combined with a frozen-core
description of the nuclei based on the PAW method.
2.2
The theory of X-ray photoelectron spectroscopy
Thus far, I have indicated that it is possible to calculate energies of atoms, molecules
and materials using density functional theory. Once the energy can be calculated,
also other properties are accessible. In the current section, I will indicate how
DFT can be used to calculate X-ray photoelectron spectroscopic binding energies
and chemical shifts (the difference between the binding energy and a chosen standard binding energy). In this way, a direct connection between a certain structure
and the chemical shift can be obtained, and this has been used to further the understanding for the HER on RuO2 and MoS2 (described further in Section 4.4).
X-ray photoelectron spectroscopy describes the process where a core electron in
a molecule or material is ionized completely by an incoming X-ray, so that the
electron is sent out and can be detected. In experiments, the kinetic energy, given
by
Ekin = hν − EB ,
(2.32)
where hν is the X-ray photon energy and EB is the binding energy of the electron
that is expelled, is measured.[181]
The binding energy determined in an XPS measurement is the same as the difference in total energy between the ground state system (without a core-hole) and the
ionized system (where an electron is removed from the core region):
EB = Ecore−ionized − Eground−state
(2.33)
In DFT calculations in the PAW basis, this difference between total energies is
accessible in a relatively simple way. A special PAW setup must, however, be
32
CHAPTER 2. COMPUTATIONAL METHODS
prepared, which includes one less electron in a certain orbital (e.g. in one of the
3d orbitals of Ru if the 3d XPS binding energies of RuO2 is to be calculated).
The calculation for the core-ionized system is carried out with this setup, while
the ground state calculation is carried out with the standard setup. This method
does not include the effects of spin-orbit coupling, the splitting of binding energies
into two components that occur in ionization of electrons from orbitals with a net
angular momentum l (all orbitals other than s). However, chemical shifts are not
expected to be significantly affected by spin-orbit coupling.
In the present thesis, X-ray photoelectron shifts have been calculated using periodic descriptions of the considered systems. To avoid unphysical interactions
between core-holes, periodic cell sizes of at least 13 to 15 Å were used. Moreover,
the spins in the core-ionized system should be unrestricted, as the singly occupied
core level can result in polarization of the spins, affecting the total energy.
As is the case for other properties, DFT results for binding energies and chemical
shifts do not achieve chemical accuracy. Absolute values for binding energies
calculated using DFT GGA functionals have been found to differ from experiment
by up to 1 eV, depending on the functional used[182], but chemical shifts differ
much less[183].
2.3
Theoretical models for heterogeneous electrochemical reactions
Apart from using DFT to calculate XPS chemical shifts, I have also performed
DFT calculations to model heterogeneous electrochemical reactions. In practice,
theoretical studies based on DFT modeling have so far had the most success in describing heterogeneous gas-solid reactions. The reason is the relative simplicity of
such systems compared with typical electrochemical systems. In the case of gassolid reactions, understanding can be obtained even when a reaction is modeled as
occurring in vacuum since the interactions between molecules in the gas can be
assumed small. Furthermore, the rates are controlled by temperature and pressure,
which can be taken into account using well-known expressions from thermodynamics and kinetic rate theory. Another advantage is that several experimental
methods exist that give detailed information about reactions at surfaces under vacuum conditions.[109, 153]
Conversely, most heterogeneous electrochemical reactions occur in the interface
between a solid catalyst and a liquid electrolyte. The electrolyte is often a concentrated solution, with high concentrations of one or more ionic compounds. Reactants in an electrochemical reaction interact strongly with other molecules close to
the surface, leading e.g. to solvation of the reactants by water. A first principles description thus needs to include a description of not only the solid material and the
2.3. THEORETICAL ELECTROCHEMISTRY
33
reactant, but also of the electrolyte. To make matters even more complex, the electrolyte cannot be assumed to be a static conformation of molecules, but a proper
description should sample many different configurations of molecules and the energetics for reactions under all such conditions. One way of attacking this problem
is by kinetic Monte Carlo methods or (ab initio) molecular dynamics[184].
Not only is the system itself more complicated, but the reaction is controlled not
only by temperature and pressure but also by the electrochemical potential. This
means that a number of different phenomena also need to be included in a model.
These phenomena include the polarization of the electrodes, the charge transfer
between the surface of the electrode and the reactants and also the electric field
formed between the two polarized electrodes. If the polarization of the electrodes
is to be treated rigorously, a complete model should thus include not only a surface slab for a single electrode, but rather a system containing both a working
electrode, a counter electrode and the electrolyte[152]. Upon applying a bias, the
effect of the electric field in the electrolyte between the electrodes, and thus also
the charged double layers at the electrodes, could be simulated self-consistently.
However, this would require a quite large simulation cell, beyond what is practical
today. Furthermore, the non-equilibrium situation with two electrodes each having
a different potential (and thus different Fermi levels) is not possible to study with
conventional ab initio methods[152]. Not only is the size of the system a challenge, but a method of describing the polarization and charge transfer efficiently
is still lacking[184]. Several methods of modeling the polarization of the electrode have been put forward, but it is not clear which offers the best compromise
between accuracy and cost. Going back to the charge transfer process, Marcus
theory is able to describe some details of outer sphere charge transfer[185], but
its suitability for describing electrochemical bond formation or bond breaking has
been questioned.[184]
To add to all these challenges, the actual structure, under polarization, of both the
electrolyte and the surface is in general not well known experimentally (although
methods are beginning to become available [184, 186]), as the systems are more
challenging to study experimentally as well.
2.3.1
The computational hydrogen electrode method
The method that has been applied to study electrocatalytic reactions in the present
work is the computational hydrogen electrode (CHE) method of Nørskov et al.
[152]. It is not the first attempt to model electrochemical reactions at the atomic
level, but it is one of the simpler methods and, more importantly, has been shown to
have predictive power regarding electrocatalyst activities. It makes significant approximations for the electrochemical reaction and the electrolyte, but has still been
found to yield acceptable results that enable understanding of trends in electrocatalytic activity. The method is based on periodic calculations, and thus allows for
34
CHAPTER 2. COMPUTATIONAL METHODS
G / eV
1.37
1/2 Cl2 + eU=0 V
Cl*+eCl-
U=1.37 V
Reaction coordinate
al
el
ec
ic
ch
e
m
em
ic
ch
al
1.37 eV
tro
0
Figure 2.1: Illustration of the CHE method at zero applied potential and at the
equilibrium potential.
proper modeling of the solid catalyst. The model was first applied to the oxygen
reduction reaction (ORR)[152], and this reaction will be used for sake of demonstration here as well. The ORR was modeled using the following mechanism (another associative mechanism was also considered in the paper of Nørskov et al.
[152], but is not included here), where * indicates an electrode surface adsorption
site:
1
O + ∗ −−→ O∗
2 2
(2.34)
O ∗ + H+ + e− −−→ HO∗
(2.35)
HO ∗ + H+ + e− −−→ H2 O + ∗
(2.36)
The binding energies of the O* and HO* intermediates can also be calculated for
the corresponding non-electrochemical conditions:
1
H2 O + ∗ −−→ HO ∗ + H2 (g)
2
(2.37)
H2 O + ∗ −−→ O ∗ + H2 (g)
(2.38)
The model then makes the following approximations:
2.3. THEORETICAL ELECTROCHEMISTRY
35
1. The reference potential is set to that of the standard hydrogen electrode
(SHE). Then, at U = 0 V, 298 K, pH=0, and equilibrium between electrolyte
and 1 bar of H2 in gas phase the following relation applies:
1
+
−
G H +e ≡ G
H ,
(2.39)
2 2
which means that the free energy change for reaction 2.37 is equal to the
negative of the free energy change for reaction 2.35:
1
H − G (H2 O) − G (∗) =
∆G2.37 = G (HO∗) + G
2 2
−∆G2.35 = G (HO∗) + G H+ + e− − G (H2 O) − G (∗) .
(2.40)
In the same way, the free energy change for reaction 2.38 is the same as the
negative of reactions 2.35 + 2.36:
∆G2.38 = − (∆G2.35 + ∆G2.36 ) .
(2.41)
This means that the free energies of electrochemical reaction steps, where
the electron transfer is associated with hydrogen transfer, can be obtained
from the corresponding heterogeneous reactions. This is not limited to
H+ + e− , but any electrochemical reaction associated with a well known
experimental equilibrium voltage can be handled the same way. As an example, the free energy change of the electrochemical reduction of Cl2 :
1
−
−−
*
Cl + e− )
−
− Cl ,
2 2
(2.42)
which is at equilibrium at standard conditions and U =1.37 V vs SHE, can
be obtained directly from the free energy change of the associated heterogeneous reaction
1
∆G = G (Cl∗) − G (∗) − G
Cl .
(2.43)
2 2
This is illustrated in Figure 2.1.
2. The electrolyte is modeled by introducing one or more monolayers (ML) of
H2 O on the surface in the unit cell. In some situations, such as for rutile
RuO2 (1 1 0) surfaces[107], the water layer is left out as the effect on the
adsorption energies of intermediates is found to be very small[107, 114].
3. The free energy of intermediates, without applied potential, at standard state
can be calculated as
∆G = ∆Ew,water + ∆ZPE − T ∆S
(2.44)
36
CHAPTER 2. COMPUTATIONAL METHODS
where ∆Ew,water is the electronic adsorption energy, including water in the
cell, ∆ZPE is the change in zero point energy from the reactant to the intermediate state and T ∆S is the change in entropy from the reactant to the
intermediate state. The expression should also include an enthalpic temperature correction, but this contribution is small enough to disregard in many
cases2 . The electronic adsorption energy of e.g. Cl* can be obtained from
an ab initio calculation:
1
Cl2 .
(2.46)
∆Ew,water = E (Cl∗) − E (∗) − E
2
4. The effect of an applied potential U is introduced by shifting the free energy
of each reaction step that involves transfer of an electron by −ne− eU (see
Figure 2.1).
5. The effect of an external electric field on the energy of adsorbates on the
electrode surface is usually neglected as the effect is small[108], but can in
principle be included as a potential gradient extending from the surface.
6. The free energy of H+ can be calculated at pH different from 0 using the
change in free energy as a function of pH
G (pH) = −kB T ln H+
(2.47)
This leads to a model where free energies for electrochemical reactions can be
obtained without having to explicitly model neither the polarization of the electrodes nor the charge transfer process. However, the method is limited to modeling elementary electrochemical reaction steps that are directly associated with
a thermodynamic reversible potential. Still, by coupling e.g. chemical surface
reactions and electrochemical adsorption and desorption reactions, multistep processes can be studied. The advantage of not having to explicitly model polarization
or charge transfer does mean that activation energies, which can be used to obtain
rate constants by use of Transition-state theory (TST)[109], cannot be calculated
for the electrochemical steps. However, Sabatier analysis, which will be introduced in the next Section, is still possible based on energetics obtained from the
CHE model[188]. This means that predictions regarding which catalysts should be
ideal for a certain reaction can still be made, but possible differences in activation
energies (and thus kinetics) are not captured fully between different electrocatalysts.
2 As
an example, Peterson et al. [187] found that the enthalpic temperature correction,
ˆ
ˆ
∆Hcorr = (C p dT )intermediate − (C p dT )reactant ,
(2.45)
for adsorption of O on Cu (at 18.5 ◦C) was 0.015 eV, to be compared with e.g. the entropic correction
of 0.192 eV.
2.4. ELECTROCATALYSIS FROM THEORY
37
The CHE model has been applied to several systems. As mentioned, Nørskov
et al. [152] applied it to studying the ORR. Their work identified that possible
improvements in activity beyond Pt might be possible, and the work subsequently
inspired research into new Pt3 X alloys[189]. The superior activity of Pt3 X alloys
versus pure Pt has been verified experimentally[189–191]. This shows that the
CHE method can lead to results that have practical predictive power. Rossmeisl
et al. used the CHE method to study the oxygen evolution reaction (OER) from
metal [192] and oxide[107] surfaces and Hansen et al. [108] used it for studying
the chlorine evolution reaction (ClER) on rutile oxides. These studies identified a
connection between the adsorption energy of O on the rutile oxide surface and the
activities for both OER and for ClER.
To summarize, the CHE model enables reaction free energies of electrochemical surface reactions to be determined using adsorption energies obtained from
standard DFT calculations, and the results obtained have been found to compare
favorably with experiments.
2.4
Understanding of heterogeneous (electro)catalysis
from electronic structure theory
By using the methods outlined in the preceding sections, the energetics of molecules
and materials can be evaluated accurately. It has also been indicated how the CHE
model can be used to model the effects of electrode potential on adsorption energies. In the following sections, the connection between these methods and heterogeneous electrocatalysis will be made, and it will be described how these methods
can be used to understand activity trends of different catalysts. A more thorough
description can be found in the introductory textbook by Nørskov et al. [109].
2.4.1
Brønsted-Evans-Polanyi relations
The rate of a chemical reaction is determined by the activation energy for the reaction, the difference between the energy of the transition state and the energy of
the reactants. In the Arrhenius rate expression there is also the attempt rate, but
the dominating effect is the height of the barrier. In general, then, the energy of
the transition state needs to be determined to be able to predict the rate of a reaction. The description of how heterogeneous (electro)catalysis can be modeled
has so far only been based on reaction free energies and not on activation energies.
The reason why reaction rates can be understood without calculation of activation
energies is that many reactions exhibit so-called Brønsted-Evans-Polanyi (BEP)
relationships, where the activation energy of a certain reaction step is linearly dependent on the reaction free energy of the step itself (which can be evaluated in the
38
CHAPTER 2. COMPUTATIONAL METHODS
E
Cl-(aq)
Cl*M1
Cl*M2
RCl-M
Figure 2.2: A qualitative description of the relationship between the reaction energy and the transition state energy for adsorption of Cl– on two different surfaces,
M1 and M2 . M2 binds the Cl* adsorbate more strongly than M1 . ∆Ei indicate the
reaction energies and ∆Ei the transition state energies. The dotted lines indicate
the energies of the initial and final states as a function of the Cl-M distance, and the
fully drawn lines indicate the actual reaction paths. The transition state is found at
the avoided crossing between the initial state and the final state.
way previously described). This is not unexpected, as the energy of the transition
state structure is also affected by the bonding to the surface in the same way as
the energies of the intermediates, as is indicated in Figure 2.2[109]. The figure
gives a qualitative description of the relationship between the reaction free energy
of Cl– adsorption on two different surfaces M1 and M2 , where M2 binds the Cl*
adsorbate more strongly. As M2 binds the Cl* adsorbate more strongly, the reaction energy is more negative, which in turn results in a lower transition state
energy. While BEP relations have been found primarily for nonelectrochemical
heterogeneous reactions, similar relationships have also been found for electrocatalytic reactions[193]. It is assumed that BEP-relations also exist for the OER and
ClER on rutile oxides, but this has not been verified thus far.
2.4.2
Sabatier analysis
As described in the previous section, the rates of heterogeneously catalyzed reactions, including electrocatalytic reactions, are controlled by reaction energies. The
case of chloride oxidation via the Volmer-Heyrovski mechanism
2.4. ELECTROCATALYSIS FROM THEORY
∗ + 2Cl− (aq) −−→ Cl ∗ +e− + Cl− −−→ ∗ +Cl2 (g) + 2 e− ,
39
(2.48)
will be considered as an example. Different solid materials bind the Cl* intermediate with different binding energies. The fundamental reason for why this is the
case has been explained using the d-band model for transition metals[109, 194],
and similar models have been developed also for TM oxides[165]. To achieve a
high reaction rate, the catalyst needs to bind the adsorbate strongly enough to stabilize the adsorbate sufficiently to facilitate the discharge of Cl–, but on the other
hand, if the Cl adsorbate is bound too strongly, the thermodynamic barrier for the
next step in the reaction, where the Cl adsorbate is to react with solution-phase Cl–
to form Cl2 which diffuses away from the surface, becomes too high. These considerations lead naturally to the Sabatier principle, which says that the activity for
a certain catalyzed reaction shows a volcano-shaped dependence on the adsorption
energy of the intermediates (as we shall see in the next section, this is also valid
for multistep reactions). In other words, an optimal catalyst should exist for a certain heterogeneously catalyzed reaction. As was described in the previous section,
such information alone can give an understanding of both thermodynamics and
kinetics (barrier heights) of a reaction[109]. The Sabatier principle is illustrated
in Figure 2.3. In studies of electrocatalysis, the Sabatier principle is often used to
construct volcano plots based on the following precondition for activity based on
the electrode potential[108]. The precondition, in this case for the ClER involving
a Cl adsorbate, is that
U > Ueq + |∆G (Cl∗) |/e
(2.49)
for the catalyst to become active (if the precondition is fulfilled, the reaction free
energy, ∆Gr , of each elementary reaction step is negative). In equation 2.49, U is
the electrode potential, Ueq is the reversible electrode potential for the reaction and
e is the charge of the electron. The expression indicates that the electrode potential
has to be higher than the sum of the reversible potential and the potential required
to either strengthen or weaken the surface-adsorbate bond so that the thermodynamic barrier is minimized. The optimal catalyst for the ClER involving Cl* thus
has an adsorption free energy ∆G (Cl∗) = 0 eV.
2.4.3
Scaling relations
The Sabatier analysis turns out to be applicable also to multistep reactions. The
reason is that, fundamentally, similar adsorbates bind the same way to a certain
type of surface (according to the same mechanism of charge transfer between adsorbate and surface)[108, 109, 194]. An example, relevant for the ClER on oxide
surfaces, is seen in Figure 2.4. The Figure indicates how the adsorption energies
of a number of adsorbates (including e.g. Cl) depend on the adsorption energy
40
CHAPTER 2. COMPUTATIONAL METHODS
Too strong
binding
Too weak
binding
Figure 2.3: Illustration of the Sabatier principle for the case of the ClER via a Cl*
adsorbate. The optimal catalyst is the one with the Cl adsorption energy (∆E Cl* )
indicated by the dashed line.
of O, on the CUS of rutile oxides. It is seen that there exist linear scaling relationships between the descriptor ∆E (Oc ) and the adsorption energies of other
adsorbates. The existence of such scaling relationships has been found for several
different reactions (including OER on several oxides[107, 195] and ClER[108] on
rutile oxides), and make it possible to describe the activity for a certain reaction
based only on a single descriptor. This is often done as in the study of Hansen
et al. [108], by first modeling the stability of different surface intermediates as a
function of experimental parameters (like pH, temperature, and electrode potential) and constructing a general surface Pourbaix diagram (where the most stable
surface intermediate under a certain set of conditions is assumed to be the dominating one on the surface) as a function of the descriptor, and then coupling this
diagram with the precondition for activity based on the electrode potential that
was discussed in the previous section. In this way, an overall volcano plot for both
ClER and OER was constructed for rutile oxides (Figure 2.5). The volcano plot
indicates the connection between a materials property that can be calculated using
DFT (∆E (Oc )), and the electrocatalytic properties (activity and selectivity) of rutile oxides. This connection has been used in the present thesis to understand the
electrocatalytic properties of doped TiO2.
2.4. ELECTROCATALYSIS FROM THEORY
41
O2 **
6.0
TiO2
Cl(O*)2
4.0
∆ E [eV]
RuO2
PtO2
IrO2
ClO*
2.0
0.0
Cl@O*
-2.0
1.0
Cl*
2.0
3.0
4.0
∆ EO∗ [eV]
5.0
6.0
Figure 2.4: The scaling relations for Cl-containing intermediates on rutile oxide
surfaces, from the study of Hansen et al. [108]. Figure reproduced from Hansen
et al. [108] with permission of the PCCP Owner Societies.
42
CHAPTER 2. COMPUTATIONAL METHODS
1.8
1.7
U [V]
1.6
IrO2
1.5
Oc
PtO2
1.4
Occ
2 RuO2
1.3
Cl2 evolution
O2 evolution
1.2
1.1
0
1
OHc
or
c
Clc
2
3
c
∆ E (O ) [eV]
4
5
Figure 2.5: The volcano plot for OER and ClER on rutile oxides of Hansen et al.
[108]. The dashed and dotted lines enclose areas where either oxygen evolution
(blue dashes) or chlorine evolution (black dots) is thermodynamically possible according to the precondition for activity (e.g. equation 2.49 for Cl2). However,
unless the surface intermediate involved in a reaction is prevalent on the surface,
the reaction rate is predicted to be low. The thick black line encloses areas where
the chlorine evolution rate is predicted to become significant, since the precondition for activity is fulfilled and the intermediate involved in the reaction is prevalent
on the surface. Figure reproduced from Hansen et al. [108] with permission of the
PCCP Owner Societies.
Chapter 3
Experimental methods
The current section gives a brief overview of the experimental methods applied in
the present thesis. Further details are found in Hummelgård et al. [15] (paper 1)
and Sandin et al. [14] (paper 4).
3.1
Preparation of mixed oxide coatings using spincoating
In Hummelgård et al. [15], we studied the effect of modifying a conventional DSA
coating (Ru0.3Ti0.7O2) by addition of Co to obtain a coating with 31% less Ru and
the overall composition Ru0.2Co0.1Ti0.7O2 (based on the composition of the coating solution). The electrodes were prepared by depositing the coating solution
onto polished Ti discs, which were then spun at 1400 rpm on a spin-coating device
to produce an even coating layer. Afterwards, the electrode was first dried in a furnace kept at 80 ◦C for 10 minutes before being calcined at 470 ◦C for 10 minutes
in the same furnace. This process was either performed once, or repeated 3 or 7
times, before a final calcination for 60 minutes at the highest temperature. This
spin-coating method makes it possible to control the amount of coating solution
that is applied in each layer. It was estimated (based on the measured weight increase and the concentration-weighted density of the oxide coating[196]) that this
produced coatings with thicknesses of 0.5 µm, 1.45 µm or 3.4 µm. The coating was
then characterized by use of atomic force microscopy (AFM), scanning electron
microscopy (SEM), transmission electron microscopy (TEM), X-ray diffraction
(XRD) and differential scanning calorimetry (DSC) as well as by electrochemical
measurements.
43
44
3.2
3.2.1
CHAPTER 3. EXPERIMENTAL METHODS
Electrochemical measurements
Cyclic voltammetry
In the study of Hummelgård et al. [15] cyclic voltammetry (CV) was used both
for the quantification of the electrochemically active surface area (q∗ ) and to gain
further information about the processes ongoing on the electrode at different potentials. For metal oxide electrodes, the voltammogram is characterized by the
presence of broad diffuse peaks and thus cannot give the same understanding of
the surface reactions as is possible e.g. for metallic Pt[197]. However, the determination of q∗ based on the CV is a standard measurement in electrochemical studies
of metal oxide electrodes. In the present work, the q∗ is determined by measuring
the CV between two end potentials, and then integrating the current over time to
obtain a positive and a negative charge. The average absolute value of these two
charges is used as q∗ . In turn, this charge is related to the real surface area of the
electrode[97]. However, the connection is nontrivial for mixed oxide electrodes,
and q∗ is mainly utilized to compare the surface areas between different electrodes,
rather than to obtain an absolute value of the surface area in e.g. m2 /g.
3.2.2
Galvanostatic polarization curve measurements
In the study of Hummelgård et al. [15], we measured polarization curves (electrode
potential plotted versus the logarithm of the applied current density, j) for mixed
oxide electrodes. The slope of this curve is the Tafel slope. For industrial synthesis,
it is important that the Tafel slope is minimized, since high production rates at high
current densities then become achievable at a low total electrode voltage and total
applied power.
While polarization curves at low applied current densities can be measured with a
relatively simple three-electrode setup, the high current densities relevant for industrial electrosynthesis (i.e. several kiloamperes per square meter) require control
of primarily two factors: the control of mass transport of reactants and products
and accurate correction for the effect of the resistance of the electrolyte on the
measured electrode potential.
In the present work, the first factor is controlled by using rotating discs as electrodes. These discs can be rotated at high rotation rates (several thousand rotations
per minute), which allows for a controlled rate of mass transfer to the surface of
the electrode.[197]
Generally, to reduce the impact of the electrolyte resistance, a Luggin capillary is
employed to position the reference electrode close to the surface of the working
electrode. However, at high current densities, the potential gradient between the
opening of the Luggin capillary and the electrode surface becomes too high, pre-
3.2. ELECTROCHEMICAL MEASUREMENTS
45
venting precise measurements of the electrode potential. This potential gradient
is known as the IR drop. The IR drop in the electrolyte will cause the measured
electrode potential to differ from that associated with the surface reaction alone.
Therefore, the IR drop has to be corrected for. In the present thesis, the current interrupt method was used to correct for the electrolyte IR-drop. The method utilizes
the fact that the voltage component associated with the electrical resistance of the
electrolyte will disappear immediately once the electrical current is switched off.
However, the electrode potential will attain a new value more slowly, as the potential decay of the electrode is associated with the conversion of surface species. This
means that if the electrode potential is followed after the current is interrupted, the
transient potential decay of the electrode can be recorded and used to determine the
electrode potential just after the current has switched off. This electrode potential
is the IR-corrected electrode potential.
If the perfect equipment for analysis of this potential decay were available, a complete transient might be measured, and the value just after the current is turned
off would be measured precisely. However, in practice, the time resolution of the
instrument is often a limiting factor. For potential decay on oxide electrodes such
as DSA, the potential decay occurs over several tens of microseconds, and it is
in general not possible to obtain accurate measurements of this decay during the
first few microseconds. This is mainly associated with the delayed response of the
reference electrode used. To reduce this time-lag, a dual-reference electrode setup
was used in the present work. The additional reference electrode was a Pt wire
immersed in the electrolyte, and connected in series with a capacitor. In turn, the
Pt wire and capacitor were connected in parallel with the reference electrode. The
response of the Pt wire then dominates during the first few microseconds[198]. In
the present work, it was found that this allows a stable signal to be read after ca 4 µs
to 5 µs rather than after 10 µs without the dual reference electrode setup. Furthermore, to reduce the noise in the measurements, the mean value of several transients
(between 24 and 72), measured upon interruption at the same current density, was
used. Between each interruption, the electrode was polarized for several ms, to
make sure that each transient would be measured under the same conditions. Nevertheless, the electrode potential just after the interruption of the current is still not
measured, so some type of fitting must be used to obtain an expression that can
yield the potential at t = 0 s. In the present work, the expression due to Morley and
Wetmore [199] was used:
t
.
(3.1)
E (t) = E (t = 0) − b × ln 1 +
τ
In this expression, E (t = 0) is the IR-corrected potential, b is the Tafel slope of
the reaction occurring during the potential decay and τ is a constant expressed by
τ=
bC
,
j
(3.2)
46
CHAPTER 3. EXPERIMENTAL METHODS
where j is the applied current density and C the capacitance (likely made up mostly
by the pseudocapacitance, but with a lesser contribution from the double layer capacitance, on oxides such as RuO2[200]). To obtain an IR-corrected electrode potential using this expression, equation 3.1 is fitted to the mean transient recorded
after interruption of the current at a certain current density, using non-linear fitting.
However, it is also possible to use more simple types of expressions, such as polynomials, and it is even possible to use the first accurately measured potential as an
approximation to the true IR-corrected potential. This was acceptable especially
at lower current densities ( j < 1 kA/m2 ).
3.3
Combined determination of gas-phase and liquidphase compositions during hypochlorite decomposition using mass spectrometry and ion chromatography
A goal of the study of Sandin et al. [14] was to quantify important products during the chemical decomposition of hypochlorite species. The measurements were
carried out in a system where time-resolved compositions of both the liquid phase
and gas phase could be determined, by using ionic chromatography (IC) and mass
spectrometry (MS), respectively. The pH of the solution was controlled automatically in the range of 5 to 10.5, and the temperature was kept at 80 ◦C. The pH
control afforded by our experimental setup allowed a detailed study of the effect
of pH on the rate of both uncatalyzed chlorate formation and uncatalyzed oxygen
formation. In contrast with industrial chlorate production, the ionic strength of the
solution was lower, but the NaOCl concentration of 80 mM (6 g/dm3 ) was similar
to the industrial concentration. An equimolar amount of Cl– was also present in
the solution.
Chapter 4
Results and discussion
4.1
Cobalt-doped Ruthenium-Titanium dioxide electrocatalysts
As has been mentioned, in the study of Hummelgård et al. [15], Co-doped RuO2TiO2 coatings were characterized using both non-electrochemical and electrochemical measurements. Only rutile peaks were found in the XRD spectra, indicating
that the Co component was either present as an amorphous phase, or as a component in the rutile lattice. The second explanation is perhaps more likely considering
the high crystallinity of the film indicated by the XRD spectrum. The later theoretical study of Karlsson et al. [17] indicated that it should be possible to dope
Co into the rutile lattice of TiO2. Moreover, it was found that the Co-doped RTO
coatings had increased crack densities, mainly for the thinnest coating. Furthermore, the grain size, 10 nm, (estimated using the Scherrer equation1 [201]) of the
Co-doped coating was smaller, than the 20 nm of the RTO coating. Both the increased crack density and the decrease in particle size are indicative of increased
strain in the coating due to lattice mismatch between the CoOx component and the
Ru-Ti oxide component (RuO2 and TiO2 having very similar lattice dimensions).
The difference in particle size resulted in differences in q∗ , indicating that the Co1 The
Scherrer equation is written as
Kλ
,
(4.1)
β cos θ
where L is the mean size of the particle, β is the width at half-maximum intensity of the peak (e.g. the
(110) XRD peak at approximately 2θ =28° observed in RTO) and K has a value of approximately 1.
This empirical expression gives a relationship between particle size and diffraction peak width, based
on the observation that X-ray beams diffracted through randomly oriented crystals become increasingly
broadened as the crystal particle sizes decrease.
L=
47
48
CHAPTER 4. RESULTS AND DISCUSSION
Figure 4.1: The effect on q∗ of the number of coating layers and of Co-doping for
RuxTi1–xO2 coatings in the study of Hummelgård et al. [15].
doped RTO had an increased electrochemically active surface area. Furthermore,
q∗ also increased with the number of coating layers applied (Figure 4.1).
Anodic polarization curve measurements were conducted in 5 M NaCl and cathodic polarization curve measurements in 1 M NaOH (Figure 4.2). Both types
of coatings exhibited similar kinetic parameters, with Tafel slopes close to typical
values for RTO and RuO2. The Co-doped coatings exhibited lower overpotentials
at the higher current densities. This was found to be related to the higher q∗ of
these coatings, which decreases the local current density, and thus the electrode
potential, at a certain total current density. These results show one way that modification of the coating by doping can be used to alter the amount of electrical
energy needed for chlorine or hydrogen evolution. Additionally, Co is a much less
expensive noble metal than Ru, and the study thus indicates a way that DSAs can
be made using less expensive materials. While no selectivity measurements were
performed in the study, it is likely that the Co-doped coatings would have exhibited
a lower Cl2 selectivity due to the increased surface area[1]. However, the electrode
would likely be less suitable for chlorate production, as Co is a well-known catalyst for decomposition of HOCl/OCl– to form O2[1].
4.2
The selectivity between chlorine and oxygen in
chlor-alkali and sodium chlorate production
4.2.1
The uncatalyzed and catalyzed decomposition of hypochlorite species
In the study of Sandin et al. [14], we have investigated the kinetics for both uncatalyzed and catalyzed decomposition of hypochlorous acid species, for the reaction
4.2. SELECTIVITY BETWEEN CL2 AND O2
1.2
7
7
3
3
1
1
E / V vs Ag/AgCl [sat. KCl]
1.18
1.16
layers, no Co
layers, Co
layers, no Co
layers, Co
layer,no Co
layer,Co
49
Number of layers
1.14
1.12
1.1
1.08
1.06
1.04
1
10
100
1000
j / Am-2
(a) Anodic polarization curves.
−0.9
E / V vs Hg/HgO (1M KOH)
−1
−1.1
7 layers, no Co
7 layers, Co
3 layers, no Co
3 layers, Co
1 layer, no Co
1 layer, Co
−1.2
Number of layers
−1.3
−1.4
1
10
100
1000
-j / Am-2
(b) Cathodic polarization curves.
Figure 4.2: Polarization curves for Co-doped and undoped RTO coatings with
different numbers of applied layers.
50
CHAPTER 4. RESULTS AND DISCUSSION
100
ClO−3 Calculated
Ci / mM
80
CH Calculated
60
ClO−3 Data
40
CH Data
20
0
0
20
40
60
80
100
time / min
Figure 4.3: The overall rates of hypochlorous acid species decomposition and
sodium chlorate formation, together with third-order rate expressions using fitted rate constants. cH is the total hypochlorous acid species concentration (cH =
cHOCl + cOCl– ) as determined using IC. Reprinted with permission from Sandin
et al. [14]. Copyright 2015 American Chemical Society.
pathway resulting in oxygen formation and the pathway resulting in chlorate formation. A limited study of the effect of the ionic strength was carried out, where
it was found that increasing the NaCl concentration to 2.7 M had no effect on the
uncatalyzed rate of oxygen formation.
Focusing first on the uncatalyzed reactions, the overall rate of hypochlorite decomposition was found to be described by the following third-order rate equation:
r = −k [HOCl]2 [OCl–] .
(4.2)
The value of k was found to be (2.39 ± 0.12) M−2 s−1 in agreement with the value
obtained by Knibbs and Palfreeman [75]. The third-order rate constant for chlorate formation was found to be (0.731 ± 0.035) M−2 s−1 . The agreement between
data and the third-order rate equations is seen in Figure 4.3. Surprisingly, it was
found that also the uncatalyzed oxygen formation reaction was best described using a third-order rate expression, as can be seen in Figure 4.5. It was found that
both chlorate formation and uncatalyzed oxygen formation exhibit rate maxima at
approximately the same pH (Figure 4.4), indicating that both HOCl and OCl– are
reactants in both reactions.
These similarities in the pH dependence and reaction orders indicate a possible
connection between the reactions resulting in oxygen and chlorate formation. This
was first suggested for alkaline conditions by Lister and Petterson [202]. Adam
et al. [203] proposed a mechanism for the decomposition reaction in the pH range
5-8 involving formation of first Cl2O:
2 HOCl
Cl2O · H2O
(4.3)
4.2. SELECTIVITY BETWEEN CL2 AND O2
51
/ mM ·min -1
0.12
a)
0.1
0.08
FO
2
max
0.06
0.04
0.02
0
4
5
6
7
8
9
7
8
9
pH
-1
4
Ri,ClO - / mM ·min
b)
3
3
2
1
0
4
5
6
pH
1
rd
2
CO / mM
Figure 4.4: a) The maximum rates of uncatalyzed oxygen (estimated based on
flow of oxygen at the MS detector) and b) chlorate formation (based on the initial
rate). Both processes have similar pH dependence, signaling a possible connection
between the two reaction pathways. Reprinted with permission from Sandin et al.
[14]. Copyright 2015 American Chemical Society.
3 order
0.5
2nd order
Data average
0
0
20
40
60
80
time / min
Figure 4.5: The experimental data for accumulated oxygen (per electrolyte volume) together with fitted second- and third-order rate expressions indicating that
a third-order rate expression is a better description of the data. Reprinted with
permission from Sandin et al. [14]. Copyright 2015 American Chemical Society.
52
CHAPTER 4. RESULTS AND DISCUSSION
and then formation of HCl2O2– or H2Cl2O2 according to
OCl– + Cl2O · H2O
HOCl + HCl2O2–
(4.4)
HOCl + Cl2O · H2O
HOCl + H2Cl2O2
(4.5)
or
The formation of the HCl2O2– or H2Cl2O2 intermediates is third order with respect to hypochlorite according to the reactions above. As the uncatalyzed oxygen
evolution rate could be described as third order with respect to hypochlorite, the
formation the intermediate is possibly the rate determining step also for oxygen
formation. Adam et al. [203] then proposed a number of reaction steps through
which ClO3– might form. In general agreement with the suggestion of Lister and
Petterson [202], we propose that a small amount of the HCl2O2– or H2Cl2O2 intermediate might instead be decomposed to form oxygen, in a pathway with a lower
rate than the main chlorate-forming reaction.
Additionally, a variety of potential catalysts for the decomposition of hypochlorite were evaluated, including AgCl, Al2O3, CeCl3 · 7 H2O, CoCl2, Fe3O4, FeCl3
· 6 H2O, IrCl3 · xH2O, Na2Cr2O7 · 2 H2O, Na2Mo4 · 2 H2O, RuCl3 · xH2O and RuO2
· xH2O. The amounts of the potential catalysts were chosen to achieve a solution
concentration of 10 µM, except for the solid materials AgCl (100 ppm) and Al2O3
(9 ppm). Of these materials, only CoCl3 and IrCl3 · xH2O were active catalysts for
oxygen formation by hypochlorite decomposition, with the Ir compound also being a catalyst for chlorate formation. Both materials precipitated upon addition to
the electrolyte, indicating that the effects might be due to heterogeneous reactions.
Co was found to be an active catalyst for oxygen formation at pH 3, 6.5 and 10.5.
The effect of iridium chloride addition was dependent on pH, with the selectivity
for chlorate formation being higher at pH 10.5 than at pH 6.5. Furthermore, the
behavior of the oxygen evolution rate at both pH 6.5 and 10.5 indicates a possible
connection between the selectivity and activity and an Ir redox reaction such as the
following,
IrO2 + 2 H2O
IrO42– + 4 H+ + 2 e ,
(4.6)
which would be controlled both by pH and the oxidative potential of the electrolyte. Further work on the properties of Ir as a potential chlorate catalyst is
warranted. Neither the Ru chloride nor RuO2 was active for hypochlorite decomposition. This indicates that heterogeneous decomposition of hypochlorite on industrial electrodes might not be an important source of oxygen in industrial chlorate production. Nevertheless, as both concentrations of NaCl and NaClO3 were
much lower than in the industrial process, further experimental work under more
process-like conditions is necessary to draw a final conclusion.
4.2. SELECTIVITY BETWEEN CL2 AND O2
53
4.2.2
Theoretical studies of the connection between oxide composition and ClER and OER activity and selectivity
4.2.2.1
Computational details
We have performed a fundamental study of the effect on electrocatalytic properties
of doping TiO2 with both conventional dopants such as Ru or Ir, and also 36 other
dopants, including all fourth, fifth and sixth-row transition metals[17]. A deeper
study was performed of the TiO2-RuO2 mixed oxide[16]. In both cases, DFT on
the RPBE-level[150] was used to calculate the adsorption energy of O on the CUS:
∆E(Oc ) = E (Oc ) − E (c ) − E(H2O) + E (H2) ,
(4.7)
This adsorption energy has been found to serve as a descriptor for both OER and
ClER[108], meaning that this adsorption energy can be used, together with the volcano plot of Hansen et al. [108] (Figure 2.5), to predict the activity and selectivity
of different rutile oxide materials for ClER and OER. The selectivity was evaluated based on the difference between the electrode potential U required for OER
and the one required for ClER, for a certain material. We considered both binary
and ternary combinations of TiO2 and other oxide materials, as well as the distance
dependence of the effects. This was done by calculating ∆E(Oc ) with one or two
dopants placed into a unit cell with 8 cations (in the binary systems) or either 8
or 16 cations (in the ternary systems). The positions in the 8 cation-cell where
dopants could be placed are indicated in Figure 4.6, where the three-dimensional
structure of the surface is also indicated.
4.2.2.2
The self-interaction error
A key challenge in modeling mixed oxide materials is the self-interaction error
(SIE), which is related to the failure of presently available density functionals to
completely correct for the classical self-interaction (see Section 2.1.3.2). This error
causes the electronic density to delocalize in an unphysical way, and this is a particular problem for the description of dopants and defects[159, 164, 204]. The most
popular way of correcting for this error in theoretical studies of (electro)catalysis
is the Hubbard-U method, where the energy of the d orbitals of different atoms
in the solid are altered to describe the electronic density correctly. However, this
method is usually applied in a non-self-consistent manner, where the U value is
determined in some way and then kept constant during calculation of e.g. adsorption energies. Two main methods of determining the U involve either fitting the
value to recover some experimental thermochemical data[165, 205, 206], or fitting
the value to recover the correct curvature of the total energy with changes in the
number of electrons in a system (the so-called linear response U[207–210]). Still,
in neither of these methods does the resulting U value correct the description of
54
CHAPTER 4. RESULTS AND DISCUSSION
a
b
1cus
1br
2a
2b
O
O
c
c
O
Ti
Ru
3a
3b
4a
4b
Figure 4.6: a) The model systems studied, as well as the different positions where
the dopant could be placed in the studies of Karlsson et al.[16, 17]. b) A threedimensional image of the (110) surface of an Ru-doped TiO2-slab.
all physical properties, and it has been found that the U-value needed to e.g. correct the band gap of a material might be too high to recover the correct enthalpy
of formation of the material[165]. Furthermore, it is not straightforward to determine which orbitals should be altered with a U-value[209]. It is possible that a
method where the U-value is determined self-consistently using the linear response
method[207], at every step in e.g. a relaxation calculation, and for all orbitals outside of the frozen core, might correct for these deficiencies, but this is not practiced
at present. There are other methods that might approach chemically accurate energies (energies with an error of less than 1–2 kcal/mol≈0.04–0.08 eV), including
calculations where the exchange energy is evaluated exactly using the Kohn-Sham
orbitals (EXX) and the correlation energy is evaluated using the random-phase
approximation (RPA)[211] and quantum Monte Carlo (QMC)[212], but both of
these methods are presently too computationally demanding to be used in screening studies.
In the present studies[16, 17], we have used the Hubbard+U method, applied on
3d or 4d states of either dopants, Ti cations, or both, to ascertain the importance of
a partial SIE correction. For the RuO2-TiO2-system, we first found that as long as
spin polarization is neglected, which is acceptable for nonmagnetic materials such
as DSA, the addition of a U value, in the range of 1 eV to 6 eV, on the Ti d states
resulted in only minor effects on adsorption energies[16]. However, in the later
4.2. SELECTIVITY BETWEEN CL2 AND O2
55
U = 2 eV on Ti
U = 2 eV on Ti and dopant
U = 2 eV on dopant
4
3
∆E(Oc )U−∆E(Oc )U =0 / eV
2
1
0
1
2
3
Sc
Y
Ti
Zr
Hf
V
Nb
Ta
Cr
Mo
W
Mn
Re
Fe
Ru
Os
Co
Rh
Ir
Ni
Pd
Pt
Cu
Ag
Au
Zn
Cd
Hg
Ga
In
Tl
Ge
Sn
Pb
As
Sb
Bi
Se
Te
4
dopant in 1br
Figure 4.7: The effect of Hubbard U on the adsorption energy of O.
screening study[17], several dopants which are often associated with magnetism
(including Fe, Co and Mn) were included, and all calculations were therefore spin
polarized. We then studied the effect of applying a U = 2 eV, similar in magnitude
to the optimal found by García-Mota et al. [165], to either dopants, Ti or all dstates. The effect was here found to be larger for some dopants, see Figure 4.7.
The largest effects of applying a U were seen for Sc, Y, Cu, Cd, and Hg. For
Cd and Hg, the effect was associated with an instability of the dopant in the TiO2
structure. This is seen in Figure 4.8, where the distance between the surface CUS
site and the second layer 2a site for a doped TiO2 (see Figure 4.6) is compared with
the same distance in pure TiO2. In fact, Hg was unstable as a dopant regardless
of the U value. However, the large effects of U on the adsorption energies on
Sc- or Y-doped TiO2 do not appear to be associated with a change in structure. It
has been found previously that keeping a high enough number of electrons in the
valence (outside the frozen core) is necessary to achieve accurate results for these
atoms[213, 214], but the adsorption energies were hardly affected by repeating the
calculations with a setup that had a smaller frozen core. The results for these two
dopants are thus hard to rationalize. For Cu, it is possible that altering the energy
of the d-states has a large effect on the d-state filling, which seems to result in an
artificial activation of the dopant.
For a number of dopants, including Cr, Mo, Mn, Re, Fe, Ru and Os, the application
of a U value results in smaller changes in the adsorption energy and a weakening
of the oxygen-surface bond. The effects are similar to those noted by García-Mota
56
CHAPTER 4. RESULTS AND DISCUSSION
et al. [165]. The effect for a number of other dopants, such as In, Ge, Sn, Pb, As,
Sb, Bi and Se, are quite small.
However, our results indicate that the application of a U =2 eV might result in
a less accurate description than that obtained at the RPBE level. First off, the
application of a U value changes the magnetic properties of the materials in a
way that is not expected. It is known that Co, Cr, Fe, and Mn-doped TiO2 are
magnetic[215, 216]. For the bare surfaces (without O adsorbate), both RPBE+U
and RPBE calculations predict these materials to be magnetic, while for the surfaces with an Oc adsorbate only the RPBE-level calculations predict that the materials are magnetic. At the same time, the RPBE+U calculations predict that
O-covered In- or Cd-doped surfaces should become magnetic, even though both
RPBE and RPBE+U calculations predict that bare Mn- or Fe-doped TiO2 surfaces have the highest magnetic moments. RPBE gives a consistent description
of the magnetic properties, for both bare and O-covered slabs. Apart from magnetic properties, the application of a Hubbard U results in much larger changes in
adsorption energy of O for pure TiO2 than what is found by carrying out an actual self-interaction correction using the method of Perdew and Zunger[204, 217]
(PZ-SIC). Applying a PZ-SIC when calculating the adsorption energy of O on
the (110) surface of rutile TiO2, Valdés et al. [217] found that the adsorption energy was changed by only 0.03 eV. In contrast, the application of a Hubbard U
on Ti d-states results in a weakening of the adsorption energy of ca 0.3 eV, a
ten times larger change. It is expected that the SIE should be small for systems
with low d-orbital occupancy[218, 219], and this is reflected in the PZ-SIC result, but not in the Hubbard-U result. The U that is found through linear response
calculations[209] is about 5 eV, an even higher value than what we considered
here. Such a high value allows the value of the band gap for TiO2 to be calculated
correctly, but it actually results in a much worse value for the enthalpy of formation
of Ti2O3 from TiO2 (it changes the enthalpy of formation by more than 1.5 eV and
changes the sign of the formation energy)[165], and, as we indicate, it is much too
high to be consistent with a correction for the SIE. These results, combined with
the good performance of RBPE for chemical energetics[147], lead me to conclude
that the usage of RPBE without a Hubbard U correction results in more consistent
(and possibly more accurate) results for adsorption energies, even for doped TiO2.
I suggest that higher-level calculations, e.g. using RPA+EXX or QMC, are needed
to benchmark GGA, GGA+U and hybrid DFAs before a final conclusion about the
applicability of Hubbard U methods for chemisorption energies can be made. Finally, the important observation that linear scaling relations obtained at the GGA
level are usually maintained at the GGA+U level indicates that our results should
be qualitatively correct regardless of the results of such a comparison[165, 209].
The results presented hereafter are from calculations on the RPBE level.
4.2. SELECTIVITY BETWEEN CL2 AND O2
57
5
U = 2 eV on Ti
U = 2 eV on Ti and dopant
U = 2 eV on dopant
Without U
d1cus−2a,U−d1cus−2a,U =0,TiO2 /
4
3
2
1
1
Sc
Y
Ti
Zr
Hf
V
Nb
Ta
Cr
Mo
W
Mn
Re
Fe
Ru
Os
Co
Rh
Ir
Ni
Pd
Pt
Cu
Ag
Au
Zn
Cd
Hg
Ga
In
Tl
Ge
Sn
Pb
As
Sb
Bi
Se
Te
0
dopant in 1br
Figure 4.8: The effect of Hubbard U on the distance between the Ti cations in 1cus
and 2a (see Figure 4.6a). The distance parameter, d1cus−2a , is plotted relative to that
of pure TiO2 and without an applied U, indicating deviation from the rutile surface
structure. For most dopants, the relaxed structure is still rutile-like. Significant
systematic deviations (not depending on the chosen U) are clear for Hg. Significant
deviations are seen also for Cd when a U value is applied on the Cd dopant or when
no U value is used. For Zn and In, deviations are clear when a U value is applied
on the dopant. Ag, Au, Bi, and Te display lesser deviations regardless of U.
58
Te
Se
Bi
Sb
As
Pb
Sn
Ge
Tl
In
Ga
Hg
Cd
Zn
Au
Ag
Cu
Pt
Pd
Ni
Ir
Rh
Co
Os
Ru
Fe
Re
Mn
W
Mo
Cr
Ta
Nb
V
Hf
Zr
Ti
Y
Sc
1.5
CHAPTER 4. RESULTS AND DISCUSSION
Dopant as active site (1cus case)
1.0
0.5
0.0
Te
Se
Bi
Sb
As
Pb
Sn
Ge
Tl
In
Ga
Hg
Cd
Zn
Au
Ag
Cu
Pt
Pd
Ni
Ir
Rh
Co
Os
Ru
Fe
Re
Mn
W
Mo
Cr
Ta
Nb
V
Hf
Zr
Ti
Y
Sc
0.5 0.5
d1cus−2a,U−d1cus−2a,U =0,TiO2 /
Ti as active site (1br case)
0.0
0.5
1.0
1.5
d1cus−2a,U−d1cus−2a,U =0,TiO2 /
Figure 4.9: The distance parameter, d1cus−2a (the distance between the surface
CUS cation and the cation in the next layer, see Figure 4.6a), plotted relative to
that of pure TiO2, when the dopant atom occupies the active CUS site (left side)
and when the dopant atom occupies the surface br site (right side). Note that the
positive direction of the x axis is toward the left on the left side, and toward the
right on the right side.
4.2.2.3
Results of the screening study
The stability of dopants in the rutile structure is a key concern as it will determine
whether the dopant-Ti arrangements considered here can be realized in practice. To
quantify this aspect, structural changes based on the d1cus−2a (the distance between
the CUS active site and the Ti cation in the next layer, see Figure 4.6a) have been
calculated and are shown in Figure 4.9. Once more, it is seen that the stability
of Hg and Cd as dopants in rutile TiO2 is limited. However, also Pb, Tl and Y
as dopants occupying the CUS site, and Te as dopant occupying the bridge site
display deviations from the rutile structure of more than 0.5 Å. Still, the majority
of dopants display only minor deviations from the rutile structure, indicating that it
should be possible to form structures with dopants in the sites we have considered.
The adsorption energy results obtained from the screening are seen in Figure 4.10.
Before discussing the results in detail, there are two important descriptor (∆E (Oc ))
4.2. SELECTIVITY BETWEEN CL2 AND O2
59
values of the volcano plot (Figure 2.5)[108] that should be kept in mind. The first
one is the optimal descriptor value for the OER, of ∆E (Oc ) = 2.4 eV. The activity
for ClER is also maximized at that descriptor value. The second is the descriptor
value for maximized ClER activity and selectivity, at ∆E (Oc ) = 3.2 eV. There are
two descriptor values that result in optimal activity for ClER since there are two
stable surface intermediates that can result in chlorine evolution at U = 1.37 eV.
Figure 4.10 indicates the descriptor value for doped TiO2 with the dopant placed
either in the active site (the “1cus” case) or in the bridge site next to the active site.
In the first case, the dopant is the cation in the active site, whereas in the second
case, the active site is a Ti cation that might be activated by the nearby dopant.
In the left half of Figure 4.10, it is seen that the descriptor value for a number
of dopants, including As, Mn, Co and Pd, is close to optimal for selective and
active ClER (∆E (Oc ) = 3.2 eV). Sb or V in the same position results in a material
with optimal activity for OER, while Ru or Ir in that position results in a material
with electrocatalytic properties similar to those of pure RuO2. Turning next to the
situation of dopants located next to a Ti CUS site (the right hand side of Figure
4.10), it is seen that a number of dopants, including Ru, Ir, As and V all result in
a descriptor value close to the optimal one for ClER. These dopants all activate
Ti sites. Finally, a number of dopants, including Sn, Ag and Au, hardly affect the
electrocatalytic properties of TiO2.
The surface of a real mixed oxide electrode presents a multitude of sites, with
slightly varying geometry and with different arrangements of the oxide components. Some active CUS might be occupied by a dopant, and others by Ti. Thus,
the overall electrocatalytic activity of a doped TiO2 material is described by the
combination of the activity of dopant CUS sites and Ti CUS sites activated by
dopants. The sensitivity to dopant position can be found by comparing the descriptor value for the cases when the dopant occupies CUS sites or bridge sites. As
an example, on a real Ru-doped TiO2 electrode (such as a conventional DSA), Ru
CUS sites will exhibit a similar activity as pure RuO2, that is it will be active for
both ClER and OER. However, Ru bridge sites will activate Ti sites for optimum
ClER activity and selectivity. Ideally, the dopant should provide the same effect
independent of site since it is difficult to see how the dopant could be steered to
the desired site. Arsenic is thus a very interesting dopant for chlorine-evolving
electrodes, as TiO2 doped with As should exhibit optimum ClER activity and selectivity regardless of whether As occupies the CUS or the bridge site.
Figure 4.10 shows that the electrocatalytic properties of TiO2 can be modified significantly by changing the dopant. Furthermore, by performing fundamental studies of other electrochemical reactions on rutile oxides, it is possible that the results
in Figure 4.10 could be applied also to understand the electrocatalytic activity of
doped TiO2 in other reactions, such as e.g. direct hypochlorite or sodium chlorate
production. The recent study of Viswanathan et al. [220], indicating that rutile
60
Te
Se
Bi
Sb
As
Pb
Sn
Ge
Tl
In
Ga
Hg
Cd
Zn
Au
Ag
Cu
Pt
Pd
Ni
Ir
Rh
Co
Os
Ru
Fe
Re
Mn
W
Mo
Cr
Ta
Nb
V
Hf
Zr
Ti
Y
Sc
6
CHAPTER 4. RESULTS AND DISCUSSION
Dopant as active site (1cus case)
5
4
3
2
∆E(Oc ) / eV
1
0
Te
Se
Bi
Sb
As
Pb
Sn
Ge
Tl
In
Ga
Hg
Cd
Zn
Au
Ag
Cu
Pt
Pd
Ni
Ir
Rh
Co
Os
Ru
Fe
Re
Mn
W
Mo
Cr
Ta
Nb
V
Hf
Zr
Ti
Y
Sc
1 1
Ti as active site (1br case)
0
1
2
3
∆E(Oc ) / eV
4
5
6
Figure 4.10: The descriptor value for doped TiO2, when the dopant atom occupies
the active CUS site (left) and when the dopant activates the Ti active site (the 1br
case, right). The vertical lines correspond to (from left to right on the left-hand
side, and from right to left on the right-hand side) the descriptor value for pure
TiO2 (thin green), the optimal descriptor value for ClER selectivity and activity
(thick red), the descriptor value for maximum OER activity (thick blue) and the
descriptor value for pure RuO2 (black). The positive direction of the x axis is
toward the left on the left side, and toward the right on the right side.
4.2. SELECTIVITY BETWEEN CL2 AND O2
61
oxides with a descriptor value of ∆E (Oc ) =3.8 to 3.9 eV2 should be selective for
electrochemical H2O2 production, is one example.
Another way of controlling the activity of TiO2 is to use more than one dopant element. Industrial DSAs often contain not only Ru and Ti, but also other components
such as Ir[221]. An example of the effect on the descriptor value of combining two
dopants, where dopants are arranged in the same surface layer, to either side of the
Ti CUS, is seen in Figure 4.11. It is seen that by surrounding a Ti active CUS site
with two dopants, the descriptor value of that site can be modified to a value that is
close to the mean value of the descriptor values of the respective binary dopant-Ti
oxides. In this way, two dopants can be combined to further control the activity
of doped TiO2. However, how to accurately control the atomic-scale arrangement
of these dopants is an open question. If dopants are simply combined in a coating
solution that is then calcined to yield a final oxide, the final result is most likely
an oxide which exposes a variety of different active sites with different electrocatalytic activities and selectivities. New preparation methods where the atomic
structure can be controlled are needed to fully profit from the results shown in
Figure 4.10.
Unless the arrangement of the oxide components is controlled on the atomic level,
the dispersion of dopants in the material might be more or less random, and
dopants might thus be present further down in the oxide layer rather than only in
the surface layer. Furthermore, on real electrode surfaces, the coverage of adsorbates will vary with the process conditions of the electrolysis (e.g. current density
and reactant concentration). The effect of altering surface coverage, as well as
the effect of moving a Ru dopant further from the surface of a doped TiO2 slab,
is seen in Figure 4.12. It is clear that surface coverage has a strong effect on the
electrocatalytic process. However, there are still arrangements of Ru and Ti that
result in optimal selectivity for ClER, although the exact arrangement depends on
the surface coverage. It is also seen in the figure that Ru dopants in the second
and third layer also activate Ti surface sites, but that the effect has essentially disappeared once the dopant has been moved past the fourth layer. The effect is thus
short ranged, and dopants must be present in the near surface region. This trend is
likely similar for the other dopants considered in Karlsson et al. [17].
We also found that this activation effect might be specific to TiO2, or possibly to
oxides of cations with low d-electron numbers (such as e.g. ZrO2 or TaO2). Formally, Ti in TiO2 is d 0 . A comparison between the change in descriptor value with
dopant position for two oxide models, either for Ru-doped TiO2, as has already
been shown in Figure 4.12, or for Ti-doped RuO2 is seen in Figure 4.13. Also
seen in the figure is the same trend for a model system with an overall Ru:Ti stoichiometry close to that in DSA. The DSA model system behaves similarly to the
Ru-doped TiO2 system, indicating the applicability of our results also to materi2 Heine
A. Hansen, personal communication, 2015.
62
CHAPTER 4. RESULTS AND DISCUSSION
6
Average of the adsorption energies of the binary systems
Calculated adsorption energy
5
∆E(Oc ) / eV
4
3
2
1
0
Pt-Ti-Ir
Mo-Ti-Cu
Re-Ti-Zn
Figure 4.11: The effect of surrounding a Ti active site with two different dopants.
The bright bars are adsorption energies estimated as the mean value of the adsorption energies of the two binary systems, while the dark bars are the calculated
adsorption energies on the ternary slabs. The horizontal lines correspond to (from
top to bottom) the descriptor value for pure TiO2 (green), the optimal descriptor
value for ClER selectivity and activity (red), the descriptor value for maximum
OER activity (blue) and the descriptor value for pure RuO2 (black).
4.2. SELECTIVITY BETWEEN CL2 AND O2
63
Electronic adsorption energy / eV
5
4
3
2
∆E(Oc ), 50% surface coverage
∆E(Oc ), 100% surface coverage
Optimal ∆E(Oc )
∆E(Oc ) on pure TiO2
∆E(Oc ) on pure RuO2
1
1cus
1br
2a
2b
3a
Ru position
3b
4a
4b
Figure 4.12: The effect on the Oc adsorption energy at the CUS site of Ru-doped
TiO2. The position of the Ru atom is varied. The system is either a 1x1 system
(100% surface coverage) or a 1x2 system with adsorption of O only on one CUS
site (50% surface coverage). The dashed lines for the adsorption energies on pure
materials are from calculations on 1x1 surface systems. Figure reprinted from
Karlsson et al. [16], with permission from Elsevier.
als with higher concentrations of dopants near the active site. Furthermore, it is
seen in the figure that while the doped TiO2 material is strongly affected by the Ru
dopant, the RuO2 material is hardly affected at all.
Some insight into the inherent stability of different dopant-Ti arrangements can be
gained by comparing the total energies of the two arrangements in Figure 4.10. The
results are seen in Figure 4.14. It is seen that for O-covered surfaces, the situation
of the dopant occupying the surface CUS site is often more stable. This indicates
that during OER, there is a driving force for moving dopants towards the surface
CUS. Such driving forces also exist for several of the bare surfaces. The existence
of such a driving force could be an explanation for the gradual removal of Ru from
DSA electrodes during use[222, 223]. Experimental studies have shown that the
degradation of DSAs is directly tied to the rate of OER[224–228], during which
64
CHAPTER 4. RESULTS AND DISCUSSION
5.5
5.0
Electronic adsorption energy / eV
4.5
∆E(Oc ) on Ru-doped TiO2
∆E(Oc ) on Ti-doped RuO2
∆E(Oc ) on DSA
Optimal ∆E(Oc )
∆E(Oc ) on pure TiO2
∆E(Oc ) on pure RuO2
4.0
3.5
3.0
2.5
2.0
1.5
1.0
1cus
1br
2a
2b
3a
dopant position
3b
4a
4b
Figure 4.13: Oxygen adsorption energy trends, at 100% surface coverage, as the
dopant atom position is varied in Ru-doped TiO2, Ti-doped RuO2 and in the DSA
model system. The O adsorbate is allowed to adsorb on the “1cus” site in all
cases, so that adsorption occurs directly on the dopant atom only in the “1cus”
case. Figure reprinted from Karlsson et al. [16], with permission from Elsevier.
4.2. SELECTIVITY BETWEEN CL2 AND O2
Te
Se
Bi
Sb
As
Pb
Sn
Ge
Tl
In
Ga
Hg
Cd
Zn
Au
Ag
Cu
Pt
Pd
Ni
Ir
Rh
Co
Os
Ru
Fe
Re
Mn
W
Mo
Cr
Ta
Nb
V
Hf
Zr
Ti
Y
Sc
2.0
Bare surfaces
1.5
1.0
0.5
0.0
0.5
E(bare)1cus−E(bare)1br / eV
1.0
Te
Se
Bi
Sb
As
Pb
Sn
Ge
Tl
In
Ga
Hg
Cd
Zn
Au
Ag
Cu
Pt
Pd
Ni
Ir
Rh
Co
Os
Ru
Fe
Re
Mn
W
Mo
Cr
Ta
Nb
V
Hf
Zr
Ti
Y
Sc
1.5 2.0
65
Surfaces with O
1.5
1.0
0.5
0.0
0.5
E(Oc )1cus−E(Oc )1br / eV
1.0
1.5
Figure 4.14: The energy difference between the situations where the dopant occupies the 1cus position and the 1br position, for surfaces without O (left side) and
with O (right side). Positive values signify that the situation where the dopant occupies the 1br position is more stable than the situation where the dopant occupies
the 1cus position, indicating a driving force for movement of the dopant from the
1cus to the 1br position.
Oc intermediates are present on the surface[107, 108]. The driving force during
chlorine evolution is determined by the intermediates present on the surface during
ClER, such as OHc or Clc [108] for materials with a descriptor value E (Oc ) >
3 eV. Our calculations indicate, again using Ru-doped TiO2 as an example, that
the driving force for moving the dopant to the CUS site when these adsorbates are
present on the surface is either very small or negative. Further studies are necessary
to fully understand the segregation of dopants in TiO2, but this indicates that it is
possible that also the other dopants considered will be stable in e.g. the bridge site
during ClER.
Finally, we used Bader charges[229] to attempt to explain the activation of Ti
surface sites through dopants. The case of Ru as a dopant was again examined,
and it was found that the Ru dopant transfers charge to the Ti surface site, which
allows the CUS Ti-Oc bond to become stronger.
These results require experimental verification. Therefore, we also suggested a
model electrode that should be possible to manufacture using some method allowing atomically precise control of deposition of monolayers of atoms[16]. The two
model systems that we suggested are seen in Figure 4.15. What we found was that
the descriptor value (oxygen adsorption energy) on the TiO2-covered RuO2 sur-
66
CHAPTER 4. RESULTS AND DISCUSSION
(a) The O-covered RuO2 overlayer model
electrode.
(b) The O-covered TiO2 overlayer model
electrode.
Figure 4.15: The two model electrodes studied in Karlsson et al. [16]. Green balls
are Ru, grey balls are Ti and red balls are O.
face should be close to that optimal for ClER activity and selectivity. On the other
hand, the RuO2-covered TiO2 surface should have electrocatalytic properties very
similar to those of pure RuO2. By preparing such model electrodes and performing
electrolysis experiments in chloride-containing electrolyte, it should be possible to
compare the selectivity for ClER that the two model electrodes exhibit. We predict that the ClER selectivity of the monolayer TiO2 on RuO2 electrode should be
higher than that of the monolayer RuO2 on TiO2 electrode, and that both electrodes
should be active. This same theoretical prediction was also made independently
by Exner et al. [112] at almost the same time.
4.3
Semiclassical modeling of rutile oxides
There is now a large literature indicating the value of theoretical modeling based on
DFT. DFT is now a mainstream tool for materials science and catalysis. However,
the computational cost of DFT calculations still makes some studies, specifically
on the mesoscale, prohibitive. One such problem, that is expensive to tackle on the
DFT level, is the study of the segregation of Ru and Ti in DSA. As has been indicated in previous sections[16, 17], the local structure of a mixed oxide electrode
coating decides the electrocatalytic properties of the electrode. If the local structure is to be controlled, tools to model the changes in local structure over time, both
during preparation of the electrode and during use, are required. DSA coatings are
made up by nanoscale particles of typically tens of nanometers size[37], so the
model needs to be able to simulate particles of such size. A RuO2 particle with a
4.3. SEMICLASSICAL MODELING OF RUTILE OXIDES
67
Table 4.1: The Buckingham cation-oxygen shell potentials found through fitting
to experimental structures of rutile RuO2 and TiO2.
Cation
Ru
Ti
A / eV
172038.5
2665.7
ρ / eVÅ−1
0.17
0.28
C6 / eVÅ−6
0.49
0
rcut / Å
10.0
10.0
size of 10 nm contains several hundred thousand atoms. To put this number into
perspective, the largest systems modeled with DFT in the present work contained
about 1000 atoms. As the increase in computer time required for a DFT calculation
on a system with N times more atoms usually scales by more than N in practice,
it is clear that less computationally demanding methods are needed to simulate
discrete particles of realistic size. Therefore, we attempted to fit a force field that
could describe the structural and energetic properties of RuO2-TiO2 mixed rutile
oxides[18]. All calculations using FFs in this thesis were carried out using the
General Utility Lattice Program (GULP)[118, 230–232].
The fitting started from the Buckingham force fields of Bush et al. [119] and Battle et al. [233], which are both based on the same transferable oxygen-oxygen
potential. By refitting both the Ti-O potential and Ru-O potential, an attempt was
made at finding an improved set of parameters to describe the structures of pure
TiO2[234] and RuO2[235]. The approach was based on simultaneous fitting (fitting both interatomic parameters and O shell positions concurrently for a static
structure) as well as relaxed fitting (fitting parameters and shell positions, but also
relaxing the structure at every step). The parameters found by using this fitting
methodology are seen in Table 4.1. For the structures of the pure materials, the
new FF is an improvement (see Tables in [18]).
However, the goal of the fitting was to find parameters that could describe the
mixed oxide. For this purpose, three bulk mixed oxide systems were chosen (see
Figure 4.16). The geometric parameters (atomic positions as well as cell parameters) were relaxed at the PBEsol[236] level. A comparison between the DFT
structure and the structure obtained with the new FF is seen in Table 4.2. It is seen
that the combination of the new Ti-O and new Ru-O interatomic potentials yields a
worse result than the combination of the new Ti-O parameters with the Battle et al.
[233] Ru-O parameters. The energies of the three systems were calculated using
both PBEsol and the GLLB-SC[237] functional, which improves the description
of the exchange energy and therefore should be associated with a smaller SIE. The
energetic ordering (the energy differences between the three systems) were consistent with both functionals. The energies calculated using DFT as well as with the
new FF are found in Table 4.3. It is clear that none of the FFs recover the correct
energetics. For example, while “system 2” is the least stable system according to
DFT, it is the most stable system according to the force fields.
68
CHAPTER 4. RESULTS AND DISCUSSION
1
3
2
Figure 4.16: The three mixed Ru-Ti bulk oxide systems used to fit and benchmark
the force field parameters. Green balls are Ru, grey balls are Ti and red balls are
O.
Table 4.2: Comparison between volumes and cell parameters for mixed oxide
structures calculated using DFT and using the force field. The results when using
Buckingham potential parameters based on a GA search, using atomic position
and volumes in the DFT systems to evaluate the fitness, are also shown.
System
System 1
System 2
System 3
Param.
PBEsol
New FF
New FF for Ti-O,
Ru-O from Battle et al. [233]
FF from GA
search
V / Å3
x/Å
y/Å
z/Å
V / Å3
x/Å
y/Å
z/Å
V / Å3
x/Å
y/Å
z/Å
249.115
6.510
5.893
6.494
248.390
6.458
5.886
6.535
249.989
6.423
6.062
6.422
246.224
6.353
6.096
6.358
246.282
6.351
6.091
6.366
246.049
6.323
6.112
6.361
247.559
6.381
6.078
6.384
247.497
6.382
6.077
6.381
247.251
6.367
6.088
6.378
250.286
6.352
6.204
6.352
250.281
6.351
6.204
6.353
250.288
6.350
6.204
6.353
Table 4.3: Calculated energies (in eV) of the three mixed oxide systems.
System
PBEsol
GLLB-SC
New IP
System 1
System 2
System 3
−212.071
−211.580
−211.927
−212.233
−211.693
−212.191
−968.624
−968.649
−968.541
New FF for Ti-O,
Battle et al. [233]
parameters for Ru-O
−961.758
−961.778
−961.694
4.4. XPS MODELED USING DFT
69
It is reasonable to assume that an improved description of the Ru and Ti cations
is required to describe the energetics correctly. Both Ti and Ru are d elements,
and the effect of the spatial distribution of the d orbitals is key to describe both
structure and energetics of their oxides. We therefore attempted to add the AOM
to the force field. The first attempt was based on standard deterministic fitting.
However, the best FF that could be found in this way only managed to minimize
the energy differences between the three systems. Therefore, several attempts to
optimize the parameters for the AOM using a GA were performed. In the GA,
an array of the FF parameters was used as the chromosome, and the fitness of
each chromosome was evaluated by relaxing the structures of the three mixed oxide systems, and comparing volumes with the volumes obtained using DFT, and
the relative energies between the three systems with the corresponding relative
DFT energies. Initial testing of the GA indicated that it was able to find Buckingham potentials of comparable quality to those obtained using deterministic fitting.
However, when using the GA to fit also the AOM parameters, to attempt to find
a FF that could describe both structural parameters and energetics, we found no
set of parameters that recovered the correct energetic orderings. Again, the best
solution found was a set of parameters that minimized the energy differences between the three systems. Although we failed to find a FF that could describe the
energies and structures of RuO2-TiO2 mixed oxides, the new FF (that is the combination of the new Ti-O parameters and the Battle et al. [233] Ru-O parameters)
is at least able to describe the structure of such oxides more accurately than previous Buckingham-based force fields. The results also indicate that something more
than the AOM model is needed to describe the energetics correctly. It is possible
that it is necessary to depart from a constant charge description of the cations to
achieve this goal, as impressive results for the description of pure rutile TiO2 have
been obtained with charge-equilibration models[238].
4.4
Understanding X-ray photoelectron spectroscopy
using density functional theory
The HER has been studied on two materials, MoS2 and RuO2, based on comparisons between experimental and calculated XPS chemical shifts. In the first
study[20], we used this method to further the understanding of the mechanism of
HER on MoS2, a material that has been devoted study as it is active for HER even
though Mo is not a noble metal[48, 49]. In the second study, we calculated XPS
binding energies for RuO2 and for a variety of hydrogenated RuO2 structures and
Ru oxides and hydroxides to reconsider conclusions drawn in previous work[51]
regarding structural changes in RuO2 during HER.
70
4.4.1
CHAPTER 4. RESULTS AND DISCUSSION
The active site for HER on MoS2
The study used an ambient pressure XPS setup, as shown in Figure 4.17. The
setup consisted of a Nafion membrane, which separated the cathode side, on which
amorphous nanoparticulate MoS3 catalysts were dispersed on a carbon support
material, from the anode side, on which a platinum-carbon catalyst was deposited.
The Mo 2p XPS measurements were performed at room temperature, with the anode side being exposed to saturated water vapor and the cathode side to a vacuum
with a leakage stream of water gas achieving a partial pressure of water of 10 torr.
The evolution of the X-ray photoelectron spectra is seen in Figure 4.18. Two main
peaks are visible, with the main peak before HER decreasing in intensity, and the
peak shifted by ca −1 eV from the main peak increasing in intensity during HER.
Comparison with the spectra for nanocrystalline MoS2 indicated that the change
during electrolysis corresponds to a conversion of MoS3 to MoS2. The inelastic
mean-free path of the electrons being ionized from the Mo sites was estimated to
be 2.2 nm, indicating that the spectral changes correspond to changes in the nearsurface region. Previous studies have shown that MoS2 nanoparticles are present
on the support as flat polygons made up of trilayer S-Mo-S structures, where the S
edges are active for HER[49]. The thickness of one S-Mo-S trilayer is about 3 Å, a
tenth of the penetration depth of the excited electrons, so it is not possible to connect the active site structure to the observed XPS shifts directly without modeling
the active site and calculating the shifts for representative structures.
To further the understanding of the active site for HER, we carried out DFT calculations to model flat trilayer structures corresponding to both MoS3 and MoS2.
Previous theoretical studies have indicated that the experimental conditions decide the S coverage on the Mo edges[239, 240]. To attempt to find the most
stable S edge coverages at HER conditions, free energy calculations were carried out to examine the stability of different Mo edge structures at varying partial
pressures of H2S. In these calculations, the electrode potential was accounted for
using the CHE model (further details of this method are found in the work of
Bollinger et al. [240]). It was found that, under HER conditions (U = 0 V vs RHE,
pH2S = 1 × 10−6 bar, the pressure chosen based on the arbitrary standard for corrosion resistance of a material[241]), the most stable Mo-S edge coverage was
θS = 0.5 ML and θH = 0.25 ML. Thus, under HER, the active edge of the catalyst
should correspond to coordinatively unsaturated Mo sites, mirroring the activity of
CUS for OER and ClER on RuO2. XPS calculations were then carried out for Mo
edges with increasing coverage of S, from θS = 0.5 ML (the CUS “MoS2” edge
structure) to θS = 1 ML (the coordinatively saturated “MoS3” edge structure).
The calculated XPS shifts for these structures are seen in Table 4.4. In cases where
multiple unique S anions are present at the edge, the XPS shifts for each unique
anion is indicated. It is seen that the XPS binding energies for the MoS3 structure
is shifted by ca 1.3 eV from the binding energy of the MoS2 structure. S in S
4.4. XPS MODELED USING DFT
71
Figure 4.17: a) The ambient pressure XPS setup used in the study of Sanchez Casalongue et al. [20]. b) A two-electrode CV performed using the setup, indicating
a cathodic current due to hydrogen evolution at cathodic voltages. Reprinted with
permission from Sanchez Casalongue et al. [20]. Copyright 2014 American Chemical Society.
72
CHAPTER 4. RESULTS AND DISCUSSION
Figure 4.18: a) The changes in the S 2p photoelectron spectra as hydrogen evolution starts and progresses. b) Photoelectron spectra measured during HER on
a nanocrystalline MoS2 material. Reprinted with permission from Sanchez Casalongue et al. [20]. Copyright 2014 American Chemical Society.
4.4. XPS MODELED USING DFT
73
Table 4.4: Calculated S 2p XPS shifts, versus the MoS2 θS = 0.5 ML structure, for
different θS .
S coverage
Coordination of ionized S atom
XPS shift vs 50% coverage
50%
62.5%
62.5%
62.5%
75%
75%
87.5%
87.5%
S monomer
S monomer (surrounded by two S monomers)
S monomer (surrounded by one S dimer and one S monomer)
S atom part of an S dimer (surrounded by S monomers)
S monomer (surrounded by S dimers)
S atom part of an S dimer (surrounded by S monomers)
S monomer (surrounded by S dimers)
S atom part of an S dimer (surrounded by S dimers)
S atom part of an S dimer
(surrounded by one S monomer and one S dimer)
S dimer
0.0
-0.05
-0.80
1.92
-1.17
1.87
-0.11
1.16
87.5%
100%
1.04
1.34
dimers (on saturated Mo cations) are associated with positive XPS shifts of 1.3
to 1.9 eV. This corresponds well with the shift between the two peaks that shift
in intensity during HER (Figure 4.18). The peak corresponding to S dimers (and
an S coverage above 50%) decreases in intensity, while the peak corresponding
to S monomers (and a 50% S coverage) increases in intensity, as HER starts and
progresses. The activity is thus correlated with the presence of CUS Mo sites on
the edges of MoS2. This analysis indicates that when an amorphous MoS3 catalyst
is used, an activation occurs during which edge sites are converted to the MoS2
structure.
4.4.2
Structural changes in RuO2 during HER
The HER reaction on RuO2 is associated with structural changes, including an
expansion of the rutile lattice as detected using XRD[51–53, 242, 243], and it
has been suggested that these structural changes are due to formation of first
RuO(OH)2 before a final reduction to Ru metal[51]. These structural changes
occur concomitantly with an activation of the electrode for HER[52, 242, 243].
The RuO2 electrode is stable while being used as a cathode in e.g. industrial chloralkali or chlorate production, but is known to degrade quickly if it is exposed to
reverse currents that can occur during shut-down of an electrolyzer[50]. Näslund
et al. [51] connected this behavior with their proposed mechanism of step-wise
reduction of RuO2 to Ru metal.
The experiments performed by Näslund et al. [51] consisted of XPS measurements performed on six different electrodes of RuO2 deposited on Ni substrates
that had been used for HER at a current density of −6.4 kA/m2 and a temperature of 90 ◦C in 8 M NaOH for either 0.5 h, 3 h, 6 h, 12 h, 18 h or 24 h. The
electrodes had been prepared by conventional thermal decomposition of RuCl3
74
CHAPTER 4. RESULTS AND DISCUSSION
Table 4.5: The observed Ru 3d binding energies found by Näslund et al. [51], as
well as the respective XPS shifts versus the Ru 3d 5/2 peak. The proposed identification for the peaks are also indicated. If the peak at 289 eV was due to a new Ru
oxyhydroxide phase, it would be the Ru 3d 3/2 doublet peak, and the corresponding
Ru 3d 5/2 peak would be found at approximately 285 eV.
BE / eV
ca 280
280.8
Shift vs Ru 3d 5/2 / eV
−0.8
0
282.5
1.8
284.9
4.2
ca 289
8.6
Proposed identification of peak
Ru metal[51], bare RuO2 CUS sites[246]
Ru 3d 5/2
RuO3[247–251],
unscreened core-hole[252], O on RuO2 CUS [253],
surface plasmon excitation[254]
Ru 3d 3/2
Unscreened core hole[102], RuO(OH)2[51],
CO3[244]
dissolved in 1-propanol, followed by a high-temperature calcination. The XPS
spectra obtained are seen in Figure 4.19. The proposed identification of the peaks
in the Ru 3d spectra as well as alternative interpretations are found in Table 4.5. As
is indicated, the peak inversion occurring during electrolysis, as well as the peak
appearing at ca 289 eV, was interpreted as due to a new RuO(OH)2 phase, while
the additional peak appearing at ca 280 eV was interpreted as due to formation of
metallic Ru. It should be noted that the peaks interpreted as due to RuO(OH)2 have
elsewhere been interpreted as due to carbon contamination[244, 245]. The appearance of XPS C 1s peaks due to carbon contamination at these binding energies
is widely known in literature, and is even used to calibrate XPS spectra[20]. The
changes in the O 1s spectra were, however, also interpreted as due to formation of
RuO(OH)2.[51]
In our work[19], we examined whether the interpretation of Näslund et al. [51]
could be supported by calculated XPS shifts for ruthenium (oxy)hydroxides. We
calculated XPS binding energies for a number of different structures, Figure 4.20,
including pure RuO2 and RuO4, O, H, C, CO and CO3 adsorbates on the (110) surface of RuO2, hydrogen (both atomic and molecular) introduced into bulk RuO2,
structures representative for Ru (oxy)hydroxides as well as metallic Ru and ruthenium carbide. As no accurate experimental structural data exists for any Ru (oxy)hydroxide, we used structural analogs of Ru or related transition metal compounds,
and replaced ions in the structure with O and H to find structures representing Ru
(oxy)hydroxides. These analogs were obtained from the Materials Project[255] as
well as from the Crystallography Open Database[256–258]. For these structures,
we carried out global optimization using minima hopping[259, 260] to find any
related Ru structures with higher stability. The details about the structural analogs
4.4. XPS MODELED USING DFT
75
Figure 4.19: Ru 3d (left) XPS spectra of RuO2/Ni cathodes and metallic Ru and
O 1s (right) XPS spectra of RuO2/Ni cathodes. (a) is a reference RuO2/Ni sample,
that has not been exposed to the electrolyte, (b)-(g) are measurements on electrodes
used for hydrogen evolution for either 0.5 h, 3 h, 6 h, 12 h, 18 h or 24 h, respectively
and (h) in the left figure is a spectrum from metallic Ru. Figures reprinted with
permission from Näslund et al. [51]. Copyright 2014 American Chemical Society.
76
CHAPTER 4. RESULTS AND DISCUSSION
and structural relaxations (local and global) are found in the paper[19].
Ru 3d, O 1s and C 1s XPS binding energies were calculated for all the structures
considered in the study. The analysis was based on chemical shifts calculated versus either the 3d BE of Ru in bulk RuO2, in the case of Ru 3d and C 1s shifts, and
versus the O 1s BE of O in bulk RuO2, in the case of O 1s shifts. Of the structures
considered, only the Ru 3d XPS shift of RuO4 is consistent with the 4 eV shift of
the 289 eV peak, interpreted as corresponding to RuO(OH)2[51]. However, it is
very unlikely that RuO4, a more oxidized, meta-stable, form of RuO2 with a boiling point of 40 ◦C[261] would form during cathodic polarization at 90 ◦C. While
the peak shifted by ca −0.8 eV could be due to formation of Ru metal, the presence
of other structures and structural motifs, including the bare or H-covered CUS on
RuO2 and Ru sites in the Ru Gibbsite (001) surface would also result in an XPS
peak with that shift. Experimental C 1s BEs for e.g. carboxyl groups[245] can
explain the peak shifted by 9 eV from the RuO2 3d 5/2 peak. Furthermore, the peak
inversion observed in Figure 4.19 might be explained by surface carbons such as
CO and CO3, shifted by 4.1 eV and 5.1 eV from the RuO2 3d 5/2 peak, respectively.
From these results, we are forced to conclude that the appearance of what might
look like a new spin-orbit split doublet peak at 285 eV and 289 eV might instead
be due to carbon contamination in the near-surface region of the electrode. Considering that the electrodes were prepared by thermal decomposition of precursor
chlorides dissolved in isopropanol, this might not be implausible.
Furthermore, it was found that the evolution of the O 1s XPS spectra, indicating
a peak shift of 2 eV, can be explained by the hydrogenation of oxygens in the
rutile RuO2 structure. The O 1s XPS BEs of OH groups in the rutile structures
were found to be shifted by around 2 eV from the O 1s XPS BE in RuO2. The
much smaller shift associated with introduction of molecular H2 into RuO2, 0.8 eV,
indicates that the XPS intensity changes occurring during HER on RuO2 are due
to introduction of H rather than H2 into the coating. Furthermore, upon relaxing
bulk RuO2 structures with H inside, which resulted in conversion of O groups into
OH groups, the structures exhibited similar volume increases as those observed
experimentally during HER on RuO2[51–53, 242, 243].
These results can be summarized as follows. During HER, H enters the RuO2
structure, and converts the structure into a Ru(OH)3 structure. This explains the
volume increase of the lattice, as well as the observed XPS shifts. The large positive XPS shifts observed by Näslund et al. [51] during HER are likely due to
carbon contamination, either from the environment during the electrolysis, or due
to residual carbon inside the coating from the solvent used in the preparation of
the electrode. The additional peak appearing at a slightly lower binding energy
than the Ru 3d 5/2 peak could be due to formation of metallic Ru, but it is more
likely that this peak is due to either Ru sites in hydroxidic regions of the outer
surface of the coating, or indeed to bare or H-covered CUS sites on the electrode
surface. This latter result suggests that a reason for the activation of RuO2 that has
4.4. XPS MODELED USING DFT
1
2
4
6
77
3
5
7
8
9
Figure 4.20: The relaxed bulk structures of RuO2 (1) , RuO4 (2), RuOOH (3, based
on FeOOH), RuO(OH)2 (4, based on RuOCl2), Ru(OH)3 (5, based on Al(OH)3 in
the Gibbsite structure), the second Ru(OH)3 structure (6, based on arbitrary placement of OH around Ru in a periodic cell) and its more stable form found through
minima hopping (7), ruthenium carbide (RuC, 8) and finally ruthenium metal in
the hcp structure (9). Red circles represent oxygen, white circles hydrogen, green
spheres ruthenium and grey spheres carbon. The (001) surface of the Gibbsite-like
structure (5) is obtained by making a cut between two Ru-O-H layers.
78
CHAPTER 4. RESULTS AND DISCUSSION
been noted during cathodic polarization might be a gradual removal of surface O,
remaining from the electrode preparation and exposure to air, opening new active
sites for HER.
Chapter 5
Conclusions and
recommendations
5.1
Conclusions
In the study of Hummelgård et al. [15], we found that the doping of DSA with Co
resulted in activation of the electrode for hydrogen evolution, an effect largely due
to increased active surface area.
In the study of Sandin et al. [14], we found indications for a mechanistic connection between the homogeneous chlorate formation reaction and the uncatalyzed
formation of oxygen during hypochlorite decomposition. Further understanding
of this connection might be used to control the selectivity, to enable more selective
chlorate formation. Additionally, it was found that the oxygen evolution is thirdorder with respect to hypochlorite. The rate of the reaction has a maximum at pH
6.5, indicating the involvement of both HOCl and OCl–. We also found indications
that higher oxidation states of Ir catalyze chlorate formation.
In the theoretical studies of Karlsson et al.[16, 17], we found that doping rutile TiO2 with transition metals, including Ru (as in DSA), As, Bi, Co, Ir, Mn,
Pd or Ru, activates Ti surface sites for selective electrochemical chlorine evolution. This is a fundamental explanation for the observed experimental selectivity
trends of Ru-doped TiO2 (DSA). Furthermore, the finding that doping with arsenic should result in a high selectivity regardless of whether the As dopant resides in the surface CUS active site or close to Ti sites is intriguing, as arsenic
is very cheap[17, 262]. In other cases, a higher degree of control of the atomic
level arrangement of dopants is required to achieve the results indicated by our
predictions. In general, the results show that it seems possible to adjust the elec79
80
CHAPTER 5. CONCLUSIONS AND RECOMMENDATIONS
trocatalytic properties of TiO2 from optimal OER activity to optimal ClER activity
and selectivity.
We also attempted to fit a force field for rutile Ru-Ti mixed oxides[18]. While an
improved interatomic potential for TiO2 and for the structure of the mixed oxide
was found, the energetics of the force field when it comes to modeling the relative
energies of different Ru-Ti arrangements were incorrect.
Finally, in the papers of Sanchez Casalongue et al. [20] and Karlsson et al. [19],
computational modeling was used to understand results from XPS studies of HER
on MoS2 and RuO2. In the first study, we found that during HER on MoS2, the
edge structure changes from the amorphous MoS3 structure, where two S anions
are found on each Mo cation of the edge, to an MoS2 structure. In the second
study, we found that the XPS shifts and structural changes which are noted during
HER on RuO2, can be explained by incorporation of H into the rutile RuO2 lattice,
converting O groups into OH groups. XPS shifts that had previously[51] been
attributed to formation of Ru metal are more likely associated with either formation
of such OH groups, or with removal of O from RuO2 surface sites, while large
positive chemical shifts that had been attributed to formation of a new RuO(OH)2
phase could only be attributed to surface carbon contamination.
5.2
Recommendations for future work
The results in the present thesis show that atomistic modeling, both semi-classical
and ab initio, is a tool that now has utility in studies of heterogeneous electrocatalysis. As Gurney [263] stated already in 1931,
“As quantum mechanics endows particles with entirely new properties, it enables us to deal with problems which have remained unsolved for many years. In electrolysis we have been unable to visualize the physical processes which underlie some of the most elementary phenomena. Thermodynamics gives a consistent account
of them, independent of any mechanism; but when we try to unravel
the actual processes their complexity is baffling. Quantum mechanics
provides a new line of attack.”
It is just during the last few years that it has been possible to start exploring
this “new line of attack”. By using the screening possibilities afforded by modern ab initio methods, the design of electrocatalysts can go beyond the “try and
see”[13] methodology that has characterized electrocatalyst design (and the design of heterogeneous catalysts) until now[153, 264]. Nevertheless, although the
tools are now available to be able to answer the question “What electrode composition should be used?”, the answer to the question “How can such an electrode
5.2. RECOMMENDATIONS FOR FUTURE WORK
81
be made?” still remains to be answered. Here a combination of theoretical and
experimental research is needed.
I would also like to make some detailed recommendations for further theoretical
work. First off, quantum chemical methods could be used to try to explore the
chlorate formation mechanism, with an aim to e.g. try to examine the properties
of the proposed “H2Cl2O2” intermediate. Using a detailed theoretical model for
the chlorate formation mechanism, the possible catalytic effect of ions or solids
might also be understood. However, the chlorate formation mechanism is likely a
complicated multistep reaction[203, 265], so that computational methods that do
not require human intervention to examine different reaction pathways[266] could
aid the search.
Continuing next to the computational screening of activity and selectivity of doped
oxides, the next step would be to broaden the possible applicability of the ∆E(Oc )
descriptor. It is possible that the same descriptor could be used to account also for
formation of higher chlorine oxides (e.g. hypochlorite, chlorate and perchlorate),
so that it might be possible to suggest electrodes that could be used for selective
direct production of these chemicals. As an example, Viswanathan et al. [220]
has suggested that rutile oxides with slightly higher ∆E(Oc ) than those for optimal
ClER could also be selective for H2O2 production. The other area that should be
explored is a verification of the accuracy of the results. As has been described in
some detail, the SIE in adsorption energy calculations on the DFT GGA level for
mixed oxide systems is not straightforward to estimate, and more accurate methods
such as RPA[267] (or improvements on the RPA[161, 211]) or QMC[212, 268,
269] might be needed to examine the accuracy of the results. GGA+U calculations
are not likely to correct for the SIE without resulting in a worse description of other
chemical properties. Regardless of whether GGA or GGA+U calculations are to
be used for larger screening studies, usage of higher-level methods (e.g. RPA)
for benchmarking is recommended. An area that also deserves more attention is
the effect of hydration of the electrode surface on the activity trends that have
been identified, although the studies of Rossmeisl et al. [107] and Siahrostami and
Vojvodic [114] have shown that the effect should be small.
Ab initio theoretical studies are limited by the availability of computational resources. For this reason, further attempts to use semi-empirical methods such as
force fields are motivated. However, as is indicated by our attempt to fit a new
force field for DSA[18], it is not necessarily simple to design a force field that
can describe all desired properties. Further work on automatic fitting of force
fields[130], to avoid the need for semi-manual fitting, is desired. As force-fields
clearly have limitations, another avenue of work that might hold more promise
is semi-empirical methods such as density-functional tight-binding (DFTB)[270].
The continual decrease in the cost of computational resources also means that the
cost of using accurate methods (e.g. LDA or GGA-level DFT) that are too expensive for present-day studies of e.g. nanoparticles soon might become acceptable.
82
CHAPTER 5. CONCLUSIONS AND RECOMMENDATIONS
When it comes to experimental studies, the main recommendation is that our
predictions[16, 17] regarding the selectivity and activity of doped TiO2 should be
tested in practice. For verification of the theoretical results, tests using the model
electrodes indicated in Karlsson et al. [16] should be made, while the predictions
resulting from computational screening[17] could serve as inspiration for studies
of other dopants in TiO2, including dopants that appear never to have been studied
for the purpose of ClER or OER electrolysis.
Although the present thesis has had a main focus on ab initio modeling tools that
have come into general use only recently, it cannot be said that the experimental tools that are available are the same today as they were almost 50 years ago,
when the DSA was introduced. In particular, the automatization of laboratory
equipment that is now becoming standard allows for experiments, such as might
be exemplified by those in our study[14], where changes in both liquid phase and
gas phase are measured with high accuracy and time resolution. Furthermore,
in studies of electrodes, attempts should be made to separate apparent kinetics
from true electronic effects, e.g. normalizing the apparent current density by the
electrochemically active surface area[15, 271]. This is unfortunately not the rule
even today. The possibility of using in-situ methods to study changes on electrode surfaces, as done in the study of Sanchez Casalongue et al. [20], should
also be explored further. New insight can also be gained from the rapid development of X-ray spectroscopic methods that has occurred only during the last
two decades[181, 272]. The combination of such spectroscopic studies, giving
insight into the atomic-level structure of oxides, with electrochemical studies, is
well exemplified by the study of Sanchez Casalongue et al. [20] and of Krtil and
Rossmeisl with co-workers[273–275] (although we have questioned some of their
conclusions[17]). It is regrettable that such an approach was not assumed in our
study on Co-doped DSA[15], as more interesting results regarding the possible
electronic effects of Co-doping in DSA and RuO2[273–275] might have been obtained. Finally, it should be stressed that much can be gained by combining experimental studies with ab initio simulations, as it is difficult to draw detailed
conclusions from e.g. experimental XPS results[51] without the direct connection
between structure and e.g. XPS shifts that computational modeling can supply[19].
Lastly, some recommendations for further work specifically focused on the chlorate process will also be made. In general, the overall goal can be said to be
to improve the understanding of the reactions that occur in the process, i.e. the
chlorate formation reaction itself and homogeneous and possibly heterogeneous
decomposition of hypochlorites to form oxygen. Detailed knowledge about these
reactions is presently based mainly on experiments under conditions that differ
from those of the actual process. Future studies should attempt to work under
conditions close to the industrial ones. Such work is important especially when it
comes to the effects of electrolyte contamination, as it seems that very few studies
of chlorate formation catalysis or hypochlorite decomposition (to form oxygen)
5.2. RECOMMENDATIONS FOR FUTURE WORK
83
under industrial conditions exist. Another area that requires further study is the
details of the connection between current density and the selectivity for oxygen
evolution, as there seems to be a minimum in the selectivity for O2 at the critical
anode potential[73].
84
CHAPTER 5. CONCLUSIONS AND RECOMMENDATIONS
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R. J. Dubos, Louis Pasteur - Free Lance of Science, Little, Brown and Company (1950).
Let me tell you the secret which has led me to the goal:
My only strength resides in my tenacity.
— Louis Pasteur[276]
Chapter 7
Acknowledgments
First, let me acknowledge the funding for this project, which was received from
Permascand AB and the Swedish Energy Agency, within the program “Effektivisering av industrins energianvändning”.
Many people have supported me during these last years. First off, I’d like to thank
my two main supervisors, Ann Cornell and Lars G. M. Pettersson. Your interest,
patience, direction and deep knowledge have helped me every step along the way.
It has been a pleasure to work with you, and I am thankful to have been able to
learn so much from both of you.
I would also like to thank my co-supervisors Joakim Bäckström, Göran Lindbergh
and Susanne Holmin for their support and helpful discussions. I would also like
to thank John Gustavsson, for introducing me to the experimental side of electrocatalysis research, and Henrik Öberg, for introducing me to the hands-on use of
density functional theory. I would also like to thank Christine Hummelgård for
a productive and interesting collaboration during the work on Co-doped DSAs.
Thanks also to Heine Hansen and Thomas Bligaard for hosting me for a week
at SLAC, and for teaching me much about theoretical electrochemistry. Thanks,
Richard Catlow, Alexey Sokol and Scott Woodley for being very gracious hosts
during my two visits to UCL, and I would like to thank you for an interesting
few weeks of arduous interatomic potential-tweaking. I would also like to thank
Staffan Sandin for our years of working together on “hypo” decomposition. I have
also enjoyed working with Lars-Åke Näslund on several projects.
During this project, I have gained much from regular discussions with staff at both
Permascand and AkzoNobel Pulp and Performance Chemicals. These discussions
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have made my work much more interesting by giving me a firm grounding in
what questions are interesting from the industrial point of view. I’d like to thank
Lars-Erik Bergman, Erik Zimmerman, Ingemar Johansson, Bernth Nordin, John
Gustavsson, Susanne Holmin and Fredrik Herlitz from Permascand. I’d also like
to thank Johan Wanngård, Kristoffer Hedenstedt, Johan Lif, Nina Simic, Magnus
Rosvall, Kalle Pelin, Adriano Gomes and Mats Wildlock from AkzoNobel Pulp
and Performance Chemicals.
Much of the work that has resulted in this thesis has been computational. Throughout these last few years, I have often gotten help from Peter Gille (PDC, KTH) with
all kinds of matters ranging from cryptic compiler messages to general Linuxrelated questions. Thanks, Peter! Furthermore, the helpful community surrounding the GPAW code has been a great help in my learning and usage of that code. In
connection with that, I would especially like to thank Ask Hjorth Larsen, Jens Jørgen Mortensen and Marcin Dulak. The calculations could not have been performed
without the computing resources of the HPC2N, and I would like to acknowledge
the support provided by their staff. Furthermore, I would like to thank Najeem
Lawal for providing access to the Goliat computer at the Mid Sweden University.
The friendly and supportive spirit provided by past and present colleagues at TEK
and EP, KTH and Atomic Physics and Chemical Physics, SU, is much appreciated.
You have made these past years a pleasure.
I would like to thank Sigbritt, Thore, Jesper and Mathilda for everything they have
done for me, and for all their support.
Finally, I want to express my gratitude to my wife Lina. Your support and encouragement has helped me get to this point. We’ve made it through these last nine
years at KTH hand in hand, through almost every course and “tentaplugg” and
now (soon) through our PhD studies. Tack för allt! Jag har tur som träffade dig.