The Lithium-Sulfur Battery:
Design, Characterization, and
Physically-based Modeling
Von der Fakultät für Energie-, Verfahrens- und Biotechnik
der Universität Stuttgart zur Erlangung der Würde eines
Doktor-Ingenieurs (Dr.-Ing.) genehmigte Abhandlung
Vorgelegt von
David Norman Fronczek
aus Würzburg
Hauptberichter:
Mitberichter:
Tag der mündlichen Prüfung:
Prof. Dr. Wolfgang G. Bessler
Prof. Dr. K. Andreas Friedrich
Prof. Dr. Stefano Passerini
1. Dezember 2015
Institut für Thermodynamik und Wärmetechnik
der Universität Stuttgart
2015


4
Contents
Abstract 9
1 Introduction 11
1.1 The need for better batteries . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 The lithium-sulfur system . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Approach and scope of this work . . . . . . . . . . . . . . . . . . . . . . . 15
2 Experimental work on lithium sulfide cathodes – materials and methods 17
2.1 The goal – coated lithium sulfide . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Active material preparation . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Ball milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 Chemical vapor deposition . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Other materials and cell assembly . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Anode materials and preparation . . . . . . . . . . . . . . . . . . 23
2.3.2 Separator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.3 Electrolyte materials and preparation . . . . . . . . . . . . . . . . 24
2.3.4 Cathode additives . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.5 Electrode preparation . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.6 Cell assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Analysis and characterization methods . . . . . . . . . . . . . . . . . . . 29
2.4.1 Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.2 Chemical/physical methods . . . . . . . . . . . . . . . . . . . . . 30
2.4.3 Electrochemical methods . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Dead ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Experimental work on lithium sulfide cathodes – results and discussion 34
3.1 Analysis of materials and electrodes . . . . . . . . . . . . . . . . . . . . . 34
3.1.1 Scanning electron microscopy of materials . . . . . . . . . . . . . 34
3.1.2 Scanning electron microscopy of electrodes . . . . . . . . . . . . . 39
3.1.3 Transmission electron microscopy . . . . . . . . . . . . . . . . . . 44
3.1.4 Chemical/physical tests . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.5 Electrolyte optimization . . . . . . . . . . . . . . . . . . . . . . . . 48
5
3.2 Analysis of full cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.2 Metrological considerations . . . . . . . . . . . . . . . . . . . . . . 53
3.2.3 Slow cyclic voltammetry . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.4 Impedance spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.5 Charge/discharge profiles . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.6 Cycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3.2 Possible improvements and outlook . . . . . . . . . . . . . . . . . 66
4 Modeling the lithium-sulfur battery – theory and methods 68
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Modeling approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.1 Model design and basic assumptions . . . . . . . . . . . . . . . . 71
4.2.2 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.3 Electrochemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.4 Electrical properties . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.5 Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.6 Reaction mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.7 Degradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3 Simulation methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3.1 Implementation and computational framework . . . . . . . . . . 85
4.3.2 Parametrization and validation techniques . . . . . . . . . . . . . 87
4.3.3 Quality control and error analysis . . . . . . . . . . . . . . . . . . 88
5 Modeling the lithium-sulfur battery – calibration and simulation results 91
5.1 Global two-step model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Multi-step model – literature parametrization . . . . . . . . . . . . . . . 97
5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2.2 Discharge at different rates . . . . . . . . . . . . . . . . . . . . . . 97
5.2.3 Cycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2.4 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3 Multi-step model – reparametrization . . . . . . . . . . . . . . . . . . . . 104
5.3.1 Parameters determined from independent sources . . . . . . . . 104
5.3.2 Parameters calculated from experimental data . . . . . . . . . . . 106
5.3.3 Parameters determined by equilibration . . . . . . . . . . . . . . 106
5.3.4 Parameters fitted to experimental data . . . . . . . . . . . . . . . 107
6
5.4 Multi-step model – validation attempts . . . . . . . . . . . . . . . . . . . 116
5.4.1 Rate capability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.4.2 Cyclic voltammetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.5 Multi-step model – further simulations . . . . . . . . . . . . . . . . . . . 120
5.5.1 Galvanostatic charge/discharge profiles . . . . . . . . . . . . . . 120
5.5.2 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.5.3 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.5.4 Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.6 Multi-step model – polysulfide shuttle extension . . . . . . . . . . . . . . 129
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.7.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.7.2 Shortcomings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.7.3 Possible improvements and outlook . . . . . . . . . . . . . . . . . 134
6 Conclusions and outlook 137
6.1 Summary of achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2 Limitations of this work and ideas for future research . . . . . . . . . . . 140
6.3 Outlook beyond academia . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A Appendix 143
A.1 List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A.2 Calculation of specific capacities, energy densities and state of charge . 146
A.3 List of chemicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
A.4 List of equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
A.5 Parameter sets for the models . . . . . . . . . . . . . . . . . . . . . . . . . 150
Bibliography 153
Cited publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Own publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
Academic résumé of David N. Fronczek 184
Acknowledgments 186
7
List of abbreviations
AN negative electrode (anode during discharge)
AFM atomic force microscopy
BET Brunauer, Emmett & Teller method (for determining surface areas)
CA positive electrode (cathode during discharge)
CCCV constant current – constant voltage (charging protocol)
CFD computational fluid dynamics
CTR charge-transfer reaction
CV control volume – or – cyclic voltammetry
CVD chemical vapor deposition
DAE differential-algebraic equation
DFT density-functional theory
EIS electrochemical impedance spectroscopy
EL battery separator (filled with liquid electrolyte)
elde electrode
elyte electrolyte
FVM finite volume (discretization) method
MEA membrane-electrode assembly (i.e. cathode, separator, and anode)
OCV open-circuit voltage
PDE partial differential equation
P&ID piping and instrumentation diagram
ppm parts per million
PS polysulfide(s)
RTIL Room-temperature ionic liquid
SCCM standard cubic centimeter per minute (flow rate)
SBR styrene butadiene rubber
SEI solid electrolyte interphase
SEM scanning electron microscopy
SoC state of charge
SoH state of health
TEM transmission electron microscopy
8
Abstract
The lithium-sulfur (Li/S) battery is a promising candidate for next-generation electro-
chemical energy storage. Its unique combination of electrochemical performance, cost
effectiveness, and environmental sustainability are unprecedented among battery ma-
terials. As of today, however, Li/S batteries are only used for few niche applications; a
broader adoption of this technology is impeded by the yet unsatisfactory energy effi-
ciency, self discharge, and limited lifetime. This work contributes to the advancement
of Li/S technology in two respects: First, a novel kind of positive electrode, based on
coated lithium sulfide (Li2S), was prepared, tested and optimized. Second, the under-
standing of the complex chemical and physical processes in the cell was improved by
creating and utilizing a computational model of the Li/S battery.
For the experimental part of this work, a chemical vapor deposition process was
developed to apply a carbon coating to Li2S particles. The focus of this work was on
the optimization of the process chain from commercially available chemicals to the
final coin cell in general and on the characterization of the materials and electrodes
during various processing steps in particular.
For the modeling part, an existing multiscale electrochemical modeling framework
was extended to enable full-cell simulations of Li/S batteries. The model includes a
detailed description of electrochemistry, transport, and the evolution of solid phases
in the cell, but also accounts for the electrical double layer and, in a generic fashion,
the microstructure of the electrodes. Finally, a phenomenological description of the
shuttle effect and associated cell degradation was implemented and analyzed. The
parametrization and partial validation of the model makes use of original data col-
lected for this purpose, but also data from literature.
Simulation results comprise charge/discharge profiles, cyclic voltammetry, impe-
dance spectra, and the evolution of the chemical composition of both the electrolyte
and the electrodes over time. The analysis of these results reveals limiting factors and
suggests improved operating conditions.
The apt combination of theoretical and empirical methods enabled an improvement
of the performance and cycle life of the novel cathode material, but also contributed
to a more profound understanding of the Li/S battery.
9
Zusammenfassung
Lithium-Schwefel (Li/S)-Batterien sind eine vielversprechende Technologie für elek-
trochemische Energiespeicher der nächsten Generation. Sie vereinen auf bisher un-
erreichte Weise hervorragende Leistungsdaten, Kosteneffizienz und Umweltverträg-
lichkeit. Bisher kommen Li/S-Batterien jedoch nur für wenige Spezialanwendungen
zum Einsatz. Der Eignung für ein größeres Marktsegment stehen die derzeit noch un-
befriedigende Energieeffizienz, Selbstentladung und Lebensdauer entgegen. Die vor-
liegende Arbeit trägt zur Weiterentwicklung der Li/S-Technologie in zweifacher Weise
bei: Zum einen wurde ein neuartiger Elektrodentyp, basierend auf beschichtetem Li-
thiumsulfid (Li2S), hergestellt, getestet und optimiert; zum anderen wurde durch die
Modellierung und Simulation einer Li/S-Zelle das Verständnis der komplexen chemi-
schen und physikalischen Vorgänge in der Batterie verbessert.
Für den experimentellen Teil der Arbeit wurde ein Verfahren zur chemischen Gas-
phasenabscheidung von Kohlenstoff auf Li2S-Partikeln entwickelt. Das Hauptaugen-
merk der Arbeit lag jedoch auf der Optimierung der gesamten Prozesskette ausgehend
von kommerziell erhältlichen Chemikalien bis zur fertigen (Knopf-)Zelle, insbesonde-
re aber auf der dazu notwendigen Charakterisierung der Materialien und Elektroden.
Für den theoretischen Teil wurde ein skalenübergreifendes elektrochemisches Mo-
dell so erweitert und kalibriert, dass damit Simulationen der Li/S-Batterie möglich
sind. Das Modell enthält eine detaillierte Beschreibung der elektrochemischen Reak-
tionsmechanismen, der Transportvorgänge im flüssigen Elektrolyten und der Ausfall-
und Auflösungsreaktion in den porösen Elektroden. Daneben werden die elektroche-
mische Doppelschicht sowie eine effektive Beschreibung der Mikrostruktur berück-
sichtigt. Schließlich wurde eine phänomenologische Beschreibung des Shuttle-Effekts
sowie der Degradation der Zelle implementiert und ausgewertet. Die Parametrierung
und teilweise Validierung des Modells erfolgte anhand der eigenen experimentellen
Daten sowie anhand ausgewählter publizierter Daten und Ergebnisse aus der Litera-
tur. Mit dem solchermaßen kalibrierten Modell wurden Lade- und Entladekurven, Cy-
clovoltammetrie, Impedanzspektren sowie die zeitliche Entwicklung der chemischen
Zusammensetzung des Elektrolyten und der Elektroden simuliert. Von den Ergebnis-
sen lassen sich unter anderem Rückschlüsse auf limitierende Prozesse oder günstige
Betriebsbedingungen ziehen.
Durch die Kombination von theoretischen und empirischen Verfahren konnten die
Leistung und Lebensdauer des neuen Kathodenmaterials erhöht sowie gleichzeitig ein
Beitrag zum besseren Verständnis der Li/S Batterie geleistet werden.
10
1 Introduction
Today, batteries are ubiquitous and many appliances and products we use in our daily
life would not be possible at all without portable and affordable electrochemical energy
storage. Still, further advances in capacity, lifetime, and cost of batteries will open up
new possibilities and markets. Some of the future uses have long been anticipated, but
could not yet be put into practice because of these limitations, while others may have
been unthought of before.
1.1 The need for better batteries
Since the emergence of modern batteries in the mid 19th century, their performance
has been improving continuously [1]. For decades, however, batteries were not widely
used except for small gadgets (clocks, torches, remote controls, ...) and as starting,
lighting, and ignition (SLI) automotive batteries. The situation changed with the
advent of portable electronics in the 1980s and the pace at which battery technol-
ogy evolves has increased ever since [2, chap. 2]. Durability and safety of batteries
were mostly improved by engineering and design optimizations, whereas capacity
and power density saw the biggest improvements whenever new battery chemistries
were introduced to the market. Looking at Fig. 1.1, one can see that actually most of
the improvement in terms of energy density up to date was due to new chemistries.
Especially when Li-ion batteries were introduced in the 1980s [3, 4], they enabled many
new applications including truly portable communication, sensors, and entertainment.
While the cheaper Pb-acid batteries still dominate the SLI and backup power mar-
ket, today’s high-capacity state-of-the-art rechargeable batteries are almost exclusively
Li-ion systems with graphite anodes and intercalation cathodes made from various
host materials such as cobalt oxide or olivines [5]. Even though one has to be care-
ful when comparing performance metrics for different cells without full knowledge
about data acquisition and validation [6], the upper limits in terms of energy den-
sity are commonly agreed to be well below 500 Wh/kg for Li-ion batteries, a value
that is considered to be too low for many practical applications including long-range
electromobility [7, 8]. Additionally, the cost of today’s technologies is too high for
many large-scale applications, such as grid storage [9]. Finally, there is a growing
concern about the supply, processing and recycling of battery materials both in terms
of technical availability and cost [10–13].
11
> 400 km
> 225 km
> 200 km
160 km
100 km80 km50 km
> 550 km
1000
800
600
400
200
0
Price (USD·kWh−1) 200 600 900 600 < 150 < 150 < 150 < 150
Available Under development R&D
PB-acid Ni-Cd Ni-MH Future Zn-air Li-S Li-air
Sp
ec
ifi
c
en
er
gy
(W
h·k
g−
1 )
Li-ion
Today
Li-ion
Figure 1.1: Energy density of various battery chemistries compared. The lighter re-
gions of the bars show the range of anticipated practical energy densities.
Reproduced from [14].
New chemistries are not invented out of the blue, though. From a theoretical point
of view, the properties of many potential future battery systems are well known, in-
cluding those of Li/S. The challenge is to exploit this potential without compromising
security, reliability, or durability known from established materials [15, 16]. In Ger-
many, two national goals were formulated that cannot be achieved without significant
investments in battery development, production, and deployment: the “energy revo-
lution” and the advent of electromobility. Regarding the former, 80 % of Germany’s
total electrical energy (and 60 % of the total gross domestic energy consumption) shall
be supplied by renewables no later than 2050 [17, pp. 4f]; regarding the latter, one mil-
lion electric cars shall be deployed by 2020 [18, 19]. Several national research programs
support these goals [19–23] and some noteworthy results have already been obtained
[23, 24]. In the U.S., national goals include 25 % of electricity consumed to be generated
by cost-competitive solar and wind power by 2025 as well as one million all-electric or
plug-in hybrid electric vehicles on the road by the same time [25]. Founded in 2012,
the JCESR [26] is funding and conducting research to meet these targets. Also, there
are dozens of ARPA-E1 projects dedicated to the development of advanced batteries.
Similar programs exist in many industrialized and developing countries worldwide.
Finally, there are several high-stake industrial ventures such as the USABC [27] or
KLIB [28] which support research and development of next-generation batteries as
part of their mission.
1Advanced Research Projects Agency-Energy, U.S. Department of Energy, 1000 Independence Ave. SW,
Washington, DC 20585, USA
12
140
100
60
20
Selling price
(150 USD/kWh)
Calendar Life (10 years)
Cycle Life (1000 Cycles)
Operating Temperature
Range (−40 to +50 °C)
Specific Power (300 W/kg)
Power Density
(460 W/l)
Specific Energy
(150 Wh/kg)
Energy Density
(230 W/l)
Figure 1.2: A “spider plot” showing the various requirements for batteries in all-
electric vehicles (numbers = 100 % = USABC targets). The shaded area
represents commercial Li-ion technology in 2006. Reproduced from [35].
While advances in Li-ion technology are to be expected [29, 30], their maximum spe-
cific capacity and energy density are constrained by fundamental limits, see appendix
A.2. Also, the requirements for batteries are complex (Fig. 1.2) and Li-ion based cells
may not fit every application’s profile. This is where lithium-sulfur (Li/S) batteries
come into play: They have the potential to overcome most of the aforementioned limi-
tations by providing considerably increased energy density at a lower cost [31, 32].
Although a thorough analysis of all raw materials and processing costs of this type
of cell at an industrial scale has not yet been done in detail, there is a very promising
case study by students of the Haas School of Business2, cf. Ref. [33]. Additionally,
sulfur, as well as all other components of sulfur cathodes, are relatively abundant,
environmentally benign and conflict-free [10–13, 34].
1.2 The lithium-sulfur system
The Li/S cell is a conversion cell, where sulfur is reduced from S8 to Li2S according to
the following global reaction:
S8 + 16 Li 8 Li2S+ 3400 kJ/mol .
2Haas School of Business, University of California, 2220 Piedmont Ave, Berkeley, CA 94720, USA
13
Li Li+
S8
Li2S
S2−8
S2−6
S−
S2−4
S2−2
S e
pa
ra
to
r
N
eg
at
iv
e
el
ec
tr
od
e
Po
si
ti
ve
El
ec
tr
od
e
Figure 1.3: General layout of a liquid-electrolyte lithium-sulfur cell and proposed re-
action mechanism.
Assuming pure, solid reactants and products at room temperature, the specific en-
ergy of the Li/S cell is ∼ 2600 Wh/kg [36], its energy density ∼ 2800 Wh/l [29, 37],
and its specific capacity 1675 Ah/kg, cf. appendix A.2. These values are considerably
higher than those of today’s batteries. Taking into consideration that only a fraction of
the theoretical capacity can be practically exploited, it is expected that Li/S cells can
still achieve 2–3 times the performance of the best cells in use today [30, 33, 37].
Li/S cells have been known for decades; the first patent involving Li/S technology
was filed in 1958 [38]. Initially, however, the Li/S system was investigated as a high-
temperature battery only [39, 40], i.e. operating at a temperature above 115 °C, the
melting point of sulfur. In this setup, the positive electrode of the cell is liquid so that
material transport and microstructure are of no concern. This concept is widely used
in sodium-sulfur (Na/S) batteries [41], but the development of high-temperature Li/S
batteries never reached commercialization.
In the late 1970s, first attempts were made to build Li/S cells using liquid elec-
trolytes at room temperature [42, 43]. With this kind of cell, both sulfur and lithium
sulfide are present as solids, but the reduction/oxidation reactions take place among
polysulfide species dissolved in the liquid electrolyte. While the precise reaction mech-
anism is still discussed controversially [44], it definitely includes several dissolved
species of the form Li2Sx and/or S2−x . The general layout of such a Li/S battery along
with the reaction mechanism assumed for this study is shown in Fig. 1.3.
Despite their very attractive properties, Li/S batteries have not yet been introduced
to the battery market, except for some special niches. There are only few companies
offering Li/S cells [45–47], but many more are engaged with research and development
as well as material supply. Besides issues with the commonly used lithium metal
negative electrode [15, 48, 49], there are two main problems causing degradation and
other side effects preventing widespread adoption of Li/S batteries so far: First, there
is the large mechanic stress induced by the expansion and shrinkage of the electrode
due to volume changes of the active material during cycling [50], which eventually
14
leads to structural degradation of the cathode and possibly loss of electronic contact
between the active material and the current collector. Second, there is the polysulfide
shuttle, which has long been known to cause chemical degradation as well as poor
charging efficiency, at least under certain circumstances [51]. On the other hand, the
shuttle provides an inherent means of overcharge protection – a feature sometimes
artificially added to Li-ion batteries [52]. There are many more side reactions and
degradation mechanisms; most of them have not yet been studied in detail [53].
Close to 500 articles on Li/S batteries and related technologies were published in
the past year alone, more than in the entire scientific history before3. Some of these
publications are of purely academical interest, since they require very exotic or ex-
pensive materials, processing, or analysis steps, e.g. [55–57]. Most of them, however,
contribute directly to the improvement of specific material components, electrode and
cell designs or analysis techniques, thereby bringing Li/S batteries closer to actual
applications. These findings gradually enable building better batteries, but more fun-
damental research is needed to truly understand this complex and challenging, but
also very promising electrochemical system.
1.3 Approach and scope of this work
This work contributes to a better understanding of Li/S batteries by analyzing their
properties and performance both experimentally and theoretically. For the experimen-
tal part, a new method for synthesizing carbon-coated lithium sulfide (Li2S) nanopar-
ticles for use in porous electrodes was tested and optimized. Also, various electrolyte
compositions and additives were investigated in terms of cycling performance and
compatibility with the novel electrode. The materials were studied by means of mi-
croscopy, physical and chemical analysis methods. In addition, full cells with lithium
metal counter electrodes were assembled and tested electrochemically.
The modeling part comprises the design, implementation, parametrization, and ap-
plication of a model of the Li/S cell, based on the state of the literature. Extensions
to the existing electrochemical modeling framework Denis [58, 59] were implemented,
enabling computer simulations of the Li/S battery in various kinds of experiments in-
cluding charge/discharge cycling, electrochemical impedance spectroscopy, and cyclic
voltammetry. The model was calibrated, partly validated against experimental re-
sults, and used to investigate the charge/discharge mechanism, rate performance, elec-
trolyte composition and – phenomenologically – the degradation of the cell. Besides
using data available from literature, original experiments were conducted to specifi-
cally record data for the purpose of validating the model. Contrariwise, preliminary
modeling results provided feedback to improve material and electrode design.
3data according to Thomson Reuters Web of Science, April 2014; see also Ref. [54]
15
This thesis is structured in six chapters. Following this introduction, the materials
and experimental methods are described in chapter 2. In addition to listing and de-
scribing the experiments, the rationale behind each material or methodology choice
is explained. Also, the design, setup, and optimization of the carbon coating furnace
and the corresponding protocol is treated in this chapter. The results obtained are
presented and discussed in chapter 3. Besides the successful demonstration of the
use of carbon-coated lithium sulfide in a Li/S cell, valuable information on the mate-
rial and cell level was collected and improvements to the performance and lifetime of
Li/S cells are reported. Chapter 4 contains information on the modeling and model
calibration methodology as well as a description of the Li/S model itself. Some the-
oretical background including detailed explanations of the governing equations used
for transport, thermodynamics, and electrochemistry is also part of this chapter. The
proposed reaction mechanism is discussed along with possible routes of degradation.
In chapter 5, the results of the simulations are presented, interpreted and discussed.
Finally, the last chapter summarizes and interprets the achievements of this work and
the impact on Li/S research. In the closing remarks, suggestions for future research
are discussed alongside with progress in the field of electrochemical energy storage
and its applications in general. Lengthy calculations, listings, and full bibliographic
records can be found in the appendix.
16
2 Experimental work on lithium sulfide
cathodes – materials and methods
The experimental part of this work was carried out from 04/2012 to 05/2013 at Lawrence
Berkeley National Laboratory1 under the supervision of Prof. Elton J. Cairns and in
collaboration with his research group as well as the inorganic chemistry team at The
Molecular Foundry2 and other members of the BATT program [60].
2.1 The goal – coated lithium sulfide
Lithium-sulfur cells have been studied for more than four decades. While the opti-
mization of electrolytes, additives and counter-electrodes is critical for the long-term
success of Li/S batteries, most research currently focuses on the design of the cath-
ode’s microstructure. Many different approaches for optimizing it have been tried
with some success, as discussed in several review articles [14, 30, 37, 61, 62].
Most conventional electrode designs suffer from poor cyclability, which has its ulti-
mate cause in the mechanic stress induced by the swelling and shrinking of the active
material during (de-)lithiation as well as side reactions of the dissolved polysulfides.
Thus, a straightforward approach to mitigate degradation is to prevent polysulfide
dissolution altogether. This can be achieved by a) choosing a gel or solid electrolyte
with virtually no polysulfide solubility [63–66], or b) chemically modifying the elec-
trode’s active surface so that sulfur is more strongly bound [67, 68], or c) introducing
mechanical barriers on top of the electrode [69, 70] or inside the electrode [71, 72].
For the latter approach, carbon coatings have been suggested [71, 73], but conductive
polymer coatings [72] have also been tried, as well as combinations thereof [74]. Of
the above references, however, only [73] had been published at the time this work was
started. The goal of this work is to prepare stable electrodes for Li/S cells synthesized
from carbon-coated lithium sulfide. The coating around the active material, illustrated
in Fig. 2.1, is intended to prevent direct contact between sulfur and the electrolyte,
1Environmental Energy Technologies Division, Lawrence Berkeley National Laboratory, 1 Cyclotron
Road, Berkeley, CA 94720, USA
2The Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA
94720, USA
17
ensure electronic conductivity, provide a stable, large surface area for Li+ exchange
between the electrolyte and the active material, and maintain mechanic stability all at
the same time. In order to meet these requirements, the average size of the individ-
ually coated particles must not exceed a few micrometers. For larger particles, both
the mechanical stability and the speed of diffusion inside the particles pose significant
problems.
CVD coating charge
Li+
Li2S Li2S S8
discharge
Figure 2.1: Schematic of the core-shell material concept.
Despite recent progress in terms of cycling stability [75, 76], all cathodes synthesized
from elemental sulfur suffer from a common problem: During discharge, the sulfur
is converted to lithium sulfide, the density of which is considerably lower than that
of sulfur. Whether the material is converted in place or dissolved and re-precipitated,
the volume of the fully discharged active material (Li2S) is 172 % that of the fully
charged active material (S8). If the initial structure of the electrode is formed while
(mostly) S8 is present, either the whole electrode will grow and shrink during cycling
[50] or at least some of the active material will deposit in places where there was
no sulfur initially. Regardless of how elaborate the structure of the cathode may be
prepared initially, this poses a high risk of destroying or downgrading the electrode’s
microstructure during cycling.
The alternative approach, followed in this work, is to synthesize the electrode in
the discharged state, i.e. from Li2S. While this does not prevent any degradation by
itself, it ensures that there is enough room for all the active material in the porous
structure of the pristine electrode. Besides, this approach is compatible with most
other techniques for improving the performance and durability of Li/S cells such as
nanostructured electrode design [77–80], electrolyte additives [81–83], or protective
layers on either electrode [55, 84, 85]. The downside of using Li2S is that this material
is extremely sensitive to moisture and oxygen, so that the processing needs to be done
in an inert and dry atmosphere. This disadvantage is mitigated, to some degree, by
the carbon-coating strategy outlined above: Once the coating has been applied, the
material should be considerably less reactive and could thus be processed without
special precautions.
18
Electrodes are prepared from the materials and tested in coin cells. Except for the
ready-made separator membranes, the cells were prepared from scratch. The individ-
ual steps of the preparation process are outlined in Fig. 2.2 and described in detail in
the following sections.
CVD
cell assembly
Li foil
slurry mixing
binder
analysis
comm.
Li2S
high energy
milling
non-aqueous
electrolyte
carbon
black
spreading,
drying, punching
operation /
cycling
materials
chemical processes
mechanical processes
Figure 2.2: Flowchart illustrating the basic steps of the cell preparation process.
2.2 Active material preparation
Undoubtedly, the most delicate step of this project is the preparation of the cathode
active material. To meet the requirements outlined in section 2.1, a novel coating
procedure was established and optimized. It consists of a ball-milling step, followed
by subsequent alternating steps of chemical vapor deposition and mechanical mixing.
Since Li2S is highly susceptible to trace moisture and – to some degree – to oxygen, all
preparation steps must be carried out in an inert Argon atmosphere.
2.2.1 Ball milling
As a first step, pure, commercially available lithium sulfide (Sigma Aldrich, St. Louis,
MO, USA) is ball-milled to reduce the particle size to . 2 µm. While even smaller
particle sizes are considered beneficial for both sulfur utilization and rate performance,
ball-milling provides an easy, affordable and reproducible way of reducing the particle
size to a reasonable value. The instrument used for ball-milling is a 8000M Mixer/Mill
19
(Thomas Scientific, Swedesboro, NJ, USA) rotating at 1060 RPM. For each run, its air-
tight stainless steel milling chamber is loaded with 1–2 mg of Li2S and a set of yttria
stabilized zirconia (YSZ) grinding balls (type 4039GM, Inframat Advanced Materials,
Manchester, CT, USA). The milling balls come in three sizes (3.0, 5.0, and 10.0 mm),
which are added to the Li2S powder in the ratio of 4:6:4:1 by weight (large : medium
: small : Li2S). In order to keep the sulfur content of the active material as high
as possible, no carbon black or other material is added to the milling container, even
though it is known to improve the milling performance slightly [77]. To avoid exposure
to moisture or air, the material is loaded into and collected from the milling chamber
inside the glovebox. The milling procedure itself consists of six milling rounds, 20 min.
each, with a 20 min. break between subsequent runs to avoid overheating. The ground
up material is collected inside the glovebox and stored in a sealed glass vial until used.
2.2.2 Chemical vapor deposition
In order to apply the carbon coating to the Li2S particles, a CVD furnace installation
had to be designed and set up which allows for the treatment of the material without
exposure to ambient air.
Furnace design and installation. The tube furnace used for the coating (Lindberg
Blue M Mini-Mite; Thermo Fisher Scientific, Waltham, MA, USA) consists of a quartz
tube with a diameter of 1.0 ” which can be inserted into the heating chamber so that
both ends protrude from the furnace. The quartz tube’s ends are sealed by an O-ring
seal each with a flange for connecting Swagelok tubing. A P&ID drawing of the final
CVD setup is presented in Fig. 2.3.
tube furnace exhaustgas feed & manifold
quick connectAr C 2
H
2
C
2H
4
Figure 2.3: Design of the tube furnace installation used for CVD coating. The furnace’s
quartz tube containing the sample can be disconnected from the setup at
the positions indicated by the arrows and removed without exposing the
material inside to ambient air.
20
Argon, ethylene, and acetylene are supplied to the gas manifold. The flow rates
can be adjusted individually using electronically operated mass flow controllers (58xx
series; Brooks Instruments, Hatfield, PA, USA). The gas feed ends in a Swagelok quick
connect, a special valve that is automatically sealed on both ends upon disconnec-
tion (see inset in Fig. 2.3). The residual volume for this type of quick connect is just
0.1 ml. The exhaust side of the tube has a low pressure one-way check valve attached,
preventing backflow of exhaust gas or ambient air into the tube even when it is not
pressurized or installed in the CVD setup. The entire quartz tube can thus be hermet-
ically sealed and removed from the furnace by disconnecting it at the quick connect
and behind the check valve.
Coating considerations. In order to obtain optimal coating results, it is important
to quantify the amount of carbon deposited during each run. Lacking other practical
and reliable means, this is done by weighing the sample before and after deposition.
Assuming no material loss, spherical particles, and uniform coating, the thickness of
the carbon shell can be calculated from the carbon weight and the particle’s geometry
using the following relation (see also Fig. 2.4):
mcarbon
mtotal
=
mshell
mcore + mshell
=
$C · 4/3pi
(
r3shell − r3core
)
$Li2S · 4/3pir3core + $C · 4/3pi
(
r3shell − r3core
) . (2.1)
For particles with a diameter of 1 µm, adding 2 % carbon by weight would already
result in a coating of ∼ 300 nm thickness. In order to obtain reasonably thin coatings,
the amount of carbon deposited should stay below 1.0 % for all but the smallest parti-
cles. Since the above assumptions are typically not fulfilled, this calculation can only
serve as a first approximation and further measurements are required to determine
the thickness and uniformity of the coating more precisely.
Coating procedure. Before starting the actual coating, the following preparatory
steps need to be completed: The gas manifold is assembled and flushed with Ar-
gon for several minutes. Next, the quartz tube is removed at the quick connects so
that the inlet side of the manifold is now closed, but under pressure. The caps are
removed from the quartz tube, which is then cleaned, dried and transferred into the
glovebox using its large antechamber. Inside the glovebox, the previously ball-milled
Li2S is put into a ceramic boat and weighed (Fig. 2.5); typically 500–2000 mg go in a
large boat so that its floor is only thinly covered. The boat is placed in the quartz tube
and aligned centered before the caps are attached and tightened. Next, the loaded
quartz tube is transferred out of the glovebox and installed in the gas manifold (outlet
side first). The pressurized Argon in the gas manifold immediately starts to flush the
tube. A mechanical stand is used to stabilize the tube during the following CVD run to
21
rcore
rshell
lshell
Li2S core
Carbon shell 0
200
400
600
800
0.0 1.0 2.0 3.0 4.0 5.0
20
00
nm
100
0 n
m
500 nm
250 nm
l sh
el
l
/
nm
Weight increase / %
Figure 2.4: Left: illustration of the carbon/Li2S core-shell structure. Right: Weight in-
crease (≈ carbon deposition) vs. coating thickness lshell for different particle
radii (rcore = 250–2000 nm).
avoid any forces on the O-ring seals. Once the tube is installed in the gas manifold, the
furnace is closed and quickly heated to CVD temperature. When that temperature is
reached, the Argon flow is reduced and a defined amount of hydrocarbon gas is added
to the flow. After the specified coating time has elapsed, the flow is reset to pure Ar-
gon and the sample in the tube is cooled down. In order to speed up the process, once
the hydrocarbon gas is turned off, the furnace may be opened at temperatures below
500 °C to increase convection. At 100 °C, the tube may be removed from the furnace
and transferred back to the glovebox. There, the CVD boat (cf. Fig. 2.5) is removed
from the tube and the partially coated Li2S is stirred up mechanically by grinding it
using a mortar and pestle for about 30 s. After the first round of coating, the mate-
rial usually forms a very brittle pellet, which breaks apart easily when applying the
slightest force. For subsequent runs, the material is already a powder before grinding,
but the coating may not be uniform in the direction of the gas flow as well as across
the powder from top to bottom. The sample is weighed directly before and after the
coating step in order to estimate the amount of carbon deposited on the sample. For
small samples, the amount of material lost during the grinding step is about equal to
Figure 2.5: Small CVD boat containing a sample of carbon-coated Li2S nanoparticles.
22
Table 2.1: Standard protocol for coating Li2S nanoparticles.
step duration procedure
1 30 min. CVD with 70 SCCM Ar, 10 SCCM C2H2 @ 440 °C
2 2 min. grinding with mortar & pestle
3 60 min. CVD with 70 SCCM Ar, 10 SCCM C2H2 @ 440 °C
4 2 min. grinding with mortar & pestle
5 120 min. CVD with 70 SCCM Ar, 10 SCCM C2H2 @ 440 °C
6 2 min. grinding with mortar & pestle
the amount of carbon deposited, therefore the weighing has to be repeated for each
CVD run, i.e. after collecting the sample from the mortar.
Various coating protocols were tested in order to come up with an optimal pro-
cedure before the most effective coating method was established as a standard; see
Tab. 2.1 for the full protocol. Heating, cooling and evacuation times are not included
in the list, but account for half of the total processing time: approximately 10 min.
heating, 30 min. cooling, 15 min. evacuation (transfer in the glovebox), and 10 min.
general handling time per step.
2.3 Other materials and cell assembly
Besides the carbon-coated Li2S, several other materials are needed to build a functional
cell. All raw materials used are listed in Tab. A.3; the rationale behind each choice is
outlined in the following paragraphs.
2.3.1 Anode materials and preparation
For the anode, plain lithium foil with a thickness of 180 µm is used (Lectro Max 100;
FMC, Philadelphia, PA, USA). While there are known issues with lithium metal anodes
[15], it can be safely assumed that for the cell configuration presented here, none of
these are critical. Room-temperature ionic liquids [86, 87] as well as the additive
lithium nitrate [81, 88, 89] can considerably improve the cyclability of lithium metal
electrodes, so that more than 1000 cycles could be demonstrated with an electrolyte
similar to the one used in this study [76]. The lithium foil is stored inside the glovebox
at < 1 ppm oxygen and virtually no water vapor. Directly before use, it is cut into 5/8 ”
discs, roughened using a plastic brush, and firmly pressed onto a stainless steel plate
by hand.
23
2.3.2 Separator
The separator is composed of a single porous polypropylene membrane (type 2400;
Celgard, Charlotte, NC, USA). A large number of suitable separators is commercially
available for lithium batteries [90]. While there are some interesting developments in
the separator regime lately [91, 92], the material chosen for this work is known to work
well from previous studies [77, 93, 94]. Also, assuming it can be wetted properly, the
choice of separator does not affect the performance of the cell much — at least for
low to medium discharge/charge rates, as can be told from the resistances reported
in [95]. Therefore, the optimization of the separator membrane is beyond the scope of
this work.
2.3.3 Electrolyte materials and preparation
The separator as well as the porous electrode are filled with a liquid electrolyte. Un-
fortunately, carbonate based electrolytes, which are commonly used for Li-ion cells
[96, 97] are not suitable for Li/S batteries because of their reactivity with dissolved
polysulfides [98]. Instead, popular choices have been glymes, ethers and dioxolane-
based electrolytes [99–103]. Only recently, the first review on electrolytes for Li/S
batteries was published by Scheers et al. [103]; a generic scheme for the classifica-
tion of various electrolytes can also be found in Ref. [104]. For this work, electrolytes
based on room-temperature ionic liquids (RTILs) are used, which are known to work
well with Li/S batteries [104–107]. By selecting and chemically modifying the anion
or cation, RTILs can be designed to have properties matching the requirements of
a specific application. Various combinations have been used for energy applications
[12, 108]. In general, RTILs are known for their good ionic conductivity as well as
their excellent thermal and electrochemical stability. Additionally, they offer technical
and environmental benefits including a very low vapor pressure, nonflammability, low
toxicity and synthesis methods which are both scalable and environmentally friendly
[108, 109]. Recent reports indicate that RTILs can even contribute to a better reversibil-
ity of lithium metal electrodes by dynamically forming a favorable, protective SEI
layer at the lithium|electrolyte interface [87]. If required, these electrolytes can also be
turned into gels easily, as reported by [110].
For this work, several different compositions of liquid electrolytes were tested, all
of them based on 1-butyl-1-methylpyrrolidinium bis(trifluoromethanesulfonyl)imide
(Pyr14TFSI) with 1.0 mol/kg LiTFSI dissolved. Tab. 2.2 lists the compositions tested
and Tab. 2.3 shows the chemical structure of the components used. Liquids are stored
in sealed vials together with activated type 4 Å molecular sieves (TRICAT, Hunt Val-
ley, MD, USA) in order to reduce the amount of trace water. Salts are dried under
24
vacuum at 140 °C overnight before use. Electrolytes are prepared in an Argon-filled,
water-free (dew point: −70 °C) glovebox (UNILab/LabMaster 130; MBraun, Garching,
Germany). All ingredients are added to a brown 20 ml glass vial at once and stirred
with a magnetic stir bar in the glovebox bar until the salt is entirely dissolved (at least
24 h for the saturated solutions). The water content of the liquids as well as of the final
electrolyte is tested by a Karl Fischer Coulometer (DL39; Mettler Toledo, Greifensee,
Switzerland).
Table 2.2: List of electrolytes tested
# composition (liquid) salt additive
1 PEGDME, Pyr14TFSI 1:1 1 mol/kg LiTFSI 0.3 mol/kg LiNO3
2 TEGDME, Pyr14TFSI 1:1 1 mol/kg LiTFSI 0.3 mol/kg LiNO3
3 TEGDME, Pyr14TFSI 2:1 1 mol/kg LiTFSI 0.3 mol/kg LiNO3
4 TEGDME, Pyr14TFSI 2:1 saturated with LiTFSI 0.3 mol/kg LiNO3
5 DOL, DME, Pyr14TFSI 1:1:2 by vol. 1 mol/kg LiTFSI 0.5 mol/kg LiNO3
6 DOL, DME, Pyr14TFSI 1:1:2 by vol. saturated with LiTFSI 0.5 mol/kg LiNO3
7 PC, EC, DEC 5:1:1 1 mol/kg LiPF6 –
Unless otherwise noted, all following experiments use electrolyte #5 (see Tab. 2.2),
a mixture of DOL, DME, and Pyr14TFSI, 1:1:2 by volume with 1.0 mol/kg LiTFSI and
0.5 mol/kg LiNO3 dissolved.
2.3.4 Cathode additives
Unlike the negative electrode, the positive electrode cannot be composed of solid bulk
material because of the poor electronic conductivity of both sulfur and lithium sulfide.
Even though the active material is already coated with carbon, additives are needed
to build a more durable and reliable electrode.
First, additional carbons provide increased and more stable conductivity: Carbon
black (Super C65, Timcal, Bodio, Switzerland) is interspersed between the active ma-
terial particles to increase the contact area. Multi-walled carbon nanotubes (MWCNT;
Cheap Tubes, Brattleboro, VT, USA) are added to enhance long-range electronic con-
ductivity [111] and to obtain a more uniform, well-distributed cathode morphology
[112]. Even though the electronic conductivity of carbon-coated particles themselves is
sufficiently high, the total electronic resistance would be unacceptably high for thicker
electrodes without added carbon because the current would have to pass through
countless point contacts between adjacent carbon-coated particles. Prior to their first
use, the carbon additives are dried under vacuum at 130 °C for at least 16 h to remove
residual water.
25
Table 2.3: Chemical structure of materials used in the liquid electrolyte. Full names,
properties, and sources of the chemicals are listed in the appendix (Tab. A.3).
component structure
DME H3C
O
CH3
DOL
O O
Pyr14TFSI
⊕
N
F3C
S
O
O
	
N
S
O
O
CF3
TEGDME
H3C
O
O
CH3
[ ]
4
Second, a binder ensures mechanical stability and structural integrity of the porous
matrix. Conventionally, polyethylene oxide (PEO), polyethylene terephthalate (PET),
or polyvinylidene flouride (PVDF) have been used as binders in sulfur cathodes [113–
116]. However, these binders are not very elastic and thus struggle to accommo-
date the large volume changes associated with charging and discharging the sul-
fur electrode [117, 118]. Several alternatives were suggested and evaluated, among
them alginates [118], gelatin [119], sodium carboxymethyl cellulose (CMC) [120, 121],
styrene-butadiene rubber (SBR) [77], and combinations thereof [122, 123]. Besides
the improved elasticity, these advanced binders are known for their better dispersive
properties. They are also considerably cheaper than PVDF and finally, they facilitate
recycling of the electrode materials after use since they can be easily removed from a
waste electrode by pyrolysis. For this work, plain SBR (Sigma Aldrich, St. Louis, MO,
USA) was chosen as a binder, because of its reported compatibility with the electrolyte
used, easy processing, and previous studies indicating good performance for a similar
system [77].
2.3.5 Electrode preparation
While electrodes for coin cells can be prepared individually with very small amounts
of electrode slurry, for this work a batch of electrodes was prepared at once, yielding
higher loadings and better reproducibility among the electrodes of that batch. The
only downside of this approach is that individual electrodes cannot be weighed before
and after application of the slurry; implications of this issue will be discussed below.
26
The typical batch size used in this work is 90× 180 mm2, resulting in 20–30 coin cell
sized electrodes.
The slurry and electrode preparation can have a big impact on the electrode’s per-
formance [124], therefore this procedure needs to be analyzed and optimized care-
fully. The electrodes are prepared from a slurry mixed according to the composition
in Tab. 2.4. A stock slurry is prepared in advance as follows: 120 mg of carbon black
are predispersed in 3000 mg of toluene by vigorously stirring with a magnetic stir
bar (120V Stirrer; Corning Inc., Corning, NY, USA) overnight at room temperature.
Next, 30 mg of MWCNT are added while stirring continuously. After another 90 min.,
1500 mg of a toluene solution containing 10 wt.% SBR are added. The stock slurry is
kept on the stir plate until used up. The actual slurry is made by dispersing 72 mg of
active material in 550 mg of toluene, then adding 270 mg of the previously prepared
stock slurry. Two different techniques for optimizing the dispersion of the final slurry
were evaluated: a short treatment with an ultrasonic tip homogenizer (Digital Sonifier
450; Branson, Danbury, CT, USA) or sonicating in an ultrasonic bath (2510, Branson)
for 20–30 min. with the vial being held upright and partly immersed into the bath.
Either way, once the slurry is properly dispersed, it is transferred onto a primed, i.e.
carbon-coated, aluminum foil in the glovebox and cast into a film of 200 µm thickness
using a hand-operated doctor blade. The film is dried in the glovebox for a few min-
utes until it looks blunt and dry. The coated foil is then enveloped between two sheets
of weighing paper and subsequently cut into individual 1/2 ” disc electrodes using a
mechanic punch. Initially, an additional drying step (4 h at 60 °C on a heat plate) was
applied. It turned out, however, that this step does not affect the performance of the
electrodes and hence it was skipped for all the experiments presented in this work.
There is even a report which indicates that small amounts of residual toluene may be
beneficial for both rate capability and Coulombic efficiency [125] – no such effects were
observed, though. During the casting and drying process, the circulation purifier of
the glovebox is deactivated in order to spare the copper catalyst. Afterwards, the box
is purged for 5 min. and circulation is reactivated. The cut-out electrodes are stored in
a 20 ml glass vial which is sealed and kept in the glovebox until used.
2.3.6 Cell assembly
Except for some special experiments, coin cells were used throughout this work. The
components assembled into a cell are shown in Fig. 2.6. The stainless steel cell casing
parts (National Research Council Canada, Montreal, QC, Canada) are dried overnight
at 140 ◦C under vacuum before use to avoid contamination of the electrolyte by resi-
dual water. Cells are assembled in a glovebox filled with Argon 5.0 (Praxair, Danbury,
CT, USA), compressed using a custom-build Argon-driven pneumatic press and rest
27
in the glovebox for at least 12 h at room temperature before being tested. Since the
amount of electrolyte can have a great effect on the performance of the cell [126, 127],
its dosing is done extra carefully: Each cell contains 50–60 µl of liquid electrolyte; in
order to ensure optimal wetting of all surfaces, the electrolyte is added to the cell in
two portions: the first one is applied on top of the cathode before adding the separator,
the second one on top of the separator before adding the anode. The separator is larger
than both the anode and cathode and the amount of electrolyte is slightly more than is
needed to fully soak the separator. Thus, there is a surplus of liquid electrolyte which
causes an adhesive force between the separator and the stainless steel casing around
the electrodes and also serves as a reservoir – in case some electrolyte is decomposed
or otherwise lost during cell operation.
Figure 2.6: Components of a CR2032 cell (left to right): Stainless steel bottom cas-
ing, positive electrode (Li2S composite on Al foil), separator (Celgard 2400
membrane), negative electrode (Li foil), stainless steel spacer, bell washer,
rubber seal, stainless steel top casing. Ruler: inches.
Table 2.4: Standard composition of the positive electrode slurry.
compound amount
wt.% of solids
total amount
mg
active material 80 72
carbon black 10 9
binder 10 9
toluene (~900) 804
28
2.4 Analysis and characterization methods
2.4.1 Microscopy
To check materials and electrodes, Scanning Electron Microscopy (SEM) was used pri-
marily. Li/S type samples are somewhat difficult to image because both S8 and Li2S
are insulators. Nevertheless, for most samples, the microscope used (JSM-7000F; JEOL,
Akishima, Tokyo, Japan) allows for magnifications of 10 000–50 000× while still pro-
ducing reasonably clear and crisp images. Also, energy dispersive X-ray spectroscopy
(EDX) can be performed simultaneously. Sample preparation is straightforward: elec-
trodes or powders are fixed on top of an aluminum cylinder by means of conductive
carbon tape. To prepare cross-section images of an electrode, it is ripped apart in the
glovebox, bent into an L-shape, and affixed to the SEM sample holder so that one of the
edges is pointing upwards. For powders, it is important to shake off any material that
is not sticking firmly in order to avoid pollution of the microscope and other samples.
The aluminum cylinder is installed into a sample holder while still in the glovebox,
which is subsequently installed in the microscope via an evacuatable antechamber.
The entire sample holder is kept in a sealed container until the antechamber is ready.
Then, it is quickly installed in the corresponding adapter; the chamber is closed and
pumped down immediately. If exposure to air must be absolutely avoided, a glovebag
may be installed around the opening of the antechamber and flushed with Nitrogen
or Argon prior to installing the sample.
Using SEM, the particle size, agglomeration state, and uniformness is investigated,
but the presence of cracks, defects or other types of degradation can also be detected.
To estimate the composition and purity of the samples, they are also analyzed by
EDX while in the microscope. EDX detects the characteristic X-ray emissions [128] of
the atoms in the sample which are excited by the incident electron beam. From the
spectrum and the known absorption cross-sections of each element, the composition
of the sample can be calculated [129, chap. 7.2]. For very light elements such as H or
Li, however, both the absorption cross-section and the energy of the emitted X-rays are
too small to produce accurate results, given the spectral resolution of the instrument.
Additionally, Transmission Electron Microscopy (TEM) images can be recorded to
analyze the structure of the carbon-coated Li2S particles. TEM images were recorded
at the facilities of the National Center for Electron Microscopy3.
3National Center for Electron Microscopy, Lawrence Berkeley National Laboratory, 1 Cyclotron Road,
Berkeley, CA 94720, USA
29
2.4.2 Chemical/physical methods
Coating test. A special test was developed in order to test the integrity of the carbon
coating and its properties as a barrier for (lithium) polysulfides. The idea behind this
test is that polysulfides are intensely colored [104, 105] so that their presence in a
clear liquid can be easily detected. Also, because of disproportionation reactions, they
are formed readily when dissolved S8 and Li2S are mixed. Therefore, in a suitable
solvent which is able to dissolve sufficient amounts of both S8 and Li2S, the presence
of Li2S can be detected quickly and easily. A selection of solvents including toluene,
tetrahydrofuran, carbon disulfide, tetraethylene glycol dimethyl ether, 1,3-dioxolane,
and N-methyl-2-pyrrolidone (NMP) was screened for suitable components (data not
shown). Also, different mixing protocols were evaluated.
The most suitable combination turned out to be a mixture of NMP and toluene:
First, 10 mg/ml of S8 are dissolved in pure toluene to create a “test solution”. Next,
about 1 mg of the material to be tested is dispersed or immersed in 200 µl NMP and
incubated for 1 min. so that any Li2S exposed to the solution will be dissolved (at least
partly). Finally, 50 µl of the test solution are added with a pipette. If dissolved Li2S
is present, polysulfides will be formed immediately, coloring the solution. For this
combination of solvents, the intensely blue S•−3 radical is most prominently visible. If
there is no Li2S, the solution will stay clear; otherwise it will turn blue. Even though
the intensity of the color does vary depending on the amount of Li2S present, this test
is by no means quantitative.
There may be more simple methods for detecting Li2S, e.g. by adding water to the
sample (vigorous reaction) or trying to detect the smell of H2S, which is formed when
Li2S is exposed to air. The advantage of this special test is that it is highly sensitive,
reliable, safe, and quick all at once.
Beaker cell. A transparent beaker cell was built from a 3 ml glass vial, see Fig. 2.7.
This cell contains significantly more electrolyte than a regular coin cell, and is gener-
ally not very suitable for quantitative electrochemical measurements [127]. However,
the formation and movement of polysulfides can be directly visually analyzed with
this kind of cell [130].
2.4.3 Electrochemical methods
For analyzing full cells, a range of electrochemical methods is exploited using several
different potentiostats.
Impedance spectroscopy. Initially, all cells are tested for their open-circuit voltage
(OCV) and their electrochemical impedance spectrum (EIS) [131]. Besides providing
30
Figure 2.7: A transparent beaker cell for optically investigating polysulfide formation
and transport.
important information on the cell, this data can be used to identify faulty cells and
to estimate the uniformity of a batch of electrodes. Impedance spectra are recorded
using a 10 mV sinusoidal excitation around OCV in the frequency range of 1.0 MHz to
50 mHz with a VSP potentiostat (BioLogic, Claix, France). For some cells, impedance
was also recorded during stabilization or intermittently during cycling. For the latter,
the discharge was stopped 15–60 s before the EIS recording so that the cell voltage can
relax (almost) to OCV. The experimental results are plotted in Nyquist representation
[132, chap. 16.1].
For all tests except EIS, cells are stored in controlled temperature chambers (TEC1;
TestEquity, Moorpark, CA, USA) at 30 °C in order to minimize the effect of circadian,
seasonal, or weather-dependent temperature changes.
Cyclic voltammetry. In order to study electrolyte stability and to evaluate differ-
ent charging procedures, slow-sweep cyclic voltammetry (CV) was employed, using
a VMP3 potentiostat (BioLogic, Claix, France). “Slow” in this context means that the
voltage is changed quasistatically, i.e. the rate at which the voltage changes is slow
compared to the time it takes for the current to stabilize at the new voltage. Practi-
cally, it means that the SoC of the cell is following the voltage so that one CV cycle
corresponds to one full charge/discharge cycle. In the cyclic voltammetry (CV) plots,
the current (i.e. charge passed per time) is plotted vs. the externally applied voltage.
Because of the slow and steady voltage ramp, the current is approximately propor-
tional to the total charge which would be passed if the cell was held at that voltage
indefinitely. A typical CV experiment, utilizing a voltage ramp in the range of 10–
25 µV/s, runs for several days.
31
Cycling. All cycling experiments start with an initial slower activation cycle followed
by the regular cycling, which consists of an initial rest time, a constant-current (CC)
discharge to 1.5 V, a variable rest time, a CC charge to 2.8 V, and finally a constant
voltage (CV) charge which ends once the current drops below a certain threshold,
typically 5 % of the CC charging current.
2.5 Dead ends
More materials, techniques and procedures were tested and applied, but not found
to be particularly helpful or constructive. For completeness, this information is also
included here, but the corresponding results are not presented or discussed. These
“dead ends” include:
• Thermogravimetric analysis (TGA): The measurements did not work out very
well because of the high reactivity of Li2S which decomposes rapidly in ambient
air. TGA samples need to be sealed in aluminum containers in the glovebox,
which are then opened by the TGA machine right before the heating starts. Still,
the drift is quite high and there is considerable weight loss even before the tem-
perature starts to rise.
• Impedance of symmetrical cells (both Li/Li and S/S): The former turned out
to be highly dependent on the Li pretreatment, an effect which is deemed to
be irrelevant for real cells after completing the first cycle. For the latter it was
very difficult to obtain reproducible results, especially for SoC 6= 0 %. The most
promising method is to remove a positive electrode from a cell cycled to the de-
sired SoC, cut it into half, and build a new cell using the two halves as electrodes.
However, it is tricky to cut and align the two parts perfectly symmetrically. Also,
most of the original electrolyte is lost (even if the separator is reused) and has to
be replaced, which causes a significant loss of polysulfides and thus a change in
SoC.
• Baking electrodes after fabrication: In addition to drying, electrodes were baked
for 2–4 h at 70–100 °C. While the electrodes look rather similar in SEM images
(except for some cracks), their electrochemical performance and lifetime was
always significantly worse than that of non-baked electrodes.
• Atomic Force Microscopy (AFM) was used as an analysis method complemen-
tary to SEM. An environmentally protected holder including the AFM tip can be
loaded with the sample in the glovebox, and directly installed in the instrument
(MFP-3D, Asylum Research, Santa Barbara, CA, USA). While the resolution is
32
typically slightly better than that of the SEM, the morphological information
which can be extracted out of the images is roughly similar. However, AFM
imaging takes considerably more time and also, conducting and insulating areas
cannot be told apart easily. Therefore, AFM was no longer used after the initial
testing, which confirmed that the SEM images look very similar to the actual
structure as detected by AFM.
• Three electrode cell measurements: A Swagelok-type three-electrode cell was
used to obtain impedance spectra and discharge curves relative to a Li refer-
ence potential. Besides technical complications, this approach was not pursued
further because this cell is quite different from the coin cell in many respects.
Therefore, an interpretation of the results requires many additional assumptions
which are difficult to verify.
• Spraying electrodes: Spraying can be superior to doctor blading in many re-
spects, especially for smaller batch sizes. Initial tests of spraying carbon sus-
pensions outside of the glovebox with a hand-operated nozzle worked out quite
well. To match and surpass the performance of doctor blading, however, an
automated spraying setup would have had to be installed inside the glovebox,
which was not feasible for time, cost, and glovebox space reasons.
33
3 Experimental work on lithium sulfide
cathodes – results and discussion
In this chapter, the results of the experiments and analyses outlined in chapter 2
are presented. First, the characterization of materials by means of microscopy, spec-
troscopy, and chemical methods is reported and discussed in section 3.1 with a focus
on the processing of the Li2S particles. Next, the impact of the core-shell structure
on the cells’ performance is analyzed. Electrochemical tests of full cells are presented
and discussed in section 3.2, including cycling, impedance spectroscopy, and cyclic
voltammetry. A critical assessment of the results is conducted in section 3.3 and fi-
nally, conclusions are drawn.
3.1 Analysis of materials and electrodes
Various techniques have been employed to study the materials in the Li/S cell in order
to understand and optimize the carbon coating as well as the electrode preparation
steps. In the following, these results will be presented and discussed.
3.1.1 Scanning electron microscopy of materials
Microscopy is one of the main analysis methods used in this work. SEM images and
EDX spectra of the raw materials in the positive electrode are shown in Figs. 3.1 and
3.2, respectively. One ingredient of the electrode, the binder SBR, is missing because
it is rapidly decomposed by the microscope’s electron beam, making imaging and
spectroscopy impractical. Besides giving an estimate of the materials’ purity, particle
size, and agglomeration state, these images are recorded for comparison with the
material after treatment. Thereby, the effect of each step of the preparation may be
investigated, which is of particular interest for the processing of the Li2S powder. For
the pristine, commercial Li2S, the particle size is determined to be 4.0± 2.0 µm, with
some outliers as big as 50 µm. This finding confirms that the commercial material is
not suitable for use in Li/S batteries unless the particle size is significantly reduced.
34
5 µm 0.5 µm
(a) Commercial Li2S powder
5 µm 0.5 µm
(b) Carbon black
5 µm 0.5 µm
(c) Multi-walled carbon nanotubes
Figure 3.1: SEM images of the raw materials of the positive (sulfur) electrode. Left
column: 5 000× magnification, right: 50 000× magnification.
35
0400
800
1200
0.0 1.0 2.0 3.0
C O Al
S
S: 71 %
O: 18 %
C: 11 %
C
ou
nt
s
X-ray energy / keV
(a) Commercial Li2S
0
100
200
300
0.0 1.0 2.0 3.0
C
O Al S
C: 100 %
C
ou
nt
s
X-ray energy / keV
(b) Carbon black
0
400
800
0.0 1.0 2.0 3.0
C
O Al S
C: 97 %
O: 3 %
C
ou
nt
s
X-ray energy / keV
(c) Carbon nanotubes
Figure 3.2: EDX spectra of the raw materials of the positive (sulfur) electrode. All spec-
tra are recorded at 15 keV incident electron beam energy. Labels indicate
the position of the Kα1 line of each detected element and atom percentages
for prominent elements.
0
400
800
1200
0.0 1.0 2.0 3.0
C
O Al
S
S: 71 %
O: 16 %
C: 13 %
C
ou
nt
s
X-ray energy / keV
(a) after ball-milling
0
500
1000
1500
2000
0.0 1.0 2.0 3.0
C
O Al
S
S: 64 %
C: 22 %
O: 13 %
C
ou
nt
s
X-ray energy / keV
(b) carbon-coated using C2H4 at
700 °C
0
500
1000
1500
2000
0.0 1.0 2.0 3.0
C
O Al
S
S: 61 %
C: 29 %
O: 10 %
C
ou
nt
s
X-ray energy / keV
(c) carbon-coated using C2H2 at
400–450 °C
Figure 3.3: EDX spectra of the active material at various steps during the preparation
process. All spectra are recorded at 15 keV incident electron beam energy.
Labels indicate the position of the Kα1 line of each detected element and
atom percentages for prominent elements.
From the recorded EDX spectra, presented in Fig. 3.2, the composition of the sam-
ples can be calculated. Note that H and Li are not detected by EDX, cf. section 2.4.1.
Judging from the spectra, the carbon components are indeed very pure. Small amounts
of oxygen may have been introduced during the installation of the samples in the mi-
croscope, e.g. in the form of adsorbed trace water. For the Li2S, however, the situation
is different. A pure Li2S sample should ideally show up as 100 % sulfur. The additional
carbon content might be caused by stray carbon black particles, which are easily spilled
across the sample holder during the evacuation step. Also, the carbon tape used to
affix the powder to the aluminum sample holder may be detected by EDX. There is
no similar explanation for the high amount of oxygen in the specimen, though. The
EDX signature of oxygen is relatively small so that the composition is only accurate to
plus/minus several percent. Nevertheless, it has to be assumed that the Li2S particles
have been partly converted to LiOH and LiO2 at least at their surfaces. Unfortunately,
36
the same result was obtained for all Li2S samples tested (from three different batches,
data not shown) and there is no practical way of purifying them or even determining
the precise composition of the impurities. Therefore, the material was used for the
experiments nevertheless, keeping in mind that the composition of the final electrode
may differ from what is intended, cf. section 3.2.2.
The material is also imaged during the refinement process. Fig. 3.4a shows Li2S
particles after high-energy ball milling. It becomes clear that the particle size is suc-
cessfully reduced during that step. More precisely, it averages to 1.2± 0.8 µm. Even
after ball-milling, the particle size distribution is not very uniform and outliers as large
as 4 µm can be found regularly and up to 20 µm rarely.
Finally, the coated material is analyzed. An immediate effect of the conductive
coating is that the quality of the SEM images is visibly improved because the sample is
no longer charged during imaging, cf. [129, chap. 15]. The coating smoothly conforms
to the particle shapes, which are still not very regular (compared e.g. to synthesized
nanoparticles [133]). Unfortunately, two types of agglomeration can also be observed:
First, coated particles stick to each other, creating a cross-linked network. This kind
of agglomeration seems to be ubiquitous after coating, but does most likely not pose
a problem with regard to the electrochemical performance. On the contrary, it might
even improve the long-range electronic conductivity of the electrode and is thus not
considered harmful. Second, however, particles which stick to each other before or
early during the carbon deposition are coated collectively and are thus trapped under
a common “carbon veil”. This may well have a negative impact on the material’s
performance because less area is exposed to the electrolyte, causing longer diffusion
distances and thereby higher overpotentials in the particles. As a consequence, sulfur
utilization is expected to be lower. During the optimization of the coating protocol,
it was observed that this kind of agglomeration can be broken up partly during the
intermittent grinding steps. Also, it can be decreased by reducing the continuous
coating time. Still, even for coating times of only 30 min. per run, agglomeration was
detected (data not shown). To improve the situation, a rotary furnace may be needed,
as discussed in section 3.3.2.
37
5 µm 0.5 µm
(a) Li2S powder after high-energy ball-milling
5 µm 0.5 µm
(b) Li2S powder after ball-milling and carbon coating with ethylene at 700 °C
5 µm 0.5 µm
(c) Li2S powder after ball-milling and carbon coating with acetylene at 400–450 °C
Figure 3.4: SEM micrographs of ball-milled and carbon-coated Li2S. Left column:
5 000× magnification, right: 50 000× magnification. Note how coating at
elevated temperature causes agglomeration (b), whereas coating at lower
temperatures (c) nearly preserves the particle size distribution of the ball-
milled, uncoated material (a). For comparison, the pristine material is
shown in Fig. 3.1a.
38
When looking at Fig. 3.4, it also becomes obvious that treating the material at el-
evated temperatures significantly increases agglomeration. The samples shown in
Figs. 3.4b and 3.4c are prepared from the same batch of ball-milled Li2S, coated with
roughly the same amount of carbon. Because of different carbon precursors (Fig. 3.4b:
ethylene, Fig. 3.4c: acetylene), the CVD process was run at different temperatures. At
700 °C, the secondary particle size is increased almost tenfold compared to 400–450 °C.
When using methane, which decomposes at around 900 °C, the agglomeration is even
worse (data not shown).
EDX confirms that neither the oxygen nor the carbon content is increased during
the ball-milling step, compare Figs. 3.2 and 3.3. As expected, an increased carbon
content is detected after the CVD coating. In principle this increase could be used
to calculate the amount of carbon deposited and thereby the thickness of the coating
according to Fig. 2.4. However, measurements of various coating runs report 3–16 %
increase in carbon content, yielding a carbon coating with an average thickness of
200–900 nm, depending on the particle size distribution. This result is way beyond
the intended amount of coating and does not agree with the visual impression of the
SEM images. The reason for this discrepancy is the limited penetration depth of the
incident electron beam [129, chap. 6.3]. Electrons are preferentially scattered in the top
layers of the particles, which of course inflates the share of the carbon detected, since
most of the bulk Li2S does not contribute to the EDX signal.
3.1.2 Scanning electron microscopy of electrodes
Besides the handling of the active material, the optimization of the electrode prepa-
ration was the processing step which needed the most tweaking and tuning. In the
following section, SEM micrographs of sulfur/carbon electrodes are presented, illus-
trating problems encountered and solutions found.
The newly developed active material requires optimized handling, processing, and
examination. In particular, the mixing and doctor blading of the electrode slurry
needed adjustments in order to achieve good results. Because of the susceptibility
of the material and the small batch size, no automatic coating machine is used, but
all electrodes are doctor bladed by hand. In this context, SEM images are used for
quality control and process optimization. First, electrodes prepared from differently
mixed slurries are compared in Fig. 3.5. An electrode prepared according to the recipe
published in Ref. [77] is depicted in Fig. 3.5a. This electrode is prepared from a very
thin, low-viscosity slurry. Even though it looks nicely solid and uniform, the load-
ing of 0.1–0.5 mg/cm2 is considerably lower than the target range of 1.5–2.5 mg/cm2.
While alternating steps of coating and drying could build up a thicker electrode, this
procedure is associated with various problems ranging from poor mechanical stability
39
1 mm
(a) Very low loading
1 mm
(b) Strongly non-uniform deposition
1 mm
(c) Cracked electrode
1 mm
(d) Proper electrode
Figure 3.5: SEM micrographs of differently prepared electrodes, illustrating various
issues with the preparation process. Magnification: 50×. Details see text.
to loss of electronic contact. Therefore, the slurry composition had to be changed in
order to deposit more electrode material in one coating run. Subsequently, the active
material content was increased from 68 % to 80 %. Simultaneously, the consistency of
the slurry was increased to a solid:solvent ratio of 1:7, compared to an initial 1:20.
Efforts to further increase the ratio, e.g. to 1:4, were abandoned, since the high vis-
cosity causes severe problems with the doctor blading step (data not shown). When
subjected to the same processing as before, however, the improved slurry yielded
poor electrodes, shown in Fig. 3.5b. A strongly non-uniform deposition was observed,
caused by poor dispersion of the active material powder and the carbon black. Disper-
sion is considerably improved if the mixing step is longer and more powerful. If the
mixing is too powerful, however, it may not be suitable for this material despite the
superior dispersion. Fig. 3.6 shows an electrode whose slurry was mixed in a 20 ml
40
2 µm
0
200
400
600
0.0 1.0 2.0 3.0
C
ou
nt
s
X-ray energy / keV
C
O
Al
SC: 70 %
O: 22 %
S: 8 %
Figure 3.6: SEM micrograph of an electrode prepared from an electrode slurry mixed
with the ultrasonic tip. Magnification: 5 000×. The large sulfur particle is
obviously broken. Inset: EDX spectrum of a selected region next to the
broken particle, which contains significant amounts of sulfur.
glass vial with the ultrasonic tip for 1 min. at a power of only 40 W. The large particle
in Fig. 3.6 is obviously broken and its sulfur content spilled all over the electrode even
before it got in contact with the electrolyte. As an immediate conclusion of this find-
ing, the ultrasonic tip was no longer used. Instead, the much more gentle magnetic
stirrer and ultrasonic bath were favored, at the cost of prolonged mixing times. After a
discussion with the supplier of the carbon black material (Timcal, Bodio, Switzerland),
the procedure described in section 2.3.5 was established. Indeed, the dispersion and
hence the spreading are significantly improved over the previous approach as shown
in Fig. 3.5c. This electrode, however, suffers from another issue associated with the
higher loading: if a thick electrode is dried is too rapidly (e.g. at elevated tempera-
ture) tension builds up in the material which can cause cracks. These cracks in turn
are related to irreversible structural degradation [134]. To overcome this issue, the
electrodes are cut into coin cell sized pieces immediately once they are dry enough,
i.e. no longer sticky, and collected in a sealed vial. This measure slows down drying
by lack of convection and the accumulation of the toluene solvent in the vial’s atmo-
sphere, thus effectively preventing the formation of cracks, see Fig. 3.5d. In addition,
the initial drying can be slowed down by placing the electrode inside a closed petri
dish directly after the doctor blading. Despite the increased drying time, however, this
did not seem to further improve the quality of the electrodes (data not shown).
Another electrode prepared according to the recipe optimized for uniformity, cov-
erage, and sulfur loading is presented in Fig. 3.7 at a higher magnification in top-
down and cross-section view. While its thickness does vary, it is obvious from the
cross-section that the composition of the electrode is uniform. No sedimentation or
additional agglomeration is observed. At the bottom of the cross-section image, the
aluminum current collector with a thin layer of carbon priming is discernible.
41
5 µm 10 µm
Figure 3.7: SEM image of an optimized electrode. Left: top down, 5000×; right: cross-
section, 2500×.
SEM was not only used for quality control and optimization, but also to investigate
changes to the electrode structure during cycling. Micrographs of the sulfur electrode
recorded at different SoC are presented in Fig. 3.8. More precisely, these are images
of several similar electrodes removed from identical cells at different times during the
formation and first discharge. The changes to the morphology are apparent: After the
initial charge (activation) a film has formed on the entire electrode (Fig. 3.8b). During
the first discharge, the film is temporarily removed to a great extent (Fig. 3.8c), but
re-deposited before the end of the discharge (Fig. 3.8d). The results are comparable
to those of Refs. [134, 135], both of which used a cell design without any polysulfide
confinement. This is a strong indication that – despite the carbon coating – some of
the sulfur can actually move around freely in the cell. In fact, this behavior can be
explained nicely by the model as presented on pages 93ff.
42
5 µm
(a) pristine electrode
5 µm
(b) after 1st charge
5 µm
(c) during discharge
5 µm
(d) after 1st discharge
Figure 3.8: SEM micrographs of Li2S electrodes during cycling. All electrodes are from
the same batch, built into similar cells. The cells are stopped and the
electrodes removed at different SoC during the first cycle. Magnification:
5 000×.
43
3.1.3 Transmission electron microscopy
To confirm that a core-shell structure is actually formed when coating the Li2S parti-
cles, transmission electron microscopy (TEM) is used. The resulting micrographs were
provided by Xiangyun Song1 and can be found in Fig. 3.9. The images suggest that
the Li2S core particles are mostly crystalline (pattern on the right hand side of Fig. 3.9)
and coated by a thin and rather uniform carbon layer. Based on the TEM images, its
thickness is 16–29 nm. The material used for these images has an average particle size
of 1.1± 0.3 µm (determined by measuring the diameter of numerous particles in a low
magnification SEM micrograph) and contains 1.9± 0.9 wt.% of carbon coating (deter-
mined by weighing during the CVD process). Choosing mcarbon : mtotal = 1.9 % and
rcore = 1.1 µm in Eq. (2.1), the coating’s thickness is calculated to be 370± 140 nm. This
tremendous discrepancy can be explained by two effects: either the average particle
size is considerably smaller than determined or carbon is trapped in the agglomerates.
To match the results, however, the particles need to be as small as 100 nm or > 80 % of
the carbon needs to be unavailable for coating, both of which seems unlikely. Another
possible but equally unlikely explanation is that the carbon is preferentially deposited
on the CVD boat instead of the Li2S particles, forming carbon black. Since TEM mi-
crographs are the most direct way to examine the coating, the figure extracted from
these images is considered to be more accurate than the calculations based on weight
and geometry.
5 nm100 nm
x
Figure 3.9: TEM images of carbon-coated Li2S particles. Left: core-shell structures,
250 000×; right: close-up of the Li2S core, 1 000 000× (at the position marked
by “x”). The arrows indicate the carbon coating.
1Senior technician, Environmental Energy Technologies Division, Lawrence Berkeley National Labora-
tory, 1 Cyclotron Road, Berkeley, CA 94720, USA
44
3.1.4 Chemical/physical tests
Raman spectroscopy. Using TEM, one can only take a close look at a few random
samples of the material. In order to further analyze the effect of the coating, Raman
spectroscopy is used which can simultaneously or subsequently scan comparatively
large amounts of material, averaging over hundreds of particles. Raman spectra were
kindly provided by Simon F. Lux2. Fig. 3.10 shows spectra of three samples of carbon-
coated Li2S powder. The samples were coated for 1 h, 2× 1 h, and 3× 1 h, respectively,
and ground up in the glovebox in between subsequent coating runs as described in
section 2.2.2.
0 500 1000 1500 2000
Li2S
CD
CG
In
te
ns
it
y
/
a.
u.
Raman shift / cm−1
1 h CVD
2 h CVD
3 h CVD
Figure 3.10: Raman spectra of carbon-coated Li2S recorded after different coating
times. The main Li2S signal is labeled as well as the carbon’s D and G
bands.
The powders look uniformly dark gray after 1 h and black after 2 or 3 h coating time.
As expected, the Raman signal of Li2S is damped for longer coating times. Note that
the Raman activity of Li2S is considerably higher than that of carbon. In all three cases,
the carbon shell constitutes the majority of the volume analyzed, which is confirmed
by the virtually unchanged carbon signal. Spectra recorded at different positions look
very similar (data not shown). Even after 3 h of coating, however, there is still some
Li2S signal left. While Raman spectroscopy is often called a surface-sensitive tech-
nique, the incident laser beam (with a wavelength of 488 nm) has a finite penetration
depth of a few hundred nanometers [136, 137]. Since the carbon coatings can be even
2Electrochemist post-doc fellow, Environmental Energy Technologies Division, Lawrence Berkeley Na-
tional Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
45
thinner, the underlying bulk material does still contribute to the Raman signal, even if
it is not exposed. Unfortunately, it cannot be told whether the remaining signal after
3 h of coating is due to the faint contribution of the bulk Li2S or whether some par-
ticles are not perfectly coated. Additionally, because of the way Raman spectroscopy
works, the results do not rule out the possibility that the coating is uniformly applied,
but meso- or microporous. In that case, it could still be penetrated by the electrolyte
and dissolved polysulfides.
Chemical coating test. Since both TEM and Raman spectroscopy cannot convincing-
ly answer the question whether the carbon coating can be penetrated by dissolved
polysulfides or not, the test described in section 2.4.2 is applied to the material. This
test examines the properties of the carbon coating as a barrier for polysulfide move-
ment in a realistic setting. Images of a test are presented in Fig. 3.11, whereas Tab. 3.1
lists the sample compositions and the corresponding test results.
Table 3.1: Chemical test of the carbon coating as a polysulfide barrier. Composition
and test result of each sample shown in Fig. 3.11 are listed. For the elec-
trodes, the size of the patches is chosen such that they contain about 0.5 mg
of active material.
label composition result
0 NMP and test solution only negative
- 5.0 mg carbon black negative
+ electrode prepared from non-coated Li2S (according to [77]) positive
++ 0.5 mg Li2S positive
1 electrode prepared from coated Li2S positive
2 0.5 mg coated Li2S powder positive
3 electrode prepared from coated Li2S (extra thick coating) positive
In order to build confidence in the test, four controls are included in the series: First,
pure testing solution is added to NMP in order to confirm that the sulfur contained
in the testing solution does not cause any coloration by itself. Second, plain carbon
black is tested as a negative control. Third, an electrode containing Li2S without any
coating is tested as a positive control and finally, pure Li2S is dissolved in NMP in
order to confirm that Li2S actually causes the coloration. All of the controls result
in the expected outcomes. There is a difference in the intensity of the blue color for
samples “+” and “++”, which is most likely caused by different amounts of dissolved
Li2S; nevertheless the interpretation of the results is unambiguous.
Three samples are tested in this experiment: one electrode prepared according to
46
(a) control “0” (b) control “−” (c) control “+” (d) control “++”
(e) sample 1 (f) sample 2 (g) sample 3
Figure 3.11: Chemical test of the carbon coating as a polysulfide barrier. Blue color
indicates that polysulfides are present in the solution. The composition of
each sample is reported in Tab. 3.1. Details see text.
the standard recipe (1), the corresponding active material powder (2), and another
electrode (3) prepared from Li2S powder with an extra-thick coating (total CVD time
5 h at 400 °C; 6.5 wt.% carbon). Unfortunately, dissolved polysulfides were detected for
all three samples (as for all other Li2S-containing samples, data not shown). Therefore,
despite the uniform black look of the coating as well as the encouraging Raman and
TEM results, it has to be stated that the coating does not prevent polysulfide formation
and dissolution in the liquid electrolyte – at least not completely.
In the full cell, the situation might actually even be worse, since the coating may
also be damaged during the Li extraction at elevated voltages during activation, cf.
section 3.2.3. Also, there is considerably less solvent in the cell than in this test, there-
fore the concentrations of the polysulfides may be quite high, even if only a small
fraction of the active material is leaked into the electrolyte.
Beaker cell runs. The failure to prevent polysulfide dissolution is also evident when
looking at the beaker cell during operation. The photos in Fig. 3.12 picture the cell at
47
various stages during the first charge. A constant current is applied (corresponding to
a nominal C/4 rate) by the potentiostat until the cell voltage reaches 4.0 V. Intensely
colored polysulfides are visible in the electrolyte already at moderate SoC and poten-
tial, as reported for conventional electrodes without sulfur retention in Ref. [130]. Of
course, the geometry of this cell is rather particular and the amount of electrolyte used
is very high; nevertheless, there is no reason to believe that the leakage would be any
less severe in a coin cell.
Figure 3.12: Images of the beaker cell taken at various times during the first charge.
From left to right: 0 h, 1 h, 2 h, end-of-charge (∼ 4 h).
While these findings are somewhat discouraging, they do not necessarily mean that
the approach followed in this work is generally unsuitable. The coating still serves
other purposes such as providing electronic conductivity, a large reactive surface area,
and (micro-)mechanical stability, cf. section 2.1. It does mean, however, that polysul-
fides will be present outside of the carbon shells and hence the electrolyte is important
for the performance of the cell beyond providing a means of Li+ transport. Therefore,
the electrolyte’s composition and its optimization will be discussed next.
3.1.5 Electrolyte optimization
Initially, it was planned to use the ionic liquid based electrolyte reported in Ref. [109]
for this work because of its thermal stability, good ionic conductivity, and safety. Using
this electrolyte, however, the best cycling performance could not be obtained. Inves-
tigating the issue, it turned out that the PEGDME component – which additionally
contains a polymerization inhibitor and up to 4000 ppm trace water – can disintegrate
and dissolve significant amounts of metallic lithium, see Fig. 3.13.
When drying the solvent using molecular sieves, or adding salts to the solution,
this problem can be alleviated greatly. Nevertheless, this finding set off a search for
48
Figure 3.13: Lithium foil incubated in different solvents for 24 h at room temperature.
From left to right: Pyr14TFSI ionic liquid (Li intact, dark), PEGDME (Li
completely dissolved), electrolyte according to Ref. [109] (Li intact, dark),
EC:DEC 1:1 (Li intact, shiny).
alternative electrolytes. As discussed in the previous section, it simultaneously became
clear that – despite the coating of the active material – there will still be dissolved
polysulfides during operation. Therefore, most electrolytes used for conventional Li-
ion batteries, such as mixtures of ethylene carbonate (EC) and diethyl carbonate (DEC),
can be ruled out right away. Of course, the new electrolyte should maintain as many
of the attractive properties reported in Ref. [109] as possible. Therefore, the ionic
liquid portion, responsible for most of the benefits, is not changed. Instead, different
combinations are evaluated as listed in Tab. 2.2 on page 25. Various properties of these
electrolytes are compared against the base electrolyte (#1). Since the density and the
mean molecular weight of the PEGDME used are very similar to that of pure TEGDME
(i.e. PEGDME with n = 4), this solvent is a natural choice. In addition, the mixture
of DOL/DME is evaluated because various references reported good performance
of either the pure solvent or various blends: according to Ref. [103], DOL/DME is
actually the most commonly used electrolyte for Li/S batteries. Finally, a mixture of
PC, EC, and DEC, an electrolyte typically used for Li-ion cells, is tested for comparison.
The conductivity is calculated from high frequency impedance measurements. Elec-
trochemical stability was analyzed by cycling the electrolyte in cells with two stainless
steel electrodes in the voltage range of 0.0–5.0 V. The column “wetting” relates to
the time it takes for the electrolyte to soak the separator, which ranges from ∼ 1 s for
DOL/DME to about 30 s for the PEGDME based electrolytes. The cycling performance
was evaluated at various rates by Min-Kyu Song3 (data not shown). The data was ob-
tained using a different kind of sulfur cathode, as reported in Ref. [76]. Nevertheless,
the results are considered to be qualitatively applicable for this work as well.
3The Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA
94720, USA
49
electrolyte conductivity
electrochem.
stability
wetting/
viscosity
cycling
rate
capability
#1 0.044 S/m ?†–4.4 V poor poor decent
#2 0.051 S/m 1.5–3.7 V poor good decent
#3 0.096 S/m 1.5–3.6 V poor good decent
#4 —† 1.6–3.4 V poor good decent
#5 0.199 S/m 1.3–4.2 V good good good
#6 0.047 S/m 1.4–4.0 V good good good
#7 0.232 S/m 1.5–4.2 V very good very poor —‡
Table 3.2: Summary of the properties of various electrolytes evaluated as a replacement
of electrolyte #1. The composition of each electrolyte is listed in Tab. 2.2.
Annotations: †data not available. ‡cell capacity smaller than instrument
noise.
Because of the combination of good wetting and rate capability, the mixture con-
taining DOL/DME is chosen as the new standard electrolyte. Another benefit of the
DOL/DME mixture is its low dielectric constant (er = 7.2), which correlates to a lower
voltage hysteresis compared to otherwise similar electrolytes [138]. Since the higher
salt concentration does not seem to provide any advantages, the variant with the lower
concentration (#5) is chosen. Of course, DOL and especially DME has a higher vapor
pressure and lower flashpoint than PEGDME, thereby partly compromising the inher-
ent safety of the electrolyte. A systematic study investigating exactly this mixture was
reported by Matic et al. [139], unfortunately only after this work was completed. Their
findings indicate that the optimum mixture contains only 5–15 % DOL/DME. Already
a small addition of organic solvent to the ionic liquid can result in considerably lower
viscosity, better wetting, and higher ionic conductivity while still maintaining high
safety, see also Ref. [140]. Nevertheless, the electrolyte identified in this study works
very well with the novel cathode material, as can be seen in the following sections.
3.2 Analysis of full cells
While the investigations at the material and electrode level presented above are most
important for understanding and improving the composition and structure of the cell’s
components, it is the analysis of full cells that matters most from the application’s point
of view. Therefore the performance at cell level is a key criterion for the evaluation of
battery materials.
50
3.2.1 Introduction
Many different properties of a battery can be studied by means of various electro-
chemical full-cell measurements. The most common and most direct way to analyze
a battery is the recording of charge/discharge traces, either voltage vs. time or volt-
age vs. capacity. A characteristic charge/discharge profile is shown and annotated in
Fig. 3.14. Several distinct stages are visible in the plot: Starting from a fully charged
cell, the voltage stabilizes at an upper plateau (I), while solid S8 is available and ac-
cessible [141]. The concentration of polysulfides increases during that phase, both
because of the dissolution of S8 and because of the reduction and disproportionation
of polysulfides, forming dissolved S2−x species with 8 > x ≥ 4, cf. Refs. [141, 142].
These processes are balanced until the supply of S8 diminishes. Once it cannot keep
pace with the reduction reactions anymore, the average length x of the dissolved poly-
sulfides starts to decrease. The solubility and the molar Gibbs free energy are lower
for shorter polysulfides [143], which directly translates to a lower free energy of reac-
tion ∆Gq for the reduction steps forming shorter polysulfides. Since the equilibrium
potential is proportional to ∆Gq (see e.g. [144, chap. 1.2]), this corresponds to a de-
creasing cell voltage during this stage (II). When x approaches 4, the formation of
even shorter polysulfides (x = 1–3) is enhanced [142]. Their solubility is considerably
lower [145], so that the saturation concentration of S2− will be reached quickly and
precipitation of Li2S starts to occur. Whether or not S2−2 may also precipitate as Li2S2
is a current subject of debate, beyond the scope of this explanation [146, 147]. De-
pending on the cell’s design and composition, a significant level of supersaturation is
required to induce nucleation of Li2S, though. Once a sufficient amount of precipitate
is formed, a dynamic equilibrium is established, buffering the concentrations of the
short polysulfides [141, 148, 149] and hence the cell voltage (III). Most of the discharge
capacity is actually obtained at this low-plateau phase during which the polysulfides
are continuously converted to solid Li2S (or maybe also other solids, see note above).
The discharge ends if a) there is no pathway for electrons to meet with the dissolved
ions anymore [150], e.g. if the entire carbon surface is covered by insulating material
[151, 152], or b) the electrode is clogged by precipitate so that any remaining active
material is not accessible to the electrolyte [153], or c) all active material is actually
used up [99]. Depending on a cell’s composition, the rate of discharge, and the SoH,
either of these factors may trigger the end of discharge.
During charging, the situation is different: Dissolved S2− is oxidized along with the
other dissolved polysulfides, so that its concentration quickly drops below the satura-
tion concentration and Li2S is re-dissolved. The solubility of dissolved S2−x increases
with increasing x [154]. As a result, the concentrations of all polysulfide species are
replenished more or less simultaneously, as confirmed by simulations in section 5.2.3.
51
1.5
2.0
2.5
3.0
0 250 500 750 1000
I
II
III
IV
V
C
el
lv
ol
ta
ge
/
V
Capacity / Ah/kgS
discharge
charge
I formation of PS
II reduction of PS
III
precipitation of
Li2S
IV
oxidation of PS
and formation
of S8
V
depletion of
dissolved PS
Figure 3.14: Representative discharge/charge profile of a liquid-electrolyte Li/S cell.
Characteristic stages are labeled and described in the text.
When the concentration of dissolved S8 becomes high enough, solid S8 can be formed
again [155, 156]. The average length of the dissolved polysulfides increases steadily
and therefore, plateaus are hardly discernible (IV). Since the precipitating S8 occupies
a smaller volume than the corresponding amount of Li2S, pore clogging is usually not
an issue during charging. Instead, the end of charge is triggered by the depletion of
the electrolyte of polysulfides, whether all material is used up [148] or the dissolution
of Li2S cannot keep up with the oxidation reactions [157]. Either way, this causes a
steep rise of the cell voltage. In order to fully recharge the cell, a constant voltage
phase is applied at the end of charge (V), during which the current density drops
steadily, but not immediately, until it reaches the order of magnitude of the leak cur-
rent. This technique is known as CCCV charging and also used for many other types
of batteries, cf. [158]. This way, additional capacity is recovered without the need for
hazardously high voltages or excessively long charging times. Whether or not Li/S
cells are also suitable for boost charging [159] has not yet been investigated.
Of course, the specific shape of the profile does depend on the composition of the
cell. For dissimilar systems, e.g. with solid electrolytes, the charge/discharge profiles
look rather different. For liquid-electrolyte Li/S cells, however, the profile shown in
Fig. 3.14 is very typical and was observed for all cells in this work.
52
3.2.2 Metrological considerations
For comparison with other published results, the most useful figure besides the cell
voltage is the specific capacity of the cell, i.e. the charge transferred per unit mass. It is
measured in Ah · kg−1, either specified with respect to the mass of sulfur (1675 Ah · kg−1
being the maximum theoretical capacity), or with respect to the total electrode mass,
see also section A.2 in the appendix. As already mentioned, there are various issues
associated with the determination of the true capacity of each individual coin cell. Be-
fore results will be reported in the following sections, an attempt is made to quantify
the error by summing up the sources of deviations and uncertainties:
• First, the amount of LiOH, LiO2, and impurities in the sample is estimated to be
10± 8 % of the Li2S weight as determined by EDX.
• A fraction of the Li2S particles is very large even after ball milling. While there
are only very few large particles, they contain a disproportionately high fraction
of the Li2S, estimated to be 4± 2 %. Some of these particles will be scraped off
the electrode in the doctor blading step, but anyway, they do not contribute to
the cell’s capacity, since the bulk material of those large particles is not electro-
chemically accessible because of high diffusion overpotentials.
• During CVD coating, the amount of additional carbon is only known to ±1 %.
• Weighing and mixing precision during slurry preparation is ±4 %.
• Weighing of the final electrodes is precise to ±2 %.
• The electrodes are always weighed including the current collector, which ac-
counts for the largest part of the electrodes’ weight and is not perfectly uniform.
The fluctuation is determined to be ±0.13 mg by a dedicated series of measure-
ments (data not shown). This corresponds to an uncertainty in the capacity of
±9 % for a 10 mg electrode containing 1.5 mg of active material.
In summary, the true capacity is expected to be 14± 13 % higher than specified. This
uncertainty is unpleasantly high, but also difficult to improve. Nothing is really known
about the impurity of the initial Li2S powder and without exchanging the material or
equipment, the other sources of uncertainty cannot be eliminated. Except for the
weighing precisions, however, the above errors affect all cells – or at least all cells of
one batch – in a similar way, so that a comparison of different results within this work
is still possible with reasonably good accuracy. The absolute values reported most
probably underestimate the true capacity, however.
53
3.2.3 Slow cyclic voltammetry
As reported in Ref. [78], positive electrodes made from Li2S need to be activated in
order to access the full capacity of the material. This effect is also observed for the cells
presented here. Without activation, the capacity obtained for the first cycle is consid-
erably lower. It may slowly rise during the following cycles, but even the maximum
capacity – typically reached within the first 50 cycles – is lower than that of similar
cells after activation (data not shown). Therefore, a first activation (or “formation”)
cycle is run with a different protocol to access the full capacity. Cyclic voltammetry
was used to optimize this activation protocol.
In general higher voltages, higher temperature, slower charge rates, and longer rest
times at elevated voltage can increase material utilization during the activation cycle.
To begin with, a very simple approach is tested: The cell is charged up to 4.0 V, i.e.
to the edge of the window of electrochemical stability of the electrolyte (cf. Tab. 3.2).
Starting from the pristine (discharged) state, the cell was cycled three times at a fixed
rate of 25 µV/s in the range of 1.5–4.0 V, see Fig. 3.15.
-100
-50
0
50
100
150
1.5 2 2.5 3 3.5 4
C
ur
re
nt
/
µA
Voltage vs. Li/Li+ / V
1st
2nd
3rd
Figure 3.15: The first three cycles of a cell. Voltage sweep rate: 25 µV/s. Charging up
to 4.0 V is needed to activate all of the Li2S. Only during the first cycle,
however, a significant amount of charge is passed above 2.8 V.
Since the voltage changes quasistatically and with a constant sweep rate, peaks in
Fig. 3.15 correspond to voltage plateaus in a galvanostatic discharge profile, as in
Fig. 3.14. During the discharging step, two cathodic peaks are apparent which are
well separated, with a small peak or shoulder in between. After activation, i.e. for the
second and third cycle, two anodic peaks are detected during the charging step, which
are relatively close to each other and overlap significantly. Since in this representation,
one peak corresponds to one electrochemical reaction (or one set of simultaneously
54
occurring electrochemical reactions), it becomes immediately clear that the chemistry
of the cell is dominated by different reactions during the early and late discharge and
charge, respectively. Regardless of the specific cell geometry or composition, this is a
characteristic feature of the Li/S system.
The results confirm the behavior reported in Ref. [78]: The first charge looks distinc-
tively different from the following charges. Two conclusions can be drawn from the
results presented in Fig. 3.15: First, during the initial charge most of the capacity is
obtained at voltages above 2.8 V, i.e. above the charge cutoff voltage used throughout
this work. Therefore, in order to access as much capacity of the cell as possible, it is
necessary to exceed 2.8 V during activation. Second, for the following cycles, almost all
of the capacity is recovered within the peaks at 2.3 and 2.5 V. While a little capacity is
lost from the second to the third cycle, the shape of the CV profiles remains the same.
There is virtually no benefit in using a higher voltage for a second time. Actually, it
is not even clear whether the little remaining capacity above 2.8 V is reversible. Op-
erating the cells at high voltages is potentially harmful, since it increases stress in the
electrode material. Also, it might open up pathways for electrochemical degradation
reactions, causing a leak current which must not be attributed to Li2S oxidation. Thus,
for subsequent tests, lower voltages are used instead, but for prolonged times. After
the initial activation, the cells are charged up to 2.8 V only.
An improved activation protocol is presented in Fig. 3.16a: The voltage is increased
to 3.4 V instead of 4.0 V, but at a lower rate of 10 µV/s. While almost the same capac-
ity is obtained, a significant current is still passed at 3.4 V. In order to collect all the
charge associated with that current, but without further lowering the sweep rate, the
following protocol was tested: After the upsweep, the voltage is held at 3.4 V for 20 h.
This way, even more capacity can be accumulated, cf. Fig. 3.16b. The standard Itot vs.
E plot of this experiment looks unintuitive, though. To better analyze the situation,
the plot is changed from the traditional Itot vs. E to a Qtot and E vs. t plot as presented
in the right column of Fig. 3.16. Here, the cell voltage E (red line) is enforced by the
potentiostat and can be interpreted as the “input”, whereas the total accumulated ca-
pacity Qtot (black line) is the “response”. In this representation it becomes obvious that
at the time the charge cutoff voltage is reached, the capacity is not yet at its maximum,
see Fig. 3.16a. Even while the voltage is swept down, the capacity still increases until
E drops below the cell’s OCV. In Fig. 3.16b, however, the capacity levels off during the
constant voltage phase, meaning that all charge available at that potential is actually
collected.
As a result, the activation protocol presented in Fig. 3.16b is established as a stan-
dard for the following experiments. As a slight variation, this protocol is usually
preceded by a constant-current charging step.
55
-100
-50
0
50
100
1.5 2 2.5 3 3.5
C
ur
re
nt
/
A
h/
kg
S
Voltage vs. Li/Li+ / V
0
200
400
600
800
1000
0 20 40 60 80 100
1.0
1.5
2.0
2.5
3.0
3.5
C
ap
ac
it
y
/
A
h/
kg
S
C
el
l v
ol
ta
ge
/
V
Time / h
(a) Upsweep to 3.4 V, immediately followed by the downsweep to 1.5 V.
-200
-150
-100
-50
0
50
100
1.5 2 2.5 3 3.5
C
ur
re
nt
/
A
h/
kg
S
Voltage vs. Li/Li+ / V
0
200
400
600
800
1000
0 20 40 60 80 100
1.0
1.5
2.0
2.5
3.0
3.5
C
ap
ac
it
y
/
A
h/
kg
S
C
el
lv
ol
ta
ge
/
V
Time / h
(b) Upsweep to 3.4 V, followed by a 20 h rest at constant voltage and a downsweep to 1.5 V. Note that in
the left representation, the extra capacity collected during the constant voltage phase is hidden in the
vertical line at 3.4 V.
Figure 3.16: Two different activation protocols, analyzed by slow cyclic voltamme-
try. Left column: Standard representation, i.e. current plotted vs. voltage.
Right: Voltage (red) and charge capacity (black) plotted vs. time.
3.2.4 Impedance spectroscopy
A typical impedance spectrum of a Li/S cell is shown in Fig. 3.17. There are several
distinct features, which can be evaluated qualitatively directly from the graphical rep-
resentation, see [132, chap. 18]: First, the high frequency intersect with the real axis
directly correlates to charge transport in the liquid electrolyte. Second, the size of the
semicircle correlates to the average charge-transfer resistance at the carbon surface.
Additional overlapping or separate semicircles may be discernible, representing dif-
ferent surfaces, e.g. the lithium electrode. Third, the low-frequency tail is governed by
material transport and changing SoC. A straightforward phenomenological approach
to interpret EIS results is the creation of equivalent circuit analogs [132, chap. 9]. While
this method is suitable for describing even complex electrode morphologies, the val-
56
0200
400
600
800
0 200 400 600 800
50 mHz
6 Hz
−Z
”
/
Ω
·c
m
2
Z’ / Ω · cm2
0
50
100
150
200
250
0 50 100 150 200 250
14 mHz
6 Hz350 kHz
−Z
”
/
Ω
·c
m
2
Z’ / Ω · cm2
0
5
10
15
20
0 5 10 15 20
350 kHz
2 kHz
−Z
”
/
Ω
·c
m
2
Z’ / Ω · cm2
Figure 3.17: Example impedance spectrum of a pristine Li/S cell at three different
magnifications, emphasizing different features of the spectrum.
ues assigned to the various circuit elements have only limited physical meaning. Of-
ten, they represent complex processes in an oversimplified fashion. Nevertheless, the
recorded impedance spectra have a very high information content and are thus equally
useful for the determination of parameters in more complex models as presented in
chapter 4.
EIS can also be performed at intermittent intervals during operation. Since the com-
position of the cell changes during cycling, so do the impedance spectra. In principle,
these results can be used to estimate the SoC of the cell. Impedance spectra recorded
at four distinct stages during the discharge are plotted in Fig. 3.18.
Unfortunately, it turned out that the impedance spectra are not very reproducible,
even when looking at cells of the same batch, see Fig. 3.19a. Also, the spectra of one
cell, recorded continuously at short intervals do not vary smoothly, but rather messily,
see Fig. 3.19b. This is either because the spectrum actually has a very sensitive and
complicated dependence on the SoC or – more likely – because the EIS measurements
themselves interfere with the cell’s behavior.
This instability was found to have its origin mainly in the lithium electrode, see
Fig. 3.19c. Here, spectra of several symmetrical cells (composed of two similar lithium
metal electrodes) are plotted. The variance is almost as large as the entire fluctuation
observed for regular Li/S cells in Fig. 3.19a. Even worse, the electrodes’ impedance
also changes to a great extent during storage and cycling, see Fig. 3.19d, where the EIS
results obtained from one Li electrode at different times are plotted. The fluctuations
were discovered to become smaller after several charge/discharge cycles (data not
shown); nevertheless, the above findings disqualify EIS as a tool for quantitative SoC or
SoH analysis for the present cells. Unstable impedance of the lithium metal electrode
is a phenomenon reported for various electrolytes in the literature, e.g. in Ref. [160].
In order to separate the contributions of the anode and the cathode, either measure-
ments with a three-electrode cell or with symmetrical cells would be needed.
57
1.5
2
2.5
0 0.5 1 1.5
•
• •
•Ce
ll
Vo
lt
ag
e
/
V
Time / h
0
100
200
300
400
0 100 200 300 400
−Z
”
/
Ω
·c
m
2
Z’ / Ω · cm2
Figure 3.18: Impedance spectra recorded at regular intervals during discharging. Left:
discharge profile. Right: corresponding impedance spectra. Colored spots
indicate the times at which each spectrum was recorded.
0
100
200
300
400
0 100 200 300 400 500 600
−Z
”
/
Ω
·c
m
2
Z’ / Ω · cm2
(a) EIS of several similar, freshly assembled cells.
0
1000
2000
3000
0 1000 2000 3000 4000
−Z
”
/
Ω
·c
m
2
Z’ / Ω · cm2
(b) EIS of a single cell, recorded every minute during
an entire charge/discharge cycle.
0
200
400
600
0 200 400 600 800
−Z
”
/
Ω
·c
m
2
Z’ / Ω · cm2
(c) EIS of several similar, symmetrical Li/Li cells.
0
500
1000
0 500 1000 1500 2000
−Z
”
/
Ω
·c
m
2
Z’ / Ω · cm2
initial
16 h rest at 40 °C
24 h rest at 40 °C
cycled (fwd)
cycled (rev)
(d) EIS of one Li/Li cell at different times.
Figure 3.19: Impedance spectra of similar cells. The purpose of this figure is not to
represent the data in detail but to illustrate issues with the reproducibility
of the EIS measurements.
58
For the reasons outlined in section 2.5, however, such data is not available. There-
fore, the value of the impedance spectra is limited and they cannot be used for model
calibration below.
3.2.5 Charge/discharge profiles
The typical shape of a galvanostatic charge/discharge profile was already explained
above. In the following, the evolution of such profiles over several cycles is analyzed.
To this end, the first ten cycles of a Li/S cell are plotted as a continuous line in Fig. 3.20.
The cell is activated at 3.4 V as discussed in section 3.2.3. Notably, almost half of its
capacity is extracted during the constant voltage phase, emphasizing both the need for
and the effectivity of the constant-voltage step. After activation, cycling is performed
at a C/10 rate. Here, it is interesting to note that – despite a little degradation during
each cycle – the plot collectively “drifts” to the left.
1
1.5
2
2.5
3
3.5
0 200 400 600
loading: 0.99 mg/cm2
composition: 80 % Li2S (coated), 10 % carbon, 10 % SBR
50 % Pyr14TFSI, 50 % DOL/DME + 1 M LiTFSI, 0.5 M LiNO3
act.
act.
9 . . . 1
9 . . . 1
C
el
l v
ol
ta
ge
/
V
Capacity / Ah/kgS
Figure 3.20: Activation and first ten cycles plotted continuously against the total ca-
pacity. Activation at C/50 (charge) and C/20 (discharge). Subsequent
cycles at C/10 (charge and discharge). The experiment starts with a fully
discharged cell, i.e. at capacity= 0. Labels indicate the cycle index.
While the reasons for this “drift” are not clear, two reasonable explanations can be
proposed: First, additional Li2S could be activated during each cycle. While not all of
the theoretical capacity is made available in the activation step, this does not necessar-
ily mean that all of the remaining Li2S is completely inaccessible. Instead, the material
may be released slowly during subsequent cycles, for example from the bulk of very
large particles or agglomerates. This effect happens predominantly during the first 50
59
cycles and can actually lead to a temporary increase of the capacity, cf. e.g. Fig. 3.24.
For reasons unknown, the SBR binder used in these cells is often associated with this
behavior [161]. Second, a small fraction of the electrons transferred may not contribute
to sulfur oxidation, but instead fuel side reactions such as the polysulfide shuttle. This
cell configuration does not suffer from the shuttle effect much and consequently, no
infinite charging [51, 113] was observed ever. Still, some of the charge capacity might
enter this pathway or other side reactions.
The voltage trace can also be illustrated differently: Fig. 3.21 shows results of an-
other cell, this time plotting voltage vs. the capacity transferred during each half cycle,
resetting the abscissa to zero at the end of each charge or discharge. In this represen-
tation, each half cycle can be directly compared to the following (or preceding) one.
This way, any discrepancy can be spotted and quantified easily. Degradation can be
observed by comparing subsequent discharges. To enhance the effect, only every 10th
cycle is plotted in Fig. 3.21.
1
1.5
2
2.5
3
0 200 400 600 800 1000
loading: 1.03 mg/cm2
composition: 80 % Li2S (coated), 10 % carbon, 10 % SBR
50 % Pyr14TFSI, 50 % DOL/DME + 1 M LiTFSI, 0.3 M LiNO3
C
el
l v
ol
ta
ge
/
V
Capacity / Ah/kgS
10th
20th
40th
60th
80th
99th
120th
140th
160th
180th
199th
Figure 3.21: Continuous capacity fade over 200 cycles. Charge and discharge at a rate
of C/10. The cell voltage is plotted against the capacity transferred during
each half-cycle.
Besides the capacity fade, the most prominent effect of degradation is the increased
voltage hysteresis, which almost doubles during the first 200 cycles for the cell pre-
sented in Fig. 3.21. The hysteresis, i.e. the difference in voltage at the same SoC during
charge and discharge, is directly proportional to the anergy, i.e. the energy lost as heat
during one cycle. The exergy, in contrast, is proportional to the integrated area under
the discharge curve and finally, the total work performed is equal to the area under the
60
charge curve. An increased hysteresis therefore means a lower charge/discharge en-
ergy efficiency of the cell. In addition, it makes it increasingly harder to fully recharge
the cell. Finally, the shape of the discharge/charge profiles alters over time, too. The
most obvious changes are the upper plateau to lower plateau ratio during discharge
and the voltage slope of the charging branch. A quantification of these (and other)
features for comparison with data extracted from reference profiles would allow for a
phenomenological SoH estimation without the need for a detailed analysis of degra-
dation effects [162].
The first and most obvious conclusion from this data, however, is that degradation
of the cell happens at a non-negligible rate, thereby limiting the cyclability of the cell.
This effect is best analyzed by looking at plots of capacity vs. cycle number, presented
in the following section.
3.2.6 Cycling
Since both cycle and calendar life of the Li/S cells presented in this work are rather
limited, the quantification of the SoH is very important. To this, the measured capacity
of a cell is plotted vs. cycle number in Fig. 3.22. The data was recorded from one of the
first batches of the novel electrodes based on carbon-coated Li2S, using the material
presented in Fig. 3.4b. The results confirm that this material is generally suitable for
Li/S cells, despite the agglomeration. However, they also indicate an unattractively
high rate of degradation. This cell was built and tested before the electrolyte was
optimized (cf. section 3.1.5). Whether or not the degradation is influenced by the
agglomeration and/or the deficient electrolyte was not investigated.
0
200
400
600
800
1000
1200
1400
1600
0 50 100 150 200
0
200
400
600
800
1000
loading: 0.57 mg/cm2
composition: 68 % Li2S (coated), 23 % carbon, 9 % SBR
66 % PEGDME, 33 % Pyr14TFSI + 1 M LiTFSI, 0.1 M LiNO3
C
ap
ac
it
y
/
A
h/
kg
S
C
ap
ac
it
y
/
A
h/
kg
el
ec
tr
od
e
Cycle
discharge capacity
charge capacity
Figure 3.22: Performance of the initial material and cell composition. Charge and dis-
charge at a C/5 rate.
61
Cells are also tested at different rates. For many applications it is a crucial property
of the battery to be able to deliver power at various rates. Results of a cell cycled with
different discharge currents are presented in Fig. 3.23. This cell uses the best active
material (see Fig. 3.4c), but the electrode preparation is not fully optimized yet (no
images available, but processing similar to the electrode presented in Fig. 3.5c). This
particular batch of cells did not perform outstandingly and the experiment was not
repeated with later batches, due to lack of channels. In addition, the previously men-
tioned effect of increasing capacity during the first couple of cycles is observed and
overlaps with the rate survey – compare for example cycles 10 and 35 (both C/20).
Nevertheless, this plot proves that the Li2S based cells with a high content of active
material can still be operated at moderately high rates, albeit with a significant perfor-
mance penalty for rates above C/5. Adding more carbon and binder would most likely
improve the rate capability further, but at the expense of the cell’s specific energy.
0
200
400
600
0 5 10 15 20 25 30 35 40
0
100
200
300
400
loading: 0.84 mg/cm2
composition: 80 % Li2S (coated), 10 % carbon, 10 % SBR
50 % Pyr14TFSI, 50 % DOL/DME + 1 M LiTFSI, 0.5 M LiNO3
C/50
C/20
C/10 C/5
C/2
1C
C/20
C
ap
ac
it
y
/
A
h/
kg
S
C
ap
ac
it
y
/
A
h/
kg
el
ec
tr
od
e
Cycle
discharge capacity
charge capacity
Figure 3.23: Rate capability test. The cell was charged with a constant rate of C/20.
Outlier suppressed.
After the various improvements to the slurry composition, coating, mixing, and
casting procedure as well as the electrolyte and activation protocol, described in the
previous sections, the performance of the cells is improved substantially, see Fig. 3.24.
This cell utilizes the active material presented in Fig. 3.4c and electrodes as presented
in Fig. 3.5d. Here, a slow cycle is added every 25 cycles to test for the maximum
charge which can still be extracted (but not at the regular C/10 cycling rate). Again,
the capacity rises slightly during the first 60 cycles, before the usually observed con-
tinuous decay takes over. While the loading is still slightly lower than intended, it is
62
considerably higher than in previous studies. The initial capacity is not significantly
improved, but the rate of degradation is reduced by more than 60 % and averages to a
mere 3.9h per cycle on average, and 6.1h per cycle at the 200th cycle.
0
200
400
600
800
1000
1200
1400
1600
0 50 100 150 200
0
200
400
600
800
1000
loading: 1.03 mg/cm2
composition: 80 % Li2S (coated), 10 % carbon, 10 % SBR
50 % Pyr14TFSI, 50 % DOL/DME + 1 M LiTFSI, 0.3 M LiNO3
C
ap
ac
it
y
/
A
h/
kg
S
C
ap
ac
it
y
/
A
h/
kg
el
ec
tr
od
e
Cycle
discharge capacity
charge capacity
Figure 3.24: Performance of the optimized electrodes. Charge and discharge at a C/10
rate; every 25th cycle at C/50.
Still, there is lots of room for improvement. This is especially apparent when look-
ing at the Coulombic efficiency for the above experiment, presented in Fig. 3.25. The
slow cycles (25, 50, 75, ...) are suppressed in this plot, but their effect is still visible:
The Coulombic efficiency is slightly increased for several cycles after most of the slow
cycles – an effect probably related to the “refreshing” character of slower cycles, dis-
cussed in the next chapter on the basis of Fig. 5.3. Disregarding these fluctuations, the
Coulombic efficiency stabilizes at around 98.5 %, a value that is considerably too low
for stable long-term operation [163, 164]. In effect, this result indicates that ∼1.5 % of
the current enters parasitic reactions, eventually causing loss of material and degrada-
tion. Unfortunately, the analysis and mitigation of these effects could not be completed
within this study and remains a goal for future efforts.
63
90
92
94
96
98
100
0 50 100 150 200
C
ou
lo
m
bi
c
ef
fic
ie
nc
y
/
%
Cycle
Figure 3.25: Coulombic efficiency of the cell presented in Fig. 3.24. Slow cycles sup-
pressed.
64
3.3 Conclusions
3.3.1 Achievements
A setup and protocol for chemical vapor deposition of carbon films onto Li2S particles
was developed and optimized. To this end, the CVD process, but also the entire ma-
terial and electrode preparation was analyzed and adjusted. The coating was applied
to several batches of ball-milled Li2S. The resulting material, as well as full cells built
from the material, were characterized. Considering the results reported in section 3.1,
the coating method developed in this work is falling short of its primary goal, i.e.
the prevention of polysulfide dissolution. Despite this drawback, the material is well
suited for advanced Li/S batteries. While the cycling performance is still far from
perfect, it is considerably increased compared to published results of Li2S-based elec-
trodes, cf. Fig. 3.26. The interpretation of the plot is somewhat hindered by the fact
that the data reported was recorded at different charge/discharge rates. Still, Fig. 3.26
clearly makes the point of showing the capability of the novel material to achieve state
of the art cycling performance.
0
400
800
1200
1600
0 50 100 150 200
C/72
C/10
C/5
C/10
C
ap
ac
it
y
/
A
h/
kg
S
Cycle
uncoated (Ref. [77])
coated (Ref. [71])
coated (this work, initial)
coated (this work, optimized)
Figure 3.26: Comparison of the cycle life of various Li2S based electrodes.
Since these experiments were reported at several conferences before similar results
were published, it is the merit of this work to demonstrate for the first time that
functional electrodes can be built from carbon-coated Li2S particles. Significant im-
provements were achieved concerning the initial capacity and cycle life. Ultimately,
this effort builds the groundwork for later achievements, e.g. Refs. [133] and [165], by
establishing the CVD coating procedure, electrode preparation, and analysis methods.
65
3.3.2 Possible improvements and outlook
This section is not to be confused with the outlook in chapter 6.3. While in the lat-
ter a look at the big picture is presented, this section elaborates on more immediate
improvements to the existing experimental methods and setup.
Processing. After the work for this thesis was completed in late 2013, the issue of
non-uniform particle size associated with the ball-milling of Li2S was solved in Elton
Cairns’ group by synthesizing particles of well-defined size and shape, see Ref. [133].
Additionally, coating uniformness could be greatly improved while simultaneously
reducing the complexity of the coating process by using a rotary furnace. This way,
the intermittent mixing steps could be omitted and the continuous stirring is expected
to reduce particle agglomeration, too. This would enable an additional annealing
step at elevated temperature after the coating is completed without causing excessive
agglomeration. These advantages were already known at the time the experiments
were conducted. However, no rotary furnace was available for testing. Therefore, a
simple hand-operated rotary furnace tube was designed and tested in the regular tube
furnace, cf. Fig. 3.27. While the properties of the coating were essentially unchanged,
initial results suggest that agglomeration could be prevented to some degree even with
this simple setup.
Actually, on grounds of these findings, a rotary furnace was set up at LBNL re-
cently, which enables a more uniform coating of the Li2S particles. As a result of these
measures, the performance of the new active material is expected to improve further.
16
m
m
25 mm
Figure 3.27: Photo of the self-made, hand-operated “rotary furnace” (left). The inner
tube is fixed inside the outer quartz tube eccentrically (right). When rotat-
ing the outer tube, the inner tube is turned upside down, stirring up the
Li2S powder inside.
Concerning the electrodes, spraying instead of doctor blading could improve both
uniformness and loading [166, 167]. Finally, calendering, a standard step in the prepa-
ration of electrodes, was omitted because no calendering press was available inside a
glovebox, but also because a low porosity might cause problems when operating the
cell at higher rates, see section 5.1. Besides reducing the porosity of the positive elec-
trode and thereby the amount of electrolyte needed, calendering is known to enhance
electronic conductivity and mechanical durability [168].
66
Materials. Concerning the choice of materials, a few possible improvements come
to mind: First, the separator is not specifically optimized for Li/S cells. Both func-
tionalized polypropylene and carbon nano-/microfiber separators are available which
could probably improve the wetting along with other benefits [92, 169–172]. Second,
concerning the electrolyte, Pyr14TFSI could be exchanged for Pyr14FSI which is not
only less corrosive, but also considerably cheaper. Furthermore, mixtures of these and
other anions are known to improve conductivity as well as low temperature perfor-
mance [173–175]. The same is true for the cations, where Pyr13, Pyr14, and Pyr15 could
be mixed as well as pyrrolidinium derivatives, like Pyr12O1, cf. Fig. 3.28. Third, more
electrolyte additives could be tested, e.g. those mentioned in Refs. [82, 83, 160, 176].
⊕
N
⊕
N
⊕
N
O
F3C
S
O
O
	
N
S
O
O
CF3 F
S
O
O
	
N
S
O
O
F
Figure 3.28: Possible improved composition of the ionic liquid portion of the elec-
trolyte as suggested by Ref. [177]. Details see text.
While the experimental results reported above constitute a noteworthy piece of work
in their own right, they also serve as input to the modeling study presented in the
following chapters.
67
4 Modeling the lithium-sulfur battery –
theory and methods
4.1 Background
Modeling has become an established, if not essential part of the toolset of electrochem-
ical analysis methods. Not only can it save considerable cost and time in the research
and development of batteries, but also it enables a more profound understanding than
possible by mere experimentation. Therefore, modeling has penetrated all aspects of
battery development and design, ranging from system layout to atomistic reaction
kinetics [178], see also Fig. 4.1. Small-scale models typically perform ab initio calcu-
lations, which make use of quantum mechanics and fundamental physical constants
to determine parameters of materials and reactions without experimental input [179].
Despite some simplifying assumptions, this approach is computationally very inten-
sive and thus limited by the size of the system (typically to a few dozen atoms). It
needs to be complimented by other techniques, e.g. Monte-Carlo simulations, in order
to bridge the gap to electrochemical applications [180, 181]. Large scale, system level
simulations on the other hand mostly use an equivalent circuit [91] or black box rep-
resentation of the battery, which is described by an entirely empirical set of equations,
see e.g. Ref. [182]. This approach is computationally inexpensive to the point where
the algorithms can be implemented in real-time, embedded battery management sys-
tems [183–185]. With regard to the electrochemistry, however, the insight which can
be gained by using such a model is very limited as is its extent of validity [186, 187].
An overview of the different types of models can be found in Ref. [188]. For this work,
an intermediate level is chosen both in terms of scale and computational complexity:
A physically-based continuum approach is taken to model the membrane-electrode
assembly (MEA) of the cell in detail. The system is determined by a set of differential
algebraic equations describing physical processes such as reaction kinetics, transport,
or the evolution of phases on a length scale larger than individual structural details
of the electrodes, which is why this type of model is often referred to as continuum
model.
68
Repeat unit
Single cell
Battery pack
Electrode
Particle
0.1–1 m
1–10 cm
50–200 µm
1–50 µm
50–1000 nm
0.1–10 nm Time scale
Le
ng
th
sc
al
e
Atomistic
structure
LiLi+
e− Li+
nu
mb
er
of
pa
ram
ete
rs
co
mp
ut
ati
on
al
co
st
Figure 4.1: Hierarchy of modeling detail and system size. Adapted from [189]; battery
pack image reproduced from [190].
The modeling activities were carried out in the group of Prof. Wolfgang G. Bessler at
the German Aerospace Center1 (DLR) as well as the group of Prof. Arnulf Latz at DLR
and the Helmholtz Institute Ulm for Electrochemical Energy Storage (HIU)2. Parts of
this work have already been published in [P1–P4].
The process of building a physically-based model essentially consists of three steps:
1. choosing a set of phenomena to be described by the model
2. formulating the governing equations and connecting them in a meaningful way
to come up with a consistent description of the problem
3. assigning values to all parameters introduced in step two
This work is building on top of an existing modeling framework (see below). Hence,
for most aspects of the model step one and for several even step two have been com-
pleted in advance. Before the steps taken in this work are described in detail, however,
1Institute of Engineering Thermodynamics, German Aerospace Center (DLR), Pfaffenwaldring 38–40,
70569 Stuttgart, Germany
2Helmholtz Institute Ulm for Electrochemical Energy Storage, Helmholtzstr. 11, 89081 Ulm, Germany
69
existing implementations of Li/S models shall be discussed. The selection presented
here is strictly limited to Li/S continuum models. A rigorous derivation of nonequi-
librium thermodynamics of porous electrodes can be found in Ref. [193]; a more com-
prehensive overview of battery modeling literature can be found in Refs. [188] or [194,
chap. 2.5]. Despite this restriction, there are several noteworthy works whose achieve-
ments (and shortcomings) shall be analyzed in the following.
One of the early protagonists in the recent advancement of Li/S batteries is Sion
Power Corporation [45], which started developing a model of the Li/S cell as early
as 2003 [51]. In this model, the chemistry of the cell is described by only two electro-
chemical reactions, implemented as a Nernst equation each. Only during charging,
the polysulfide shuttle is described by one additional phenomenological parameter,
the so-called shuttle constant ks, which is in effect a measure of the speed of the reac-
tions responsible for the shuttle relative to the charge/discharge reactions. This rather
simple zero-dimensional model does not include any description of transport, double-
layer capacity, or phase changes and only a very simple temperature dependence.
In a joint project of the same company with the group of R. E. White at the University
of South Carolina, a more detailed model was developed in 2007 [195, 196]. This
model is, by design, similar to the work presented here. It differs, however, at the
choice of processes to be included in the model as well as at the formulation of the
governing equations. It does include a physically-based one-dimensional description
of the transport in the liquid electrolyte as well as electrochemical reactions among the
dissolved polysulfides and the evolution of phases. However, the model does not treat
the negative electrode explicitly, but just as a boundary condition (a source of Li+ ions).
For this reason, the model cannot accommodate processes like the polysulfide shuttle
or degradation mechanisms, which require additional chemical reactions at the lithium
electrode. Also, the model does not include a description of the Helmholtz double
layer nor a sophisticated representation of the electrodes’ microstructure. It does not
include any side or disproportionation reactions and the publications do not report
anything on the impedance of the cell, the charging process, cycling, or degradation.
Finally, the end of discharge cannot be explained conclusively. Nevertheless, Ref. [197]
rightfully states that this is the first reasonably complete model of the Li/S cell.
Only recently (2014), another effort was started to model the Li/S cell by the group
of P. Chen at the University of Waterloo [198, 199]. This model is conceptually very
similar to the model presented in [196]. In fact, only a few parameters were chosen dif-
ferently. Instead of advancing the model itself, the merit of this two-part publication is
that it presents lots of simulation results, including a rather comprehensive sensitivity
analysis of the model. Unfortunately, the report is still limited to the discharge only.
A third part was announced, but never published.
70
Another notable contribution to the field comes from the group of Y. Cui at Stanford
University [78]. Taking a close look at the charge process starting from pure, solid
Li2S, simulations using a physically-based two-dimensional model indicate that the
rate-limiting step for charging is the initial oxidation of Li2S. This process is modeled
to occur only at the triple phase boundary of Li2S, carbon and the liquid electrolyte.
Unfortunately, this particle-scale model was never extend to describe a full cell and
Ref. [78] includes only very few simulation results.
Finally, at the Technische Universität München the groups of H. A. Gasteiger and
A. Jossen are working on what might soon be the most comprehensive and complete
model of a Li/S cell so far. An original rotating disk electrode study [44] led to
the conclusion that disproportionation reactions play an important and previously
underestimated role for the charge/discharge mechanism. While the aforementioned
publication proposes a detailed novel reaction mechanism, it does not present any
modeling or simulation results. Early reports of the modeling work disclosed at the
UECT2014 conference [200] indicate that the goal is a physically-based 2D or 3D model
including the detailed chemistry proposed in [44] as well as transport in the liquid
electrolyte and multi-phase management, implemented in a way conceptually similar
to this work.
Following this brief literature overview, the design of our model and its benefits
over existing models will be discussed in detail, including explanatory notes on its
implementation as well as the sources and choice of parameters.
4.2 Modeling approach
4.2.1 Model design and basic assumptions
While by origin and design this work is not merely a re-implementation of the popular
porous electrode model published by Newman and Tiedemann in 1975 [201], it may
still be called a Newman-class model, since it shares the model layout as well as the
description of the electrical properties of the cell and large parts of the description
of transport in the liquid electrolyte. However, it also features some extensions not
present in the 1975 model such as a detailed description of multiple phases in the
electrodes or an implementation of the complex chemistry in Li/S cells, as opposed to
a single Butler-Volmer type current source term.
Our model is composed of a one-dimensional representation of the cell, discretiz-
ing the volume of the MEA in the direction of Li+ transport (hereinafter called y-
direction), see Fig. 4.2. Because of this, the structure of the porous electrodes is not
resolved in morphological or even atomistic detail, but represented by many effective,
volume-averaged variables and parameters. To keep the parameter space manageable,
71
simplifying assumptions are needed that reduce the amount of unknown parameters.
Probably the most important of these assumptions is that the system be isothermal.
While this does limit the scope of the model, it is not expected to affect the quality of
the results. For small (coin) cells, the temperature can be controlled very effectively by
putting them in an isothermal temperature chamber, so that the assumption T = const.
is justified. Especially for the rather low rates used, thermal gradients in the cell are
negligible. Other important simplifying assumptions include the lack of convection
in the liquid electrolyte and constant 100 % activity of dissolved species which will be
discussed below.
Se
pa
ra
to
r
Po
si
ti
v e
el
ec
tr
od
e
N
eg
at
iv
e
El
ec
tr
od
e
yelectrode
void
Figure 4.2: Sketch of the computational domain of the 1D model. Current collectors
and cell casing are not part of the modeling domain. Discretization is
performed in y-direction. Each CV represents the entire cross section of
the cell, potentially including void space inside the casing.
Before describing the phenomena and governing equations in detail, nomenclature
and naming conventions need to be introduced. Except for the more precise indices,
notation in this work follows Ref. [59]; a full list of the symbols used can be found
in Tab. A.1. In each control volume (CV), several interfaces (denoted by index m)
may exist, at which chemical species (index n) present in the adjacent phases (index
p) take part in reactions (index q). Note that the same interfaces, species, phases,
and reactions are by design present in each CV within one layer of the cell (cathode,
separator, anode) but may differ from layer to layer. That said, the amount of a phase,
the concentration of a species or the active surface area of an interface may still be zero
in some CVs. Unless otherwise noted, all quantities are considered functions of space
(y) and time (t), e.g. εn ≡ εn(y, t). The entire set of equations, boundary conditions,
and parameters discussed below can also be found in Tabs. 4.1 and A.5–A.7.
4.2.2 Transport
There are numerous models of transport in the liquid electrolyte [144, 202–209]. Most
models published to date cannot be easily adapted to the Li/S system, however, be-
cause they assume explicitly or by the formulation of the governing equations that
72
there be only two charged species in the liquid electrolyte, e.g. [144, chap. 12], a com-
mon situation for Li-ion intercalation cells. This restriction is imposed for simplicity,
but also because of practical constraints: For a thermodynamically correct descrip-
tion of the transport in liquid-electrolyte batteries [203], one interaction parameter
is needed for each pair of dissolved species. For a system with many intermediate
species, the determination of these numerous parameters is difficult and often not
feasible, cf. [144, chap. 14.3]. Another approach to model cells with multiple species
in the liquid electrolyte was devised by C. Delacourt [209]. Following this work, the
interaction among the less concentrated “minor” species can be neglected and only
their interaction with the conducting salt needs to be taken into account. This model
is primarily targeted at electrolyte additives (and not catholytes), therefore some as-
sumptions of [209] do not apply to the situation of the Li/S cell. Instead, for our work
the liquid electrolyte is described as a dilute solution of ions as in Ref. [144, chap. 11].
In addition, to further simplify the problem, convection in the liquid electrolyte is
neglected, i.e. v ≡ 0. As a consequence, there is no need to calculate momentum,
which completely vanishes from the governing equations so that the system is fully
described by diffusion, migration, and the conservation of mass and charge.
The conservation of species, which of course also implies the conservation of mass
can be formulated as
∂c′n
∂t
= −∂Jn, tot
∂y
+ s˙n , (4.1)
where t is the time, y the spatial coordinate and c′n and Jn, tot the volume-averaged
concentration and total flux of species n, respectively. The chemical concentration cn
within a phase p relates to c′n as c′n = εpcn, where εp is the volume fraction of the
phase in the CV. The additional term s˙n is a volume-specific term which represents all
sources or sinks of particles such as chemical reactions. For heterogeneous reactions,
the specific surface area AVm of the corresponding interface m needs to be taken into
account so that s˙n can be expressed as
s˙n =∑
m
AVm s˙n,m , (4.2)
with s˙n,m being the net production rate of species n at the interface m, see section 4.2.3.
The total flux in Eq. (4.1) is defined as the sum of diffusion and migration
Jn, tot = Jn, diff + Jn, migr
= −Dn, eff ∂cn∂y − zn FRT · cnDn, eff
∂φelyt
∂y .
(4.3)
Here, zn denotes the electric charge of species n, F is Faraday’s constant, R the uni-
versal gas constant, T the temperature and φelyt the potential in the liquid electrolyte.
73
yJn, totJn, tot s˙n
iVF
iVdl
itot itot
CV i− 1 CV i CV i + 1
Figure 4.3: Layout of the ith control volume (CV), illustrating the source terms and
fluxes for chemical species, i.e. mass (s˙n and Jn, tot), and electrons, i.e. charge
(iVF , i
V
dl, and itot).
Finally, Dn, eff denotes the effective diffusion coefficient, which is defined as
Dn, eff = Dn · εelyte/τ2elyte , (4.4)
correcting Fick’s diffusion coefficient Dn for the volume fraction εelyte and the tortuos-
ity τelyte of the liquid electrolyte phase, which is discussed e.g. in Ref. [210]. Of course,
Eq. (4.1) needs to be solved for each species n. By postulating charge conservation
and electroneutrality (at the scale of the CVs), the number of equations to be solved is
reduced by one, since the condition
0 = ∑
n,m
znFAVm s˙n,m −∑
n
znF
∂Jn, tot
∂y
(4.5)
always holds and can be used to eliminate Eq. (4.1) for one (charged) species. In words,
the flux of the nth species is determined by the flux of the (n− 1) other species so that
the total flux (in units of charge) is equal to the amount of charge generated within the
CV. Fig. 4.3 summarizes the source terms in and fluxes across the boundaries of a CV.
4.2.3 Electrochemistry
The derivation of the (electro-)chemical rate laws follows Ref. [58], which in turn is
based on Ref. [211, sec. 9.3]. The following assumptions are required for a more com-
pact formulation of the problem: pressure and temperature shall be constant and all
electrochemical reactions occur only at interfaces (whereas chemical reactions may
also occur within a bulk phase). For the species continuity equation, Eq. (4.1), a gen-
eralized source term s˙n (or s˙n) was already defined which in turn is calculated from
74
mass-action kinetics. For each reaction q, the term s˙n,q is defined as
s˙n,q = νn,q
(
kq, fwd ∏
n′∈q, fwd
a
νn′ ,q
n′ − kq, rev ∏
n′′∈q, rev
a
νn′′ ,q
n′′
)
, (4.6)
where kfwd and krev are the forward and backward rate constants, νn the stoichiometric
coefficients of species n in the corresponding reaction q, and an the activities. The
products run over all reactants n′ or n′′ participating in the forward or reverse reaction,
respectively. The total production rate of species n at interface m is simply the sum of
the contributions of all reactions q at that interface:
s˙n,m = ∑
q∈m
s˙n,q . (4.7)
The equation for bulk reactions has exactly the same form, except that the units
of activities and rate constants differ, so that s˙n is a volume-specific production/
consumption rate instead of an area-specific one. The forward rate constant is given by
a general modified Arrhenius expression (index q omitted in the following equations):
kfwd = k˜0, fwdTβ exp
(
−Eact
RT
)
exp
(
− (1− α)zF
RT
∆φ
)
. (4.8)
Since there is no reason to assume otherwise, the symmetry factor α is chosen to be
1/2 for all reactions. Also, because the model is assumed strictly isothermal, both the
temperature factor Tβ and the activation energy Eact may be eliminated so that Eq. (4.8)
is reduced to
kfwd = k0, fwd exp
(
− zF
2RT
∆φ
)
, (4.9)
with z being the number of electrons transferred, F Faraday’s constant, R the uni-
versal gas constant, T the (now fixed) temperature, and ∆φ the potential step at the
corresponding interface: ∆φ = φelde − φelyte. The pre-exponential factor k0, fwd now
incorporates the contributions of the two temperature-dependent terms in Eq. (4.8),
which are reduced to a fixed factor in the case of constant temperature. For mere
chemical reactions (i.e. z = 0), kfwd is equal to k0, fwd.
While reverse reactions rates may be specified analogously, i.e. in the form
krev = k0, rev exp
(
zF
2RT
∆φ
)
, (4.10)
this is not the case for most reactions in this model. Instead, reverse rates are calculated
75
in a thermodynamically consistent fashion using the relation
krev = kfwd exp
(
∆G
RT
)
, (4.11)
which directly follows from the law of mass-action and ensures that the net production
rates are zero in the equilibrium case. The molar Gibbs reaction enthalpy ∆Gq of
reaction q is defined as
∆Gq = ∑
n∈q
νn,qµn = ∑
n∈q
νn,qgn . (4.12)
Here, the sum n runs over all species participating in reaction q with stoichiometric
coefficients νn, chemical potentials µn, and molar Gibbs free energies gn. The values
of gn for each species are required as input parameters for the model. Note that in
this work, gn is defined to be 0 for the reference state, i.e. for pure elements. Since
this model is assumed isothermal, this further simplifies the notation compared to the
common choice of h = 0, but s 6= 0 at the reference state.
A remark on the units of k: In Eq. (4.6), the activities an of the dissolved species are
often shorthanded replaced with the concentrations cn, in which case Eqs. (4.6) and
(4.11) require different units for kfwd and krev. Instead, the true relation is
an = γn
cn
cn, ref
, (4.13)
where γn denotes the dimensionless activity coefficient and cn, ref the concentration at
the reference state at which the electrical potential is zero, cf. [212]. This potential must
not be confused with the electrochemical potential φelyte as defined below. Without
loss of generality, it may be chosen such that cn, ref = 1 mol · l−1. If additionally, all
dissolved species are modeled to be available for reactions, i.e. γ ≡ 1, the values
of an and cn are identical except for the units. This is never strictly true for ions
in aqueous or organic solutions, since they can form salts and are thus always only
partly dissociated [204, sec. 3.3]. However, the ionic liquid used in this study is not a
solvent in the strict sense, but a molten salt. The anion of the ionic liquid Pyr14TFSI
and the conducting salt LiTFSI are identical, limiting the usefulness of the concept of
dissociation. While this does not imply that Li+ is never strongly bound or coordinated
in the electrolyte, it means that there is no such thing as “non-dissociated LiTFSI” and,
furthermore, no solvation shell surrounding Li+ cations. All ions are equally accessible
for reactions, therefore the assumption γ = 1 is truly justified. The fact that the
reactivity of Li+ can be different, e.g. for different temperatures, is better represented
by adjusting the activation energies for reactions involving Li+ as well as the activation
energy of Li+-diffusion [175], which are of no concern for this isothermal study.
76
4.2.4 Electrical properties
Next, the relationship of the cell’s electrical properties to the chemistry explained
above is presented. Since electrons are treated as yet another species (with zero mass),
the Faradaic current density iVF in a CV follows directly from reaction kinetics. More
precisely, it can be obtained by summing up the electrons produced or consumed at
all interfaces:
iVF = ∑
m
FAVm s˙electron, m . (4.14)
”Free” electrons only exist in the electronically conductive materials, which are as-
sumed to be in perfect contact with the current collector. The total, externally detected
current density is defined as
itot =
ˆ LCA|AN
y=0
(
iVF + i
V
dl
)
dy , (4.15)
where LCA|AN is the thickness of the cathode or anode, iVF is the Faradaic, and i
V
dl the
double-layer current source term in the CV, which in turn is calculated as
iVdl = A
V
mCdl, m
∂ (∆φ)
∂t
, (4.16)
with Cdl, m being a constant, area-specific capacity of the interface and ∆φ the potential
step defined as in Eq. (4.9). Several different effects contribute to the double-layer
capacity Cdl, m, see e.g. [213]. Nevertheless, it can be safely treated in this simplified
fashion since the influence of its properties besides the amount of capacity are only
relevant at time scales much shorter than those studied in this work. Note that in
Eqs. (4.15) and (4.16) both electrodes must be considered for the double-layer current
as well as for the integration of the local current densities. Per definition, the resulting
value of itot is positive while discharging and negative while charging.
The total cell voltage E is simply defined as the potential difference between the
cathode’s and anode’s conductive bulk material (which are carbon and lithium metal,
respectively):
E = φCA, elde − φAN, elde . (4.17)
At rest, this voltage is identical to the OCV, i.e. the equilibrium Nernst voltage of
the electrochemical system [214]. Unlike the potential of the electrolyte φelyte, which
is calculated for each CV, φelde is the same for all CVs within one electrode and thus
just represented by a number each. Physically, this means that the Ohmic drop due
to the electric resistivity within the electrode is negligible. Optionally, however, an
77
additional “global” resistance may be included so that
E = φCA, elde − φAN, elde − ρ?itot , (4.18)
with ρ? representing the combined electric resistivity of the bulk lithium metal, carbon,
and both current collectors. While ρ? is typically constant, it may also be influenced
by the microstructure of the cell, see below.
4.2.5 Microstructure
As previously mentioned, this one-dimensional model cannot explicitly describe the
electrodes’ microstructure. Instead, each control volume is assigned properties which
are representative of its morphology, namely the volume fractions εp of each phase p
present in the control volume. From these, the tortuosities τp, effective diffusion coef-
ficients Dn, eff, and specific surface areas AVm are derived. Volume fractions may change
because of precipitation and dissolution of material, phase transitions, or (convective)
phase transport. For this work, only the first option is taken into account so that the
evolution of the volume fractions is completely described by the continuity equation
∂
(
$pεp
)
∂t
= Rp Mp , (4.19)
which has to be evaluated for each phase p. Here, $p is the density and Mp the
(average) molar mass of the phase, while Rp denotes its net production rate, which
in turn is calculated from the (electro-)chemical source terms as defined in Eq. (4.6).
Assuming that phases grow and shrink at their interfaces only, Rp can be obtained
from the relation
Rp = ∑
m,n∈p
s˙n,m AVm , (4.20)
where the s˙n,m is the net production rate of species n in phase p due to all reactions
occurring at interface m. Since in the implementation used for this work the space
occupied by each CV is fixed, the following relation needs to be fulfilled at all times:
∑
p
εp = 1 . (4.21)
While there are several more rigorous options for ensuring consistency (see e.g. [215,
pp. 53ff]), a simple, yet straight-forward approach was chosen for this work: Using
Eq. (4.21), Eq. (4.19) can be eliminated for one phase. This is done by setting the
volume fraction of the gaseous phase to fill the entire space not occupied by the solid
phases. The flow or pressure increase caused by the volume change is neglected,
78
since it does not affect either the transport in the electrolyte or the electrochemistry.
A detailed, fully consistent treatment of Eq. (4.21) would require the introduction of
convection in the model as proposed in Ref. [216] as well as a description of mechanical
stress in the electrode’s particles as proposed in Refs. [217–219].
The special case, where the gaseous phase becomes zero in a CV receives no special
treatment, since it did never occur in any simulation reported here. However, if no
gaseous phase is considered at all, e.g. for the global model, Eq. (4.21) is enforced by
adjusting the volume fraction of the liquid electrolyte εelyte to fill the remaining space
in each CV. Since there is no convection in the liquid electrolyte (cf. section 4.2.2),
no material may enter or leave the CV and hence the number of molecules in the
electrolyte phase remains constant. If the solid phases grow, this causes a compression
of the electrolyte, resulting in an increase of the concentrations of all species (including
the solvent’s). Considering the low compressibility of liquids, this assumption may
not seem justified at a first glace. However, the pressure itself is not considered in
this model and the resulting concentrations quickly equilibrate with the neighboring
compartments by means of diffusion.
Once the volume fractions are determined, they can be used to calculate specific
surface areas of each interface m according to
AVm = A
V
m
(
εp
)
(4.22)
and, in the case of the liquid electrolyte, effective diffusion coefficients Dn, eff of all
dissolved species according to Eq. (4.4). This way, a feedback of the volume fractions in
each CV on the chemical kinetics and transport is established. The functional relation
of AVm and the volume fractions represents the electrode’s geometry, e.g. AVm = const.
for a film, AVm ∝ ε
3/2 for an electrode consisting of spherical particles which grow or
shrink at the same rate, or AVm ∝ ε for particles which are consumed one after the other.
As required, these expressions may be modified to include additional effects such as
explicit nucleation or surface passivation.
4.2.6 Reaction mechanisms
Once the governing equations are defined, one still needs to plug in a reaction mech-
anism. The requirements for this mechanism are rather simple: It needs to provide
at least one reaction generating or consuming electrons and one reaction generating
or consuming dissolved ions in the liquid electrolyte at both the cathode and anode
side. In this work, two different reaction mechanisms will be discussed. Both share
the same reaction at the negative electrode,
79
Li  Li+ + e− .
The anode is assumed to be able to supply an unlimited amount of lithium, which
experimentally corresponds to adding a surplus of lithium to the cell. Its kinetics is
chosen sufficiently fast, so that the anode overpotential is essentially negligible. While
the addition of a more detailed anode model would be simple and straightforward,
this assumption is made to simplify the interpretation of the results with regard to the
sulfur electrode.
Global reaction mechanism. To begin with, the most simple reaction mechanism
shall be discussed. It consists of one precipitation/dissolution reaction
S8(ds)  S8(s) ,
accompanied by only one electrochemical reaction at the positive electrode:
1/8 S8 + 2 Li+ + 2 e−  Li2S(s) .
While technically, the electrochemical reaction takes place at a surface with a con-
stant area (implemented as described above), the physical interpretation of the situa-
tion is similar to a triple phase boundary, a concept known from the field of heteroge-
neous catalysis. As pointed out by Yang et al. [78], the “active space” around the triple
phase boundary of sulfur, carbon, and the liquid electrolyte (in which the above reac-
tion could happen) is very small compared to the carbon and sulfur surfaces. Also, the
solid Li2S formed during discharge would rapidly block access to the active sites. Fur-
thermore, this ansatz assumes that single sulfur atoms may react individually, which is
virtually impossible because of the energetically unfavorable configuration left behind
in the bulk. Not surprisingly, therefore, this reaction mechanism cannot reproduce
the experimentally observed behavior of the Li/S cell very well. Nevertheless, it is
interesting to observe its results with respect to Li+ transport or material utilization
for different geometries.
80
Multi-step reaction mechanism. To overcome the limitations of the global mecha-
nism, two modifications are needed: First, lithium sulfide needs to be allowed to dis-
solve into and precipitate from the liquid electrolyte. Second, a more detailed imple-
mentation of the sulfur redox chemistry is needed. Ref. [192] discusses why there need
to be at least three electrochemical reduction steps to reproduce the two-plateau dis-
charge profile often detected experimentally. Various publications, however, suggest
that there are at more or different intermediate steps, see e.g. Refs. [138, 142, 220, 221].
Our multi-step reaction mechanism consists of a linear chain of reactions involving
several dissolved polysulfide intermediates:
S8(s)  S8(ds)
1/2 S8(ds) + e
−  1/2 S2−8
3/2 S2−8 + e
−  2 S2−6
S2−6 + e
−  3/2 S2−4 (4.23)
1/2 S2−4 + e
−  S2−2
1/2 S2−2 + e
−  S2−
S2− + 2 Li+  Li2S(s)
Opposed to the mechanism presented in [196], the only solid products in the cell
are sulfur and lithium sulfide, which is backed by X-ray diffraction and transmission
X-ray microscopy studies [155, 156].
Surface areas are modeled to depend on the solid phases as
AVS8|elyte =
(
εS8
ε0, S8
) 3
2
, and (4.24)
AVLi2S|elyte =
(
εLi2S
ε0, Li2S
) 3
2
. (4.25)
Here, ε0, n denotes the reference volume fraction of a phase, equal to either the initial
or the maximum value of εn. These expressions represent as the surfaces of homoge-
neously expanding and shrinking particles. Since the initial volume fraction of Li2S
in the charged state is very low, this naturally generates a situation where particle
growth of Li2S is severely inhibited due to the very small AVLi2S|elyte until a certain
amount of Li2S is precipitated. This can be interpreted as an implicit, yet effective
implementation of a nucleation phenomenon.
Multi-step reaction mechanism, revised. In order to reproduce experimental results
more faithfully, a revised version of the multi-step model was prepared. Besides a
rigorous reparametrization, described in section 5.3, two new phases are introduced:
81
First, each CV in the positive electrode is modeled to include 10 % non-reactive,
compressible Argon gas. In the coin cells used experimentally, the porous positive
electrode disc has a smaller diameter than the separator and the casing, see Fig. 2.6.
In the one-dimensional model, each CV represents the entire cross section of the coin
cell, see Fig. 4.2. The gaseous phase in the revised model accounts for the gas filled
volume between the outer diameter of the electrode and the casing of the coin cell.
It is required to correctly represent the relative volumes of the positive electrode and
separator in the model. The gaseous phase is not in intimate contact with the active
material, but only surrounding the entire electrode. There is virtually no triple phase
boundary and an explicit treatment of partial wetting [222] is not required.
Second, solid “passive Li2S” may be present in the positive electrode, which is only
accounted for in the model by its volume and weight. The rationale behind intro-
ducing this additional, nonreactive form of Li2S is to account for the following ex-
perimental finding: While the total amount of Li2S is the same for all cells made
from the same recipe, not all of the Li2S does actually contribute to the cell’s initial
capacity. Instead, this figure depends on various factors, including particle size, poros-
ity, and composition of the electrode as well as the protocol used for cell formation.
The mechanisms causing this missing capacity are diverse and not well understood
in detail. The most simple and straightforward way to deal with this situation is to
introduce a constant amount of “passive Li2S”, which can be fitted in order to match
experimental results, but does not otherwise affect the behavior of the model. Origi-
nally, it was planned to adjust the values of the parameters only. However, one change
was required to the model, regarding the deposition of Li2S on the electrochemically
active carbon|electrolyte interface. While the simple interface model described above
is capable of reproducing the general behavior of a Li/S cell, it turned out to be in-
sufficient for calibrating the model to match the experimentally obtained results. The
reason for this discrepancy is the mechanism triggering the end of discharge. Besides
pore clogging and depletion of active material, discussed in section 3.2.1, there is a
third mechanism, related to the passivation of the active surface. This mechanism
dominates for all cells analyzed experimentally, but it does not easily translate into a
reduced surface area in the model. Instead, the carbon surface area itself is assumed
constant, but the surface resistivity is modeled to increase when Li2S is deposited on
the surface as proposed by Albertus et al. for Li-air cells [223]. This is achieved by
modifying the surface resistivity, which is included in ρ?, see Eq. (4.18). The ansatz
used is
ρ? = ρ?0 + l · 1Ωm · exp [ξ1 · (l − ξ2)] , (4.26)
where ρ?0 is the constant sum of all other contributions to the cell’s impedance (e.g.
current collectors), ξ1 is a dimensionless constant representing the microstructure of
82
the film, ξ2 is the distance electrons can travel through the film, e.g. by tunneling, and
l is the thickness of the Li2S film on the carbon substrate, calculated according to
l =
[(
εcarbon − εLi2S
εcarbon
)1/3
− 1
]
· rcarbon . (4.27)
The radius of the carbon black particles, rcarbon as well as the constants ξ1 and ξ2
are additional input parameters. As long as l is smaller than ξ2, there is virtually
no overpotential. Once enough Li2S is precipitated to cover the entire surface, the
resistivity increases quickly, triggering the end of discharge.
4.2.7 Degradation
A first extensive review of degradation mechanisms in Li-ion batteries was presented
in Ref. [224] already in 1998. Many refinements were added over time, see e.g.
Ref. [225]. However, with a few notable exceptions [226–231], not a lot of model-
ing work has been done concerning these issues. For Li/S cells, the situation is even
more desperate. We presented the first – and so far only – modeling study of Li/S
cell degradation in Ref. [P3]. To enable degradation in the model, two additional ef-
fects are added: First, the reduction of polysulfides may also happen at the lithium
electrode. This is implemented by enabling the same set of charge transfer reactions
(CTRs) that is assumed at the positive electrode. Since the reaction rate constants and
activation energies strongly depend on the species’ interaction with the surface, they
must be chosen differently, though. For simplicity, the forward reaction rate constants
kfwd of all CTRs are scaled by the same factor ξCTR. Second, solid Li2S is allowed to
precipitate (and accumulate) at the negative electrode according to
S2− + 2 Li+ 2 e −  Li2S(s) .
Again, since the precipitation happens at the lithium surface, the reaction rate constant
may be different than that used for the same reaction in the positive electrode. It is
scaled by a factor of ξprecip.
Additionally, the active surface area of the lithium electrode is modeled to depend
on the amount of precipitated Li2S according to the heuristic expression
AVLi|elyte = A0 ·
(
ε0, elyte − εLi2S
ε0, elyte
)ξ0
,
where the expression in brackets is the fraction of the porosity remaining after Li2S
deposition and the factor ξ0 = 3.5 phenomenologically quantifies the impact on the
lithium’s active surface area [192].
83
Table 4.1: Summary of the governing equations and boundary conditions. For a defi-
nition of symbols, refer to the above text or Tab. (A.1) in the appendix.
Transport in the liquid electrolyte
Species conservation
∂c′n
∂t
= −∂Jn, tot
∂y
+ s˙n (4.1)
Species fluxes Jn, tot = −Dn, eff ∂cn∂y −
znF
RT
· cnDn, eff
∂φelyt
∂y
(4.3)
Electroneutrality and charge
conservation
0 = ∑
n,m
znFAVm s˙n,m −∑
n
znF
∂Jn, tot
∂y
(4.5)
Electrochemistry
Rate equations s˙n,q = νn,q
(
kfwd ∏
n′∈q
a
νn′ ,q
n′ − krev ∏
n′′∈q
a
νn′′ ,q
n′′
)
(4.6)
Arrhenius rate law kfwd = k0, fwd exp
(
− zF
2RT
∆φ
)
(4.9)
Calculation of the reverse rate krev = kfwd exp
(
∆G
RT
)
(4.11)
Cell voltage and current
Faradaic current source term iVF = ∑m FAVm s˙electron, m (4.14)
Double-layer current source term iVdl = A
V
mCdl, m
∂ (∆φ)
∂t
(4.16)
Total current density itot =
ˆ LCA|AN
y=0
(
iVF + i
V
dl
)
dy (4.15)
Cell voltage E = φCA, elde − φAN, elde − ρ?itot (4.18)
Multi-phase management
Continuity equation for bulk phases
∂
(
$pεp
)
∂t
= Mp · ∑
m,n∈p
s˙n,m AVm (4.19), (4.20)
Feedback on diffusion coefficients Dn, eff = Dn · εelyte/τ2elyte (4.4)
Feedback on specific surface areas AVm = A
V
m
(
εp
)
, mechanism-dependent (4.22)
84
4.3 Simulation methodology
4.3.1 Implementation and computational framework
When looking at anything but the most trivial cases, there is no analytical solution of
the set of governing equations and boundary conditions described above. Therefore,
the model needs to be implemented in a computer program capable of solving the
equations numerically. The software framework chosen for this work and its notable
features are discussed next.
All simulations in this work were carried out using the software framework Denis,
which is shorthand for Detailed Electrochemical and Numerical Impedance Simulation.
At the core of this framework, there is a C/C++ code initially designed by W. G. Bessler
at the University of Heidelberg3 and now developed and maintained at the German
Aerospace Center4, the University of Applied Sciences Offenburg5, and the Helmholtz
Institute Ulm6. Denis is best described by Refs. [58] and [59]. A component dia-
gram of the modeling framework is presented in Fig. 4.4. Internally, it makes use of a
set of third party libraries including Cantera [232], a toolset for evaluating (electro-)
chemical reactions, muparser [233], a library for parsing and evaluating mathematical
expressions, and Limex [234], a numerical solver based on the extrapolated implicit
Euler scheme [235], capable of solving linear implicit index-1 DAEs of the form
B(y(t), t) · y′(t) = f (y(t), t) , (4.28)
where y is the solution vector containing all state variables of the system and B is a
singular matrix (see also [211, chap. 15.3.2]). The “right-hand side” of the equation,
f (y, t), is a vector containing all the physical and chemical information, i.e. governing
equations, parameters and operating conditions. The solver will extrapolate y′ to
estimate the size of the next time step and compute the system iteratively until the
required absolute and relative tolerances are satisfied.
In order to solve the system numerically, the governing equations need to be dis-
cretized, which is accomplished using the finite-volume method (FVM). This method
is starting from the integral form of a conservation equation
d
dt
ˆ
V
Φ(y, t) dV = −
ˆ
∂V
f (Φ(y, t)) ·~nCV dV , (4.29)
3Interdisciplinary Center for Scientific Computing (IWR), Universität Heidelberg, Im Neuenheimer Feld
368, 69120 Heidelberg, Germany
4Institute of Engineering Thermodynamics, German Aerospace Center (DLR), Pfaffenwaldring 38–40,
70569 Stuttgart, Germany
5Institute of Energy System Technology (INES), Offenburg University of Applied Sciences, Badstr. 24,
Offenburg, Germany
6Helmholtz Institute Ulm for Electrochemical Energy Storage, Helmholtzstr. 11, 89081 Ulm, Germany
85
Output
• steady state
• transient
• impedance
• y-profiles
• parameters
• reactions
• simulation
protocol
Input
Limex
DAE/ODE solver
Cantera
chem. kinetics
Muparser
math. parser
Denis core
• electric
• transport
• double layer
• phases
• surfaces
• ...
Figure 4.4: Schematic representation of the simulation framework.
where VCV denotes a control volume (CV) with the corresponding surface ∂VCV and
the normal vector ~nCV, f (y, t) denotes the function to be discretized, and Φ(y, t) the
corresponding conserved quantity. From the form of Eq. (4.29), “it follows that a
change in Φ can only occur in combination with a flux over the boundaries of the CV.
Thus, the solution of the PDEs follows from approximation of the net fluxes over the
boundaries of all volumes. [The] main advantage of the FVM is that it is conservative,
meaning that the flux entering a control volume is always equal to the flux leaving
the adjacent CV. It is the most common discretization method in computational fluid
dynamics [236, chap. 4]”7. The number of control volumes for the numerical dis-
cretization is chosen such that a further increase does not affect the calculated results
significantly, as described in section 4.3.3.
For impedance simulations, a step excitation technique is used as described in
Ref. [237]: A quick voltage jump is simulated, followed by an long relaxation period.
The voltage jump contains contributions of all relevant frequencies. After finishing
the simulation, the voltage and corresponding current traces need to be Fourier trans-
formed in order to obtain the complex impedance Z in the frequency domain. The
frequency range that can be analyzed is determined by the duration of the voltage
jump (highest frequency) and the total simulation time (lowest frequency). In this
work, a voltage jump of 1.0 mV relative to OCV is applied within 10−8 s, followed by
a relaxation time of 104–106 s. Note that the impedance simulation is based on the full
7cited verbatim according to Ref. [215], p. 42
86
physicochemical model, including detailed chemistry and transport, without the use
of equivalent circuits [191].
4.3.2 Parametrization and validation techniques
This section deals with the third step of the modeling process as defined on page 69:
the determination of the parameters introduced by the previously discussed governing
equations. From a theoretical point of view, there are three kinds of parameters: a)
those which are ultimately based on natural constants and for which only one true
value exists, e.g. the diffusion coefficient of Li+ in a certain solvent or the enthalpy of
formation of Li2S, b) those which define morphological or technical properties, e.g.
the porosity or the discharge current, and may take any (reasonable) value, and c)
parameters that are not part of the physical description itself, such as the size of a
control volume or the solver tolerances. In principle, parameters in group a) can be
looked up or determined by independent measurements, those in group b) need to
match the conditions of a given experiment and those in group c) just need to be set
“right”, i.e. in a way that the simulation results do not heavily depend on the value
chosen, cf. section 4.3.3.
From a practical point of view, however, it makes sense to group the parameters
differently, to be specific: according to their source. The model needs lots of input to
determine all parameters and thus one has to make use of every possible independent
source. Three categories can be formed which do not necessarily correlate with those
mentioned above:
i) parameters known from literature with sufficient accuracy. This includes dedi-
cated scientific measurements as collected in the CRC handbook [238], but also other
modeling efforts, e.g. of 3D-resovled microstructures [239] or ab initio calculations
[143].
ii) parameters determined from dedicated experiments. This includes trivial param-
eters such as the dimensions and composition of the electrodes, but also more complex
ones like the conductivity of the liquid electrolyte including all dissolved ions. Often,
the parameters detected experimentally are not exactly the ones that are required as
input by the model. In such cases, the relationship of the parameters needs to be
analyzed to establish a rule for conversion.
iii) fitted parameters. For statistical reasons, this category should be kept as small
as reasonably possible [240, chap. 8]. Since the model’s behavior is strongly prede-
termined by the governing equations, its parametrization is not to be confused with
a polynomial fit. Still, fitting too many free parameters significantly reduces the ex-
planatory power of the model. Also, finding the optimal set of parameters becomes
more and more difficult as the number of parameters increases [240, chap. 8.2]. In
87
this work, most parameters were fitted manually by adjusting parameters within a
reasonable range. Some, however, were also determined – or at least fine tuned –
by applying a least-squares fitting algorithm to the problem. To this, Denis may be
called with updated input files from the programming language Python, making use
of the denistools package provided by K. Göckelmann8. Using SciPy’s [241] fitting
and error analysis capabilities, parameters can be optimized according to
χ2 ≡∑
i
{
1
σ2i
[yi − y(xi)]2
}
, (4.30)
where χ2 is the goodness-of-fit parameter, xi and yi are the abscissa and ordinate of
the experimentally obtained data points, respectively, σi is the uncertainty in yi, and
y(xi) is the simulation result corresponding to data point i [240, chap. 8.1]. Where
necessary, the simulation output is linearly interpolated to exactly match xi, which
itself is assumed to be known precisely. This assumption is valid, since the effect of
an uncertainty in xi can always be expressed by an increased uncertainty in yi [240,
chap. A1]. Finally, because the solver’s precision is the same for all data points, σ2i is
expected to be constant and thus cancels out.
4.3.3 Quality control and error analysis
Science calls for quantitative statements and these are of questionable value when
not accompanied by a confidence interval. The only true difference between a guess
and a measurement is that for the latter, deviations are known and may be analyzed
mathematically [240, chap. 3]. The absoluteness of this statement is somewhat put
into perspective, however, since the aim of the modeling activities presented here is
to generate knowledge about and insight into the system studied, which is not to be
confused with the task of e.g. a battery management system, which needs to keep
track of and predict the internal state of a given battery as precisely as possible. Still,
this work would be incomplete without error analysis.
The simulations are carried out by deterministic machines, so that random errors
should not exist [242]. Hence a general classification of errors as proposed by Beving-
ton and Robinson [240, chap. 1.1] does not apply. Instead, all errors are systematic by
definition. According to Ferziger and Peric´ [236, chap. 11.1], three kinds of errors are
common in numerical simulations of this type, namely discretization, convergence,
and modeling errors.
Discretization errors occur because time and space are continuous (at least on a
relevant scale), but need to be discretized in order to apply the numerical solution
8Lead scientific programmer at the Institute of Engineering Thermodynamics, German Aerospace Cen-
ter (DLR), Pfaffenwaldring 38–40, 70569 Stuttgart, Germany
88
methods described above. For example, one CV may represent a region which is
not very homogeneous, but nevertheless described by one set of state variables. By
decreasing the size of the CVs, the error can be decreased systematically [243]. The
strategy followed in this work is to choose the CV size according to the expected
gradients, see also [243]. Nevertheless, as long as the CV size is larger than the length
scale of the “smallest” physical effect, the error will not vanish.
Convergence errors occur in all iterative algorithms. Except for trivial cases, the
solver will never produce an exact solution of the system of equations. Instead, the
solution is improved until the difference between two consecutive iterations is smaller
than a certain tolerance. Typically one defines an absolute and relative upper bound-
ary, which are a direct measure of the solution’s uncertainty. To improve the quality
of the solution, the tolerance can be decreased, which in turn increases computation
time. In order to limit the computational cost of finding a solution which satisfies the
tolerance requirements, either all variables in the solution vector y should be scaled to
the range [0:1] or the tolerance should be set independently for each variable in the
solution vector.
Finally, modeling errors are errors which are present in the design or implemen-
tation of the model. While this includes plain mistakes, most errors of this type are
introduced on purpose in order to reduce the complexity of the model, e.g. the as-
sumption of dilute solutions or 100% activity of dissolved species as discussed above.
Another important source of errors is the neglect of certain effects. What is tricky
about modeling errors is that there is no way to systematically check for or even de-
crease them. Instead, assumptions and the errors associated with them can only be
justified by additional studies or calculations.
Even though all errors present in modeling studies are systematical errors, it makes
sense to discriminate between precision and accuracy of the results [240, chap. 1.1].
While convergence and discretization errors only affect the precision and can be re-
duced by increasing the computational effort, there is no systematic strategy to avoid
modeling errors, which typically affect the accuracy of the results. The strategy fol-
lowed in this work is to increase the precision of a given simulation until the results
do not change significantly anymore. That way, a benchmark for the uncertainty is
available. It is, however, by no means a measure for the true accuracy, i.e. the devia-
tion of the simulation results from “reality” (however one may define this). It would
be a daunting task (probably worth its own thesis) to define and analyze the effect of
all modeling errors in this or any other reasonably complex model.
Finally therefore, it has to stated that, even though the model was carefully de-
signed, calibrated, and partly validated, there is no hard evidence that it actually
describes something “true”. Confidence in the model must be built in the context of
89
specific questions, which the model has been proven to answer faithfully. Even then,
one cannot expect this type of model to predict the outcome of any single experiment
precisely, but rather to show up trends and prospects. In other words — to cite the late
David G. Goodwin, one of the pioneers of numerical modeling of chemical processes:
„We perform quantitative modeling in order to draw qualitative conclusions“9.
9D. G. Goodwin. Cited according to W. G. Bessler, personal communication.
90
5 Modeling the lithium-sulfur battery –
calibration and simulation results
In this chapter, a detailed computational study of the liquid-electrolyte Li/S battery
is presented. The structure of this chapter follows that of section 4.2.6, where the
different reaction mechanisms have been introduced and discussed: First, the global
two-step model is used to study the influence of the cell geometry, the electrolyte
properties, and the cycling protocol on the performance of the battery. Chronolog-
ically, these results predate the experimental work and also influenced the cell and
electrode design, reported in chapters 2 and 3. Next, the results of the multi-step
model with literature parametrization are presented in section 5.2, including a techni-
cal discussion of the model’s features. The results of this work, including discharge
profiles, chemical species concentrations, and impedance spectra are analyzed and
discussed, as previously published in Ref. [P2]. The reparametrization of the model is
detailed in section 5.3, including a full description of all calibration steps, using both
data recorded experimentally and values available from literature. Note that this cali-
bration is only valid for one type of Li/S cell. The same model can also be calibrated
to describe different cells, as presented e.g. in Ref. [P4]. An attempt to validate the cal-
ibrated model is made in section 5.4 before more simulation results are delivered and
analyzed in section 5.5. Simulations with the polysulfide shuttle enabled are reported
in section 5.6. A discussion of the early results of this model extension concludes the
simulation part of this work. Finally, a critical review of the model’s features, achieve-
ments, and limitations is presented in section 5.7, including a discussion of possible
model improvements and extensions.
5.1 Global two-step model
The global two-step model, described in section 4.2.6 above, is simplistic by design
and does not claim to represent the electrochemistry of any actual cell. Instead, its
parameter set is mostly chosen intuitively. Given this limitation, this model is not
capable of nor intended to reproduce experimentally obtained data. The discharge/
charge profile shown in Fig. 5.1, for example, does not share a lot of similarity with
the profile of a real Li/S cell as e.g. presented in Fig. 3.14. Because of the simplified
91
2.0
2.2
2.4
2.6
0 5 10 15 20
C
el
lv
ol
ta
ge
/
V
Time / h
Figure 5.1: Discharge/charge profile resulting from the global model.
reaction mechanism, it lacks the typical two voltage plateaus during discharge and
the slope during charge. Nevertheless, the OCV of the cell is reasonable as are the
geometry and current densities. For this model, the typically used cycling protocol
is: Discharge at a nominal rate of C/10 until E ≤ 2.05 V, followed by a charge at
a nominal rate of C/10 until E ≥ 2.4 V. The charge and discharge cutoff voltages
are chosen empirically to maximize capacity without simulating the almost vertical
slopes at the end of discharge and charge. The results of such a cycling simulation are
presented in Fig. 5.1.
Since geometry and transport parameters are physically meaningful regardless of
the simplified chemistry, this model is perfectly apt for the study of a subset of features
of the Li/S cell. This is true in particular for the transport in the liquid electrolyte. The
main variables influencing transport are the concentration of ions and the applied
current, which are chosen to represent the situation in the Li/S cell faithfully. In order
to achieve this, dissolved polysulfides are included in the model, even though they do
not participate in any reaction. Their concentrations as well as all other parameters of
the model are summarized in Tab. A.5 on page 150.
The remainder of this section focuses on transport phenomena and cell composi-
tion. To begin with, the distribution of sulfur across the cell is plotted against time in
Fig. 5.2. In this simple model, the amount of solid sulfur is directly proportional to
the cell’s SoC, which is, in turn, proportional to the discharge time – at least during
92
a galvanostatic discharge. When looking at the total amount of sulfur present in the
cell (black line), this relation is perfectly fulfilled. The colored lines, however, which
represent the amount of sulfur at various positions in the cell, do not follow the same
pattern. The closer a control volume (CV) is to the negative electrode, the faster the
conversion of sulfur in that CV. Close to the separator, the electrode is devoid of solid
sulfur long before the end of discharge.
0.0
0.1
0.2
0.3
0.4
0.5
0 5 10 15 20
Su
lf
ur
vo
lu
m
e
fr
ac
ti
on
Time / h
total
40 µm
35 µm
30 µm
25 µm
20 µm
15 µm
10 µm
5 µm
Figure 5.2: Changes in the distribution of solid sulfur during one cycle. Colors denote
the position in the electrode: 0 µm = separator; 40 µm = current collector.
When charging, the situation is reversed: the region closer to the separator is
recharged more quickly. This process is not perfectly balanced, however, so that the
distributions of S8 and Li2S across the cell are slightly different after each cycle. The
effect adds up over time, leading to an increasingly non-uniform composition of the
cell. Due to the diffusion overpotential, the local electrochemical potential depends
on the position in the positive electrode. Because of this and the sulfur redistribu-
tion, a slightly higher voltage is needed each cycle to access all the sulfur deposited in
the previous cycle. When using a standard cycling protocol that enforces a constant
charge/discharge cutoff voltage, the sulfur utilization is decreased during each cycle.
This can be best observed when looking at the average volume fractions of active mate-
rial plotted over time, see Fig. 5.3. The utilization of active material decreases rapidly,
causing faster, but more shallow cycling.
Since there is no physical or chemical mechanism in this model which could cause
irreversible degradation, the effect discussed above is naturally reversible. While less
and less material is cycled over time, none of the S8 or Li2S is actually lost. Running a
93
slow “refresh” cycle, like the 11th cycle in Figs. 5.4 and 5.5, the material utilization can
be increased to a level almost equal to a fresh cell. This refresh cycle also restores the
uniformity of the sulfur distribution (data not shown).
Another issue directly associated with the composition of the cell still persists,
though: When fully charged, there is only very little pore space left in the electrode
(blue area in Fig. 5.3). As discussed in chapter 4, the microstructure is not simulated
explicitly in this work. Instead, each phase is assumed to occupy one continuous
volume. This assumption is not justified anymore for very low porosities. At some
point, no continuous path may exist in the electrolyte phase, long before its volume
fraction is exactly zero. Because of the above assumption, this is not an issue for the
simulations, but the very low porosity is likely to cause problems in a real cell.
0.0
0.2
0.4
0.6
0.8
1.0
0 50 100 150 200
Vo
lu
m
e
fr
ac
ti
on
Time / h
electrolyte
lithium sulfide
sulfur
carbon black & binder
Figure 5.3: Volume fractions vs. time. The cell is cycled at a nominal rate of C/10, but
the 11th cycle was run at a lower rate of C/100. The reversible degradation
caused by regular cycling can be undone by the slow “refresh” cycle.
Discharge capacities for the simulation discussed above are plotted in Fig. 5.4, con-
firming that the reversible capacity fade can be all but undone. Nevertheless, it is
desirable to reduce this effect as much as possible, which leads to the question of how
to increase the highest tolerable rate, i.e. the rate at which no significant reversible
degradation occurs due to material redistribution. It turns out that there are two op-
tions: The first and most obvious possibility to improve the situation is to choose a
more suitable electrolyte. In Fig. 5.5, the cell simulated above is compared to another
cell with a different electrolyte. In this electrolyte, the transport of dissolved sulfur
(and polysulfides) is repressed. This is achieved by dividing diffusion coefficients by
94
020
40
60
80
100
0 2 4 6 8 10 12
R
em
ai
ni
ng
ca
pa
ci
ty
/
%
Cycle
Figure 5.4: Discharge capacity of several cycles, simulated with the global model. The
11th charge was run at a lower rate of C/100.
0
20
40
60
80
100
0 2 4 6 8 10
R
em
ai
ni
ng
ca
pa
ci
ty
/
%
Cycle
standard
improved elecrolyte
CCCV charging
Figure 5.5: Mitigating capacity fade. Black: standard parameters and cycling proto-
col; red: standard cycling protocol, but lower diffusion coefficients for dis-
solved sulfur species (improved electrolyte); blue: standard parameters, but
cycling protocol with CCCV charging. Details see text.
95
ten compared to Tab. A.5. The results clearly indicate that reversible degradation can
be alleviated by the improved electrolyte to some degree.
Unfortunately, the choice of known electrolytes with a decent performance is rather
limited [83]. Also, one is usually not free to choose an arbitrary electrolyte, since ad-
ditional requirements need to be met, which are not represented in the model, namely
chemical compatibility with all materials, safety, and cost.
The second option to reduce the capacity fade does not require any changes to the
cell itself, but rather to the cycling protocol. Instead of simply stopping the charge at
the predefined cutoff voltage, a constant voltage step is added: The voltage is fixed at
the charge cutoff voltage until the current drops below 1 % of the nominal charge cur-
rent. This measure, known as CCCV charging, is not only very effective in preventing
reversible capacity fade, but also easy to implement and without negative side effects,
except for the slightly increased charging time.
Conclusions. Using the global two-step model, transport in and cycling of the Li/S
cell can be studied. The spatial homogeneity was shown to decrease during cycling, cf.
Figs. 5.2 and 5.3. Also, the effect of the electrolyte’s conductivity as well as CCCV vs.
simple CC charging was investigated, cf. Fig. 5.5. The main conclusions are that a sig-
nificant amount of porosity is needed to avoid pore clogging, and that CCCV charging
is absolutely required, even at medium charge/discharge rates. This insight helped
to avoid potential issues when designing the electrodes and cells for the experiments
as outlined in chapter 2. In addition, the findings presented in Fig. 5.3 influenced the
design of the cycling protocol in the experimental section, where a slow “refresh” cycle
was added every 25 cycles, cf. pp. 62f.
While the global model can illustrate some aspects of the Li/S battery without too
much complexity, it cannot reproduce the behavior of the cell precisely. The major
limitation of the model is that is does not describe the electrochemical processes in
sufficient detail, resulting in inaccurate results for the the cell voltage in general and
the discharge and charge profiles in particular. To overcome this issue, a more detailed
description of the electrochemical reactions is needed as presented in the following
section.
96
5.2 Multi-step model – literature parametrization
5.2.1 Introduction
The multi-step reaction mechanism presented next is described in section 4.2.6. While
the implementation of the governing equations is identical to the global model, the
reaction pathways as well as the surface areas and their dependency on the electrode
composition are modified to match experimental conditions more closely. The devel-
opment of the multi-step model happened in two separate stages: The basic design,
implementation, literature parametrization, and testing was completed in mid-2012
based on data available from literature, as described in the following section. The
reparametrization and partial validation presented in section 5.3 was performed in
2014 after having recorded and analyzed sufficient original experimental data. Finally,
an addition of the shuttle mechanism to the model was implemented in 2013 in col-
laboration with Andreas F. Hofmann at DLR, see Ref. [P3], and later applied to the
multi-step model.
In the first implementation, described in this section, no validation per se is per-
formed, but the parameters are chosen according to Ref. [196] or assumed; see chap. 4.1
for a comparison of this work with Ref. [196] and other relevant publications. The en-
tire set of parameters used for these simulations can be found in Tab. A.6 on page 151.
Where reverse reaction rate constants are specified, the rates are calculated according
to Eq. (4.10) instead of Eq. (4.11). This part of the work has already been published
and the following paragraphs are cited verbatim from Ref. [P2]:
5.2.2 Discharge at different rates
Simulated discharge behavior of the cell is shown in Fig. 5.6 for different C rates, i.e.
different discharge currents. Fig. 5.6a presents voltage-capacity curves. They show the
typical three-stage behavior also known from experiments [115, 244], consisting of (I)
a high-voltage plateau, followed by (II) a fast voltage drop (at low currents associated
with a voltage dip), and finally (III) a long low-voltage plateau. The origin of this
curve shape will be discussed below. The rate capability of the simulated cell is not
very good (considerable voltage drop at increasing current), which is typical of Li/S
batteries with high sulfur content. For all C-rates, S8 is almost completely converted to
Li2S (close to 100 % sulfur utilization). The predicted energy density is 1300 Wh · l−1
and 850 Wh · kg−1 (per cathode volume/weight). Experimentally, lower values are
typically observed due to side reactions or passivation of the cathode (e.g. by film
formation); these effects are neglected in this model.
One of the most interesting aspects of the Li/S cell is the dissolution and re-formation
of phases during cycling. Fig. 5.6b shows the volume fractions of solid S8 and Li2S
97
2.0
2.2
2.4
a)
I II III
C
el
lv
ol
ta
ge
/
V
0.0
0.1
0.2
0.3
b)
solid: Li2S
dashed: S8V
ol
um
e
fr
ac
ti
on
0.01 C
0.1 C
1 C
10−8
10−6
10−4
10−2
100
102
0 400 800 1200 1600
c)
C
on
ce
nt
ra
ti
on
s
/
m
ol
/l
Discharge capacity / Ah/kg(sulfur)
S8
S2−
Figure 5.6: Simulated discharge at various current rates: a) cell voltage, b) solid volume
fractions of S8 and Li2S, c) spatially averaged concentrations of selected
species.
98
during discharge. During stage I, solid S8 is completely dissolved. During stage II,
all intermediates are present in dissolved form only. Finally, during stage III, Li2S is
formed and continuously precipitated. This behavior is observed for all C-rates in-
vestigated, i.e. there is no coexistence of solid S8 and Li2S phases during the initial
discharge. Another characteristic feature of the Li/S cell is the difference in the total
solid volume between the charged and discharged state: The volume of the product
(Li2S) is, in fact, 176 % that of the reactant (S8) due to the large difference in density of
the two phases, cf. Tab. A.6.
Fig. 5.6c shows spatially averaged concentrations of dissolved S8(ds) and S2– species
as representative intermediates. Using these profiles, the particular shape of the dis-
charge behavior can be further interpreted. During the plateau stages I and III, the
solid S8 and Li2S phases are in equilibrium with the corresponding dissolution product
S8(ds) and S2– respectively, buffering its concentration to a relatively constant value and
thereby causing a cell voltage plateau. This is the typical behavior of phase conversion
electrodes. During the intermediate stage II, however, there are no solid phases present
and the complete electrochemistry occurs among the electrolyte-dissolved polysulfide
intermediates. Within stage II, the concentration of S8(ds) strongly decreases, while
the concentration of S2– shows a peak. The cell voltage qualitatively follows the S8(ds)
concentration profile, including the respective dip in voltage and concentration at low
C-rates, which demonstrates the dominating role of the S8(ds) species for macroscopic
discharge behavior. An interesting effect can be observed for the slower discharge
rates: For the polysulfide concentrations to stabilize, it is required that the activity of
the solid product be constant. As long as there is virtually no solid Li2S in the cell,
this is not the case: The more precipitate is formed, the higher its activity. Because the
Li2S precipitation reaction is not an electrochemical reaction, as a first approximation
the rate of Li2S formation is independent of the current applied. The voltage profile
will only show the dip if no solid Li2S is present at the beginning of the discharge and
if the discharge rate is slow compared to the rate of Li2S precipitation.
5.2.3 Cycling
Simulated cycling behavior is shown in Fig. 5.7. In panels a) and b) the transient
cell voltage and current are plotted for the duration of the numerical experiment,
consisting of CC discharge, CCCV recharge, and relaxation to OCV. Note that the
model shows quite high charging overpotentials, which become particularly obvious
at the end of the CCCV phase, when the relaxation of the cell to OCV is accompanied
by a strong drop in voltage.
99
2.2
2.4
2.6
2.8
a)
C
el
lv
ol
ta
ge
/
V
-1.0
-0.5
0.0
0.5
1.0
b)
C
ur
re
nt
/
A
/m
2
0.0
0.1
0.2
0.3
0.4
0 50 100 150 200
c)
Vo
lu
m
e
fr
ac
ti
on
Time / h
S8
Li2S
Figure 5.7: Simulated cycling with CC discharge and CCCV charge at a C/50 rate. a)
cell voltage, b) current, c) volume fractions.
100
Fig. 5.7c shows the volume fractions of the bulk reactant and product phases in
the composite cathode. The model predicts an asymmetric behavior during cycling:
While Li2S is not formed during discharge before all solid S8 is dissolved, the two
solid phases do coexist during charge. The reason for this asymmetry is the assumed
kinetics of the system in general and the speed of the two dissolution reactions in
particular. In case of a partial charge or discharge, this effect can introduce hysteresis
to the system.
10−9
10−6
10−3
100
103
0 50 100 150 200
C
on
ce
nt
ra
ti
on
/
m
ol
·m
−3
Time / h
Li+
PF−6
S2−
S2−2
S2−4
S2−6
S2−8
S8
Figure 5.8: Simulated cycling with CC discharge and CCCV charge at a C/50 rate.
Evolution of the spatially averaged concentrations of dissolved species.
Fig. 5.8 shows time traces of the spatially averaged concentrations of all electrolyte-
dissolved species. Li+ is the only cation present in the cell; due to charge neutrality,
the total Li+ concentration corresponds to the total concentration of anions in the
solution. At the beginning and end of discharge, the non-reactive PF–6 is the main
counterion, while at intermediate state of charge (SoC), polysulfide anions contribute
significantly to the total anion concentration. Both end of charge and end of discharge
are characterized by strongly dropping concentrations of dissolved polysulfides (note
the logarithmic scale of Fig. 5.8), until the reactants for the charge-transfer reactions
cannot be supplied anymore. This leads to the final drop of cell voltage at end of
discharge, and the final increase in cell voltage at end of charge.
Another interesting result is that polysulfide concentrations are considerably lower
during charge compared to discharge. Therefore the conductivity of the electrolyte
is lower and diffusion overpotentials are higher during charging. Over the complete
101
cycle, both the total amount and the composition of the ions vary to a great extent.
Hence, the electrochemical response of the cell is expected to behave differently at
different SoC. This will be further discussed in the following section.
5.2.4 Impedance
The results of impedance simulations for different SoC are shown in Fig. 5.9. The
upper two panels show a Bode representation, i.e. the real and imaginary part of the
cell’s impedance Z versus logarithmic frequency. The lower panel shows the same
information in Nyquist representation, i.e. imaginary part versus real part. The cell
shows a complicated impedance behavior, which strongly depends on the SoC. At high
SoC, a feature at ∼0.5 Hz is dominating, with an additional smaller feature at ∼10 Hz.
At increasing SoC, these features merge at an intermediate frequency. Additionally,
impedance strongly increases toward low frequencies < 10 mHz, which is typical of
batteries.
5.2.5 Conclusions
From the above results, reproduced from Ref. [P2], the following conclusions can be
drawn: First of all, the results of the multi-step model are much more realistic and
detailed in terms of voltage, current, and dissolved species concentrations compared
to the two-step model. Also, the more detailed description enables the simulation
of representative impedance spectra – with regard to Li/S batteries, this is a unique
feature as of today. Despite the slightly different reaction mechanism, the results are
otherwise similar to previous simulations by Kumaresan et al. [196]: For the operat-
ing conditions simulated, the “behavior of the Li/S cell is governed by the presence
of solid reactant and product phases. The volume fractions of S8 and Li2S in the
cathode change considerably during cycling, providing an explanation of the distinct
stages during discharge. The model also predicts an asymmetric behavior of phase
formation/dissolution when comparing discharge and charge, as well as high charge
overpotentials”1.
1cited verbatim from Ref. [P2], p. 187f
102
0100
200
300
SoC:
Z
’/
Ω
·c
m
2
100
99
75
50
25
3
0
100
200
300
10−2 10−1 100 101 102 103
-Z
”
/
Ω
·c
m
2
Frequency / Hz
0
100
200
0 100 200 300 400
-Z
”
/
Ω
·c
m
2
Z’ / Ω · cm2
Figure 5.9: Simulated electrochemical impedance spectra for different SoC. Upper pan-
els: Bode representation. Lower panel: Nyquist representation.
103
5.3 Multi-step model – reparametrization
In order to obtain more precise results from the model, proper parametrization and
validation are required. While its behavior is strongly predetermined by the form of
the governing equations, the predictions of the model still rely heavily on its many
inputs. While a reasonable set of parameters was presented above (see also Tab. A.6),
this set does not represent any actual battery, restricting its usefulness to general de-
sign considerations. In the following section, the calibration of the model is described,
including a discussion of the origin and value of each parameter.
5.3.1 Parameters determined from independent sources
For any model, it is desirable to determine as many parameters as possible a priori,
i.e. from preliminary studies, dedicated experiments, or from first principles. Unfor-
tunately, despite the rapidly growing number of publications on Li/S batteries and
materials, the data base is still very sparse and incomplete. Nevertheless, the values
of some parameters could be acquired from literature:
• Molecular weights: The molecular weight of each compound is calculated from
its known composition using the standard atom masses from Ref. [245], Tab. 4.
• Densities: The densities of the (pure) phases are available from Ref. [238]. For the
liquid electrolyte, the density is adjusted to take the mass of the dissolved species
into account. For the separator, the density of pure, amorphous polypropylene
is used as reported in Ref. [246].
• Enthalpies of formation of pure species: For the pure species S8 and Li, g0 equals
zero by definition. For Li2S, the values for h0 and s0 are taken from Ref. [247]
and converted to g0,Li2S = −429.7 kJ ·mol−1 at 30 °C.
• Enthalpies of formation of polysulfides gn: Gas phase data taken from Ref. [143]
and converted to effective values taking into account the effect of the liquid
electrolyte by fitting the Gibbs free energy shift, see below. It is known that
various polysulfide species can exist in different forms with slightly different
enthalpies [248]. To keep things simple, each species is represented by just one
form with one value for gn.
• Specific double-layer capacity of carbon: Depending on the type of carbon used,
different values are reported in the literature, ranging from 0.05 F ·m−2 to 0.5 F ·m−2
[249–254]. For this work, a specific capacity of 0.2 F ·m−2 is chosen, which
falls within the standard ranges reported for carbon in various electrolytes by
Refs. [249] and [250].
104
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5
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2S
(%
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78
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S 8
(%
of
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—
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12
C
al
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i
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cf
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ef
s.
[4
8,
26
0]
105
5.3.2 Parameters calculated from experimental data
Many parameters can be obtained from the experimental setup and results, either
because they can – at least in principle – be chosen freely within reasonable constraints
or because they can be easily measured, extracted, or calculated from the experimental
results. All the macroscopic geometric parameters, the choice of materials, and the
electrode composition fall into this category. A full list can be found in Tab. 5.1. In
addition, the following parameters are readily extracted from experimental data:
• Volume fractions of all solids in the electrodes are calculated from the volume of
the electrode, the known composition of solids (see Tab. 5.1), and their densities.
In the (hypothetical) fully charged state they are for the positive electrode: 12.1 %
for S8, 0.0 % for Li2S, 3.2 % for total carbon, 2.8 % for the binder, 70.9 % for the liq-
uid electrolyte. The remaining 10.0 % are filled with non-reactive, compressible
Argon gas in the model, as discussed in section 4.2.6 on page 81.
• The concentration of the LiTFSI conducting salt is 1.0 mol · l−1; that of the LiNO3
additive is 0.5 mol · l−1 according to the standard recipe, cf. section 2.3.3.
• The total surface area of the active material can be derived assuming that all
ball-milled Li2S particles are spherical with an average diameter of 1.1 µm (cf.
section 3.1). Their total surface area is calculated to be 1.20 · 106 m2 ·m−3. Further
assuming that the number of particles does not change when the material is
converted to S8, the surface area of all sulfur particles is 1.40 · 106 m2 ·m−3.
• The electrochemically active surface area, i.e. the carbon|electrolyte interface,
was measured to be 1.1 · 107 m2 ·m−3 in Ref. [P4] by Krypton BET. Since the
electrodes used for this work are very similar in carbon content and structure,
the same value is used here.
• The radius of the carbon black particles in Eq. (4.27) is estimated to be 40 nm
based on SEM images (see e.g. Fig. 3.1b).
5.3.3 Parameters determined by equilibration
More parameters can be determined without fitting by making smart use of the model:
Only the composition of solids is known from experiments, not the composition of the
electrolyte. The equilibrium concentrations of dissolved species, however, are readily
obtained by simulating a cell at rest for a long time. Starting with volume fractions
derived from the composition of the solid material (see Tab. 5.1) and an electrolyte
with virtually no dissolved polysulfides, a simulation is run with itot = 0 for t = 109 s
(several years). After this time, the cell is perfectly equilibrated and the final values
106
11 ms 1 s 1 h 1 year
4
0.4
R
el
at
iv
e
de
vi
at
io
n
fr
om
eq
ui
lib
ri
um
Time / h
cell voltage
cLi+
cS8
cS2−8
εS8
Figure 5.10: Initial equilibration. Time trace of a simulation with itot = 0 (in the fully
charged state). For selected quantities, the relative deviation from the final
values is reported. The latter are considered to be the true equilibrium
values and are used as a starting point for subsequent simulations.
of this simulation are used as the starting values for all subsequent simulations of the
same cell. The actual equilibration run is presented in Fig. 5.10.
5.3.4 Parameters fitted to experimental data
After making use of all known independent sources, many parameters are still un-
known. In total, there are 26 parameters left, including the Gibbs free energy of
formation of solvated species in the electrolyte ∆g0, ten diffusion constants, and ten
forward reaction rate constants as well as ξ1 and ξ2 in the expression for the surface
resistivity, Eq. (4.26), and finally, the fraction of activated Li2S. These parameters need
to be fitted to experimental data, which will be discussed in the following.
Originally, it was planned to fit several parameters at once using a multidimensional
optimization algorithm, cf. section 4.30. Because of the large number of parameters to
be fitted on the one hand as well as the very different types of reference data on the
other hand (single data points, sets of data points, x-y-curves, parameters of a family of
curves), this turned out to be a futile approach. Instead, the parameters were grouped
and each group was adjusted manually within a reasonable range. In the following,
the fitting results are complemented by parameter variation plots and corresponding
discussions. In all plots, circles represent experimental data, lines simulation results.
107
The amount of activated Li2S. As discussed in section 4.2.6 on page 82, the amount
of activated Li2S is slightly different from cell to cell, even within the same batch. Even
worse, it cannot be determined by simply looking at the initial charge capacity since it
is impossible to tell regular charging apart from irreversible reactions. These happen
predominantly during the activation cycle because of the elevated voltage, cf. section
3.2.3. Not surprisingly, this figure strongly determines the behavior of the cell in all
experiments, therefore it needs to be fitted first. This is achieved as follows:
1. Choose a fast sulfur dissolution rate and electrochemistry (i.e. rate constants).
Also, enough electrolyte is needed to dissolve the entire sulfur – though this is
less of an issue for the highly porous cell configuration discussed here.
2. Starting from the fully charged cell, simulate a discharge with a relatively slow
rate and plot voltage vs. capacity (see Fig. 5.11). The transition region between
the upper and lower plateau is now very narrow.
3. Adjust the ratio of activated and non-activated (passive) Li2S, keeping the total
amount of sulfur in the cell constant. The best value is the one where the onset of
the lower plateau matches experimental data (regardless of the end of discharge).
The best choice is a ratio of activated Li2S to passive Li2S of 2:3. From an engineer-
ing point of view, an increase in activated Li2S is highly desirable. For the model
calibration, this value is perfectly acceptable, though.
1.8
2.0
2.2
2.4
2.6
2.8
3.0
0 50 100 150 200
C
el
lv
ol
ta
ge
/
V
Capacity / Ah/kgS
ref
5:3
4:3
3:3
2:3
1:3
Figure 5.11: Parameter variation: Amount of activated Li2S. The legend indicates the
ratio of activated to passive Li2S. A dashed line marking the onset of the
lower plateau serves as a visual aid to the fitting target. Since the total
capacity of the cell is irrelevant for this fit, the capacity range is restricted
in the plot to better show the relevant region.
108
The active surface area expressions. Next, the expressions describing the active sur-
face area need to be quantified. Their general form is derived and discussed in sec-
tion 4.2.6. Only the parameters ξ1 and ξ2 in Eq. (4.26) remain to be fitted. This is done
as follows:
1. Choose a moderately fast sulfur dissolution rate and electrochemistry (i.e. rate
constants).
2. Starting from the fully charged cell, simulate a discharge with a relatively slow
rate and plot voltage vs. capacity.
3. Adjust ξ1 until the slope of the final voltage drop matches the reference data
(without paying attention to the position, i.e. capacity, at which the voltage drop
occurs).
4. Adjust ξ2 until the onset of the voltage drop matches the reference data.
A parameter variation is presented in Fig. 5.12. Lines of one color share the same ξ2,
whereas all dotted (dashed, solid) lines share the same ξ1. The best fit is obtained
for ξ1 = 1.8 and ξ2 = 5.0 nm. The value of ξ2 is too large to be identified as the
tunneling distance for electrons through the deposited, insulating Li2S film. However,
the passivating layer is not expected to grow perfectly uniformly: Instead, ξ2 may be
1.8
2.0
2.2
2.4
2.6
2.8
0 50 100 150 200 250 300 350
C
el
lv
ol
ta
ge
/
V
Capacity / Ah/kgS
ref
3.0 nm
ξ2 = 5.0 nm
7.0 nm
1.8
ξ1 = 2.0
2.2
Figure 5.12: Parameter variation: Passivation of the carbon|electrolyte interface
by precipitated Li2S. The additional surface resistivity is described in
Eq. (4.26) by the thickness of the Li2S film and two additional parame-
ters ξ1 and ξ2, which are both varied in the above plot.
109
interpreted as the average thickness of the insulating layer at the time when electrons
can no longer tunnel through its thinnest segment. This paragraph only deals with
the calibration of ξ1 and ξ2; the formation of the passivating film is analyzed in more
detail below, cf. Fig. 5.27 on page 128.
Enthalpy of formation of dissolved species. In order to correct the enthalpies of
formation of the dissolved polysulfides for the effect of the liquid electrolyte, the gas-
phase thermochemistry values taken from Ref. [143] are shifted by a fixed offset ∆g0.
The offset is determined as follows:
1. Choose fast rate constants.
2. Starting from a fully charged cell, simulate a discharge with a very slow rate and
plot voltage vs. capacity (see Fig. 5.13).
3. Adjust the offset until the lower plateau voltage for charge and discharge is
roughly centered between the charge and discharge branches of the reference
data. The actual voltage will change when calibrating the reaction rates, but the
center is virtually not affected.
Unfortunately, very slow discharges are difficult to obtain experimentally – or rather
difficult to interpret since the discharge due to the externally applied current mixes
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
0 50 100 150 200 250 300
C
el
lv
ol
ta
ge
/
V
Capacity / Ah/kgS
ref
210 kJ/mol
215 kJ/mol
∆g0 = 220 kJ/mol
225 kJ/mol
230 kJ/mol
Figure 5.13: Parameter variation: Enthalpy of formation of the dissolved polysulfides.
∆g0 denotes the fixed offset to gasphase thermochemistry data. Reference
data: discharge at C/50, charge at C/20. The different simulations may
alternatively be interpreted as different electrolytes.
110
with internal self-discharge. Therefore, the slowest regular cycling data (with a dis-
charge rate of C/50 and a charge rate of C/20) was used for the calibration of the
offset in Fig. 5.13.
The best fit is obtained for ∆g0 = 220 kJ ·mol−1. Because of the different charge and
discharge rates in the reference data, the calibration is difficult and even the best result
looks slightly off-center.
Diffusion constants. Diffusion constants for different species are expected to be dif-
ferent in reality. Unfortunately, in the literature there is no direct measurement or
even a clue to the values of Dn for all dissolved species n. The values chosen by
Refs. [196, 198, 199] may serve as guidance values. However, they lack experimental
references themselves and are assumed for different electrolytes. To handle this situ-
ation, all ten diffusion constants are assumed identical so that only one value needs
to be fitted. In this study, the diffusion constants are varied in the range of 10−10–
10−12 m2 · s−1, which has been reported for ionic liquid based electrolytes [175, 261]
and compared with cycling and impedance data in Figs. 5.14 and 5.15, respectively.
While the impact of the speed of diffusion is obvious in general, it is difficult to
quantitatively distinguish between the data sets with smaller diffusion constants in
Fig. 5.14. Therefore, impedance spectra are analyzed additionally. More precisely, the
high-frequency intercept of the spectra with the x-axis is evaluated. This feature is
related to the electrolyte’s ionic conductivity, which in turn depends on the diffusion
constants according to Eqs. (4.3) and (4.4).
EIS results are presented in Fig. 5.15. Note that the shape of the Nyquist plots
does not match very well since the reaction rates are not yet calibrated. The effect of
the diffusion constants on the Z′ intercept is clearly visible. While there is a lot of
variation in the reference data (cf. section 3.2.4), the plot still shows a clear trend and
the diffusion constants can be calibrated to at least the right order of magnitude much
more easily than on the basis of discharge profiles. The best agreement is obtained for
Deff = 1.0 · 10−11 m2 · s−1.
111
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
0 50 100 150 200 250 300
C
el
lv
ol
ta
ge
/
V
Capacity / Ah/kgS
ref
1.0 · 10−10 m2/s
3.0 · 10−11 m2/s
Deff = 1.0 · 10−11 m2/s
3.0 · 10−12 m2/s
1.0 · 10−12 m2/s
Figure 5.14: Parameter variation: Impact of the speed of diffusion in the liquid elec-
trolyte on charge/discharge profiles.
0
10
20
30
40
0 10 20 30 40
−Z
”
/
Ω
·c
m
2
Z’ / Ω · cm2
ref
1.0 · 10−12 m2/s
3.0 · 10−12 m2/s
1.0 · 10−11 m2/s
3.0 · 10−11 m2/s
1.0 · 10−10 m2/s
1
10
100
H
ig
h
fr
eq
ue
nc
y
in
te
rc
ep
t
/
Ω
·c
m
2
Figure 5.15: Impact of diffusion in the liquid electrolyte on the EIS high frequency
intercept. Since there is considerable variation in the reference data, the
spectra of several similar cells are plotted. Left: Nyquist plot (two sim-
ulations are not visible since they intercept at much higher Z′). Right:
Comparison of the x-axis intercept for different diffusion constants. The
experimentally confirmed range is shaded in gray.
112
Reaction rates. Finally, there are ten rate constants in the model, which are alto-
gether unknown. An initial scan was performed testing dozens of combinations of
rate constants for just one criterion: The solver needs to be able to calculate the ini-
tial equilibrium. This technique yielded an initial guess for the rate constants in the
range 1013–1019 mol ·m−2 · s−1 (data not shown). One exception is the rate of disso-
lution of sulfur, which is a non-electrochemical reaction that occurs at a much slower
rate. Because of the model’s high sensitivity on this value, it was chosen to be fitted
independently as follows:
1. Choose the other rate constants fast.
2. Starting from a fully charged cell, simulate a discharge with a slow rate and plot
voltage vs. capacity (see Fig. 5.16).
3. Adjust the rate of dissolution until the slope matches that of the reference data.
Even though it is somewhat off in absolute position, the best agreement for the
slope is obtained for kfwd = 3.0 · 10−4 mol ·m−2 · s−1. All other rate constants were ad-
justed by studying the concentrations of polysulfides during cycling and increasing the
rate constants for those reactions whose reactants accumulated excessively (data not
shown). Those still random, but “normalized” rates, were then scaled by a common
factor, which is fitted as follows:
1. Simulate a charge and discharge at a medium rate.
2. Adjust the factor until the voltage hysteresis, i.e. the difference between charge
and discharge voltage matches experimental data.
Even for the rather slow experiment presented in Fig. 5.17 (charge: C/20, discharge:
C/50), the reaction overpotential is considerably large. For the slowest rate constant,
the solver did not even come up with a valid solution for the entire experiment. Since
the cell is charged at a faster rate, the impact of slow electrochemical rate constants
affects the charge branch more strongly than the discharge branch. In summary, the
best fit is obtained for a scaling factor of ten.
At this point the model is fully calibrated, but not necessarily validated. In order
to come up with the final values reported in Tab. 5.2, the manual fitting process was
iterated three times, since the combined effect of all other fit parameters may signifi-
cantly affect the best value of each single parameter. For the purpose of illustrating the
process, the parameter variations shown here are selected from the second run. The
first run still contains lots of aborted simulations and some which are really far off.
During the third run, in contrast, the parameters are changed by tiny amounts only,
making it hard to discern the effect they have on the simulation.
113
1.8
2.0
2.2
2.4
2.6
2.8
3.0
0 50 100 150 200 250 300
C
el
lv
ol
ta
ge
/
V
Capacity / Ah/kgS
ref
1.0 · 10−4 mol ·m−2 · s−1
2.0 · 10−4 mol ·m−2 · s−1
2.5 · 10−4 mol ·m−2 · s−1
3.0 · 10−4 mol ·m−2 · s−1
6.0 · 10−4 mol ·m−2 · s−1
Figure 5.16: Parameter variation: Effect of the rate of dissolution for sulfur.
1.8
2.0
2.2
2.4
2.6
2.8
3.0
0 50 100 150 200 250 300
C
el
lv
ol
ta
ge
/
V
Capacity / Ah/kgS
ref
100×
10×
1×
Figure 5.17: Parameter variation: Speeding up or slowing down electrochemical reac-
tions. The legend indicates the common scaling factor which is applied to
all electrochemical rate constants uniformly.
114
It is arguably unlikely that the global optimum of the 26-dimensional parameter
space can be found this way. Also, it is obvious from the results that the match is
indeed still far from perfect. Nevertheless, this multi-step, hierarchical fitting process
allows for a systematical and deterministic calibration of the model with the limited
set of reference data available.
Table 5.2: Parameters adjusted to match experimental results in the calibrated multi-
step model. See Tab. A.7 on page 152 for a comprehensive list of all param-
eters used for the simulations.
Cathode volume fractions εn
S8 0.04140
Li2S 0.00120
Li2S (passive) 0.10549
carbon 0.03248
binder (SBR) 0.02800
electrolyte 0.69143
void (Ar) 0.10000
Surface areas A0
carbon|electrolyte 1.1 · 107 m2 ·m−3
S8|electrolyte 1.4 · 106 m2 ·m−3
Li2S|electrolyte 1.2 · 106 m2 ·m−3
Area-specific capacity Cdl
carbon|electrolyte 0.2 F ·m−2
lithium|electrolyte 0.2 F ·m−2
Surface resistivity expression Eq. (4.26)
rcarbon 40 nm
ξ1 5.0 nm
ξ2 1.9 · 109
Gibbs free energy g0 of dissolved species and initial concentrations2
Li+ 0.0 kJ ·mol−1 1.50 mol · l−1
TFSI− non-reactive 1.00 mol · l−1
NO−3 non-reactive 0.50 mol · l−1
S2− −309.7 kJ ·mol−1 6.19 · 10−25 mol · l−1
S2−2 −354.1 kJ ·mol−1 3.65 · 10−19 mol · l−1
2Note: The concentration of Li+ is not exactly 1.50 mol · l−1, but equal to the total concentration of
negatively charged ions to the 10th digit.
115
S2−4 −416.8 kJ ·mol−1 2.12 · 10−7 mol · l−1
S2−6 −432.2 kJ ·mol−1 7.16 · 10−4 mol · l−1
S2−8 −441.9 kJ ·mol−1 2.92 · 10−1 mol · l−1
S8(ds) 19.0 kJ ·mol−1 9.38 mol · l−1
Reactions, forward rate constants
S8(s)  S8(ds) 3.0 · 10−4 mol ·m−2 · s−1
1/2 S8(ds) + e−  1/2 S2−8 4.0 · 1014 mol ·m−2 · s−1
3/2 S2−8 + e
−  2 S2−6 4.0 · 1014 mol ·m−2 · s−1
S2−6 + e
−  3/2 S2−4 4.0 · 1014 mol ·m−2 · s−1
1/2 S2−4 + e
−  S2−2 4.0 · 1011 mol ·m−2 · s−1
1/2 S2−2 + e
−  S2− 4.0 · 1014 mol ·m−2 · s−1
2 Li+ + S2−  Li2S(s) 3.2 · 1012 mol ·m−2 · s−1
Molar Gibbs free energy g0 of Li2S: −429.7 kJ ·mol−1
Density of the liquid electrolyte $elyte including dissolved species: 1535 kg ·m−3
Diffusion constant Dn (for all dissolved species): 1.0 · 10−11 m2 · s−1
5.4 Multi-step model – validation attempts
Having determined all parameters of the model, an attempt is made to perform an ini-
tial validation. This measure is not to be confused with a regression model validation
used e.g. for equivalent circuit models. A rigorous stochastic validation is not possible
in this case for two reasons: First, practically, the data base available for validation is
not sufficient to achieve a reasonable significance: Only experimental data not previ-
ously used for the parametrization may be employed for the validation. Also, neither
the very best (or worst) performing cells, nor the first few cycles of any cell are used
for the validation, since they are believed to be less representative. Second, fundamen-
tally, the need to evaluate both the correctness of the assumed reaction mechanism and
the set of parameters at the same time, poses fundamental limits to a mathematically
sound validation.
Instead, simulation results are visually compared to corresponding experimental
data and checked for plausibility, taking into account the considerations on model
accuracy outlined in section 4.3.3.
116
5.4.1 Rate capability
In Fig. 5.18, simulation results are compared to the experimental data presented in
Fig. 3.23, where one cell is discharged at subsequently increasing rates. For technical
reasons, the cycling protocol for the simulation uses more restrictive charge and dis-
charge cutoff voltages (1.95–2.7 V) than the experiment (1.5–2.8 V). Especially at the
end of discharge, when the remaining active surface area of the cathode approaches
zero, the calculation time would otherwise increase disproportionately even though
very little charge is actually transferred. Experimentally, the larger voltage window
is required to access more strongly bound sulfur that would otherwise stick to e.g.
functional groups on the carbon surface or other impurities [67]. Also, note that in the
experimental data, some degradation does occur from cycle to cycle, which is not the
case for the simulated data. Additionally, an almost quasistatic discharge is simulated
at a very low rate of approx. C/1000, representing the thermodynamics of the system.
1.6
2.0
2.4
2.8
0 50 100 150 200 250 300 350 400
re
f.
si
m
.
C
el
lv
ol
ta
ge
/
V
Capacity / Ah/kgS
C/50
C/20
C/10
TD
C/50
C/20
C/10
Figure 5.18: Simulated discharge profiles at various rates compared to experimental
results. While there is no quantitative match, the simulations clearly re-
produce the correct trend. TD: Thermodynamic, i.e. quasistatically slow
discharge.
There is no full agreement between the experimental and simulated data: the in-
creasing overpotential toward the end of discharge is reflected well, as is the decreasing
discharge voltage. The charge plateau is matched correctly, even though the experi-
mental data exhibits a little variation. Also, the shift of the onset of the lower voltage
plateau and the disappearance of the voltage overshoot right before the plateau are
faithfully reproduced. For three major features, however, the agreement is rather poor:
117
While the shape of the transition region between the upper and lower voltage
plateaus during discharge does change with increasing rate, the effect is significantly
overestimated by the model. A possible explanation is that the model does not in-
clude any disproportionation reactions, which could greatly speed up effective reac-
tion rates. Due to the parallel reaction pathways, “bottlenecks” in the linear chain of
electrochemical reduction reactions can be circumvented, lowering the reaction over-
potential and thereby mitigating the impact of higher discharge rates.
The second discrepancy concerns the voltage profile at the end of charge, which
is not even qualitatively reproduced by the simulations. Where the voltage starts to
rise steeply in the experiments, hitting the charge cutoff voltage, it levels off or even
decreases slightly in the simulations. While there is no plausible explanation for this
behavior from a chemical point of view, this effect is technically related to the increas-
ing active surface area available for the precipitation of S8, cf. Eq. (4.24). The voltage
does reach the charge cutoff voltage eventually (data not shown). However, this only
happens if the cell is considerably overcharged, until the electrolyte becomes depleted
of polysulfides, as is the case with the multi-step model before reparametrization,
compare Figs. 5.7a and 5.8.
The third debatable feature is the loss of discharge capacity with increasing rate.
The effect is present in the simulations, albeit greatly underestimated. This leads to
the question whether the end of discharge is actually exclusively triggered by surface
passivation, as suggested by the simulations. One possible explanation for this dis-
crepancy would be that the mean pore size is actually smaller than suggested by the
electrode’s global porosity. If there are many very small pores, clogging might play
a significant role despite the large total pore volume. A pore size distribution anal-
ysis could help to clarify this issue; such data is not available for the reference cells,
though.
5.4.2 Cyclic voltammetry
Furthermore, a cyclic voltammetry run is simulated and analyzed. The experiment
presented in Fig. 3.16a is used as reference. The results are plotted in Fig. 5.19. Note
that this plot follows the usual convention of assigning a positive current to the anodic
sweep (charge) and a negative current to the cathodic sweep (discharge). As for the
rate capability plot, there is no quantitative match. Still, the simulation reproduces
the main features of the experiment qualitatively very well: The discharge sweep is
separated into two peaks, the upper one smaller, the lower one larger, corresponding
to the upper and lower voltage plateau during a regular galvanostatic discharge. The
charge sweep is composed of only one peak.
Still, there are three obvious differences: First, the total area under the curve (equal
to the total charge transferred) does not match. This is easily explained, since the
118
-150
-100
-50
0
50
100
150
200
250
1.5 2.0 2.5 3.0
discharge
charge
C
ur
re
nt
/
µA
Voltage vs. Li/Li+ / V
ref
sim
Figure 5.19: Cyclic voltammetry data compared to experimental results. Sweep rate:
25 µV · s−1.
experimental data was recorded from a cell of a different batch than the data used
for calibration. The performance of that batch was slightly better, therefore the total
capacity of the reference cell is expected to be larger. Second, the position of the ca-
thodic peaks is shifted compared to the reference, which most likely can be explained
by a flaw in the calibration already discussed above: It is impossible to obtain exper-
imental data of a quasistatically slow galvanostatic discharge. Lacking more precise
information about the composition of the overpotential, the thermodynamics were cal-
ibrated using the midpoint between the charge and discharge voltage of a finitely fast
discharge in Fig. 5.13. The thermodynamically correct voltage, however, is possibly
off-center and indeed, judging based on Fig. 5.19, this seems to be the case for this
system. Third, the shape of the peaks does not match precisely. In the experimen-
tal data set, both the charge peak and the lower discharge peak extend into a “tail”,
blurred over several hundred mV. As already mentioned in the discussion of Fig. 5.18,
these tails are most likely caused by sulfur that is more strongly (chemically) bound
to the electrode matrix and thus only accessed at higher overpotential during both
charge and discharge – a phenomenon which is not included in the model.
119
5.5 Multi-step model – further simulations
In the following sections, full-cell simulations with the reparametrized model are pre-
sented, illustrating various aspects of the simulated type of Li/S battery in detail.
5.5.1 Galvanostatic charge/discharge profiles
In the previous section, the parameter space was reduced to a single set of values in
order to match a single set of experimental data, mostly recorded from a single cell.
Some of the parameters – called type a) parameters in section 4.3.2 – represent natu-
ral constants and are valid for all simulations. Others – called type b) parameters –
are only valid for a certain cell configuration or even only for a single cell. Changing
those values is the computational equivalent to building a different cell. In the fol-
lowing simulation, the parametrization of the sulfur electrode is changed to represent
different activation efficiencies, which can be achieved experimentally e.g. by smaller
Li2S particles or improved activation protocols, cf. section 3.2.3. The results are plotted
in Fig. 5.20 for four different activation efficiencies, represented by different ratios of
active to passive Li2S. It is assumed that the morphology is not affected, i.e. the active
surface area scales linearly with the amount of accessible Li2S.
2.0
2.4
2.8
0 250 500 750 1000
C
el
lv
ol
ta
ge
/
V
Capacity / Ah/kgS
99 %
80 %
60 %
40 %
Figure 5.20: Voltage profiles of differently activated Li/S cells cycled at the same rate.
The activation efficiency, i.e. the fraction of activated, electrochemically
accessible Li2S, is indicated by the plot key.
As expected, the discharge capacity increases with increasing activation efficiency.
Other than that, the shape of the voltage profile is almost not affected. An interesting
120
result is that even at 99 % activation, the cell still only delivers about half of the theo-
retical capacity of 1675 Ah/kgS. In order to access the remaining capacity, changes to
the electrode design are needed, preventing the passivation of the carbon|electrolyte
interface by film formation, which eventually causes the premature end of discharge.
One such measure could be to increase the carbon content in the electrode. Not sur-
prisingly, carbon contents of 40–50 % of the positive electrode’s dry weight are often
reported in experimental literature [62]. Another approach would be to prevent the
formation of the film altogether, e.g. by preventing the dissolution of polysulfides in
the liquid electrolyte as discussed in section 2.1.
5.5.2 Transport
Transport in the liquid electrolyte is difficult to analyze experimentally. Using the
model, however, it can be studied easily by looking at the concentration of dissolved
species or the distribution of solids across the cell during discharge. To begin with,
the evolution of solids in the positive electrode is shown in Fig. 5.21.
0.0
0.5
1.0
0 50 100 150 200 250
Vo
lu
m
e
fr
ac
ti
on
Capacity / Ah/kgS
void (Argon)
electrolyte
lithium sulfide
sulfur
passive Li2S
carbon black & binder
Figure 5.21: Evolution of the average volume fractions of all phases in the positive
electrode during a C/50 discharge. For the cell design presented here,
only a small fraction of the electrode’s volume is actually affected.
Compared to Fig. 5.3, which presents similar results for a much less porous cell, the
changes to the volume fractions appear much smaller. Only a few percent of the total
volume change during the entire cycle. Another interesting feature of this cell is that
a considerable fraction of the sulfur is not present in the solid form during the entire
cycle. The ratio of active to passive sulfur for this cell is 2:3 (see above). Even in the
121
fully charged or discharged state, only about half of the sulfur is precipitated. While
this cell design is not suitable for commercialization, it is perfectly apt for the scientific
study of the Li/S system. As discussed below, this cell design does not suffer from
pore clogging, exhibits only very small diffusion overpotentials, and provides good
access to the liquid electrolyte for the entire activated Li2S. The large amount of passive
weight does not restrain the study of the system either physically or computationally.
Except for the different absolute and relative amounts of the phases, the results are
very similar to those presented in Fig. 5.6b and the corresponding discussion on pages
98–99 is still fully valid for the calibrated model.
The dissolution and precipitation of solids presented in Fig. 5.21 is tightly coupled to
the distribution of dissolved species in the liquid electrolyte, which shall be analyzed
next. To this, the average concentrations of dissolved species and their evolution dur-
ing a discharge/charge cycle are presented in Fig. 5.22. It is apparent from Fig. 5.22
that the concentrations of the various species are different by orders of magnitude.
10−12
10−9
10−6
10−3
100
103
0 50 100 150 200 250
C
on
ce
nt
ra
ti
on
/
m
ol
·m
−3
Capacity / Ah/kgS
Li+
TFSI−
S2−
S2−2
S2−4
S2−6
S2−8
S8
Figure 5.22: Evolution of the concentrations of dissolved species during charge
(dashed lines) and discharge (solid lines) at a C/20 rate. The concen-
tration of S2− is so low that it is entirely outside the plot range.
122
Li+ is most concentrated because it serves as the counter ion for all other charged
species. Its concentration peaks at several mol ·m−3, which is easily dissolved by the
ionic liquid based electrolyte. All polysulfides S2−x with x < 8 get more concentrated
during discharge and less concentrated during charge. The opposite is true for S8(ds)
and S2−8 which get depleted during discharge and replenished during charge. Espe-
cially the concentration of S8(ds) (orange line in Fig. 5.22) changes dramatically from its
saturation concentration at around 4 mol ·m−3 to a mere 10−8 mol ·m−3. The concen-
tration of S2−2 and especially S
2− stays very low during the entire cycle. This is a major
improvement over the initial literature parametrization, since the results are consis-
tent with experimental findings, namely the fact that these two species are virtually
insoluble in common electrolytes [133, 141].
0.0
1.0
2.0
3.0
4.0
5.0
0 50 100 150 200 250 300 350 400
0
100
200
300
400
500
S8(ds)
S2−x
C
on
ce
nt
ra
ti
on
S 8
(d
s)
/
m
ol
·m
−3
C
on
ce
nt
ra
ti
on
S2
− x
/
m
ol
·m
−3
Capacity / Ah/kgS
TD
C/50
C/20
C/10
Figure 5.23: Concentrations of S8(ds) (dashed lines) and all S2−x polysulfide ions (solid
lines) during discharge for four different rates. TD: Thermodynamic, i.e.
quasistatically slow discharge.
While Fig. 5.22 contains lots of information, its interpretation is not straightforward.
Therefore, only the concentration of S8(ds) and the sum of all polysulfide ions (solid
lines) are plotted in Fig. 5.23, this time for different discharge rates. This plot illustrates
the situation in the liquid electrolyte of a Li/S cell much more clearly than Fig. 5.22.
The concentration of S8(ds) represents the source of new sulfur species. For the qua-
sistatic discharge, at a rate of approx. C/1000, it is buffered by the presence of solid S8
and only decreases (but sharply so) once all accessible sulfur is dissolved. The higher
the discharge rate, the lower the concentration of S8(ds). As long as there are significant
amounts of dissolved S8(ds), the production of new dissolved species by dissolution can
123
at least keep up with the consumption by electrochemical reduction. Hence, there is
a constant flow of sulfur in its highest oxidation state into the electrolyte, stabilizing a
steady-state equilibrium, which in turn results in a constant cell voltage. If the reduc-
tion is much faster than the dissolution, the steady-state value for the concentration
of S8(ds) is close to zero. More sulfur may still enter the system, but it will be reduced
almost immediately. In that situation, the voltage decreases continuously as does the
average oxidation state of the dissolved S2−x polysulfide species. As indicated by the
solid lines in Fig. 5.23, their concentration increases more than fourfold during the first
half of the discharge. This increase is fueled by both the dissolution of solid sulfur and
the subsequent reduction of S2−x species producing shorter, but also more numerous
molecules. Once no more sulfur dissolves into the electrolyte (or only at comparatively
low rates, see above), the total concentration of polysulfides levels off. The continuous
precipitation of Li2S from the electrolyte is balanced by the progressive reduction of
longer S2−x species down to S2−4 . The shorter polysulfides are very short lived; they
are fully reduced to S2− and rapidly precipitate as solid Li2S. Later during the dis-
charge, when less longer species are available, most reduction products precipitate
immediately, reducing the total amount of dissolved polysulfides. Depending on the
discharge cutoff voltage, the total concentration may either become as low as that of a
fully charged cell or remain close to the plateau level. Only for the quasistatically slow
discharge the plateau does not exist, since the ratio of each pair of S2−x species is always
equilibrated, i.e. constant. Therefore, each electron extracted from the system directly
lowers all concentrations simultaneously, in the end precipitating 1/16 molecule of S8
per electron. As outlined by the above discussion, Fig. 5.23 contains the most relevant
information about the liquid electrolyte system, which is also confirmed by various
experimental characterization techniques, see e.g. Ref. [P4].
Next, the distribution along the y-axis (MEA cross section) of the cell is analyzed.
Here, the concentration of Li+ is plotted. Since Li+ is the only cation in the model, its
flux is directly proportional to the ionic current in the cell. The y-profiles represent
different stages during the discharge, as indicated by the markers in Fig. 5.24.
It is obvious from Fig. 5.25 that there is little to no change in the concentration of Li+
across the cell. The average concentration increases significantly during the first half
of the discharge and decreases steadily thereafter, as discussed above, on the basis
of Fig. 5.23. Even when the change is fastest, i.e. in the transition region between
the upper and lower voltage plateau and at the end of discharge, the concentration
gradient is hardly discernible. It amounts to only a few percent across the entire
cell, less than 1 % across the cathode. Because of charge neutrality, the gradient of
negatively charged species is equally small. What is more, this is also true for the
individual gradients of the soluble S2−x species, i.e. x > 2 (data not shown). While
124
the concentrations and volume fractions may change greatly during cycling, there is
virtually no spatial variation across the electrode, which means that all active material
can be accessed easily.
5.5.3 Impedance
Electrochemical impedance spectra simulated at different SoC are presented in Fig. 5.26.
Starting from a fully charged cell, a regular galvanostatic discharge is simulated and
stopped at different SoC. The cell is put to rest (i.e. itot = 0) for 15 min. before the
spectra are simulated using the step excitation technique described in section 4.3.1
and analyzed in the frequency range of 1 mHz–1 MHz.
The spectra simulated at different SoC look qualitatively rather different. Given
that the composition of the sulfur electrode is very different, as are the prevalent elec-
trochemical reactions, this does not come as a surprise. At the very beginning, but
especially at the end of discharge, a long “tail” is apparent, related to the high over-
potentials seen in the charge/discharge profiles. Other than that, one large semi-cycle
dominates the appearance of the Nyquist plots. A smaller cycle is discernible in the
high-frequency region of some spectra which is associated with the lithium|electrolyte
interface at the negative electrode.
Because of the volatility of the experimental measurements, the spectra could not be
truly validated. The trend, however, is similar to that observed experimentally for the
cells presented in section 3.2.4: The charge transfer resistance is smallest at medium
SoC, i.e. at the end of the upper and beginning of the lower voltage plateau. As already
discussed on the basis of Fig. 5.9, this can be attributed to the high concentrations of
dissolved Li+ and S2−x ions in the electrolyte. The cell could not fully relax to OCV
within 15 min. in the transition region and at the end of discharge. The appearance of
the corresponding spectra does depend on the relaxation time to some degree (data
not shown), but the general behavior as described above is ubiquitous.
Regardless of the precise values, these results indicate that the spectra are highly
sensitive to the internal state of the battery. Thus, EIS could prove to be a powerful
tool for SoC and SoH analysis in Li/S batteries.
125
1.8
2.0
2.2
2.4
2.6
2.8
0 50 100 150 200 250
• •
•
• •Cel
l v
ol
ta
ge
/
V
Capacity / Ah/kgS
Figure 5.24: Overview of the discharge analyzed in Figs. 5.25 and 5.26. The nominal
discharge rate is C/20.
1500
2000
2500
0 10 20 30 40 50
postive electrode separator neg. el.
Lo
ca
lc
on
ce
nt
ra
ti
on
of
Li
+
/
m
ol
·m
− 3
Distance across MEA / µm
Figure 5.25: Concentration of Li+ across the cell at different times during a discharge.
Colors indicate the time each profile was simulated, see Fig. 5.24.
126
0100
200
Z
’/
Ω
·c
m
2
0
100
200
10−3 10−2 10−1 100 101 102 103 104
-Z
”
/
Ω
·c
m
2
Frequency / Hz
0
100
0 100 200
-Z
”
/
Ω
·c
m
2
Z’ / Ω · cm2
Figure 5.26: Electrochemical impedance spectroscopy during cycling. Upper panels:
Bode representation. Lower panel: Nyquist representation. Simulated
frequency range: 1 mHz–1 MHz. Colors of the spectra indicate the time
the spectrum was simulated, see Fig. 5.24 on the previous page. The tail of
the orange spectrum, simulated at the very end of discharge, ends outside
the plotted area at approx. (400+ 600i)Ω · cm2.
127
5.5.4 Microstructure
Since the model does not include an explicit description of the microstructure, the
following section is rather short. The composition of the cell was already analyzed in
section 5.5.2 above. Besides the trivial expressions in Eqs. (4.24) and (4.25), represent-
ing the S8|electrolyte and Li2S|electrolyte interfaces, there is only one feature in the
model dealing with the microstructure: the formation of a passivating Li2S film on the
positive electrode’s carbon matrix. This film is studied in Fig. 5.27.
Once Li2S starts to precipitate, the film grows almost linearly with time, reaching
close to 10 nm at the end of discharge. This nicely matches the experimental findings
presented in Fig. 3.8d on page 43. The film thickness cannot be measured directly in
the SEM images, but it can be estimated: The smallest structures visible in the electrode
without the film (Fig. 3.8c) are about 20 nm across. After deposition of the film, the
smallest structures are about 100 nm in size. This means that the precipitated Li2S can
fill gaps of several tens of nanometers, but not hundreds. Therefore, a thickness 10 nm,
as suggested by the simulations, is within the right order of magnitude.
In contrast to the film thickness, its resistivity does not grow initially, but remains
very small as long as the film is thinner than the threshold set by ξ2 in Eq. (4.26).
Beyond that value, the resistivity increases sharply, causing a correspondingly high
overpotential toward the end of discharge.
0
5
10
15
0 50 100 150 200 250
0
200
400
600
800
1000
ξ2 threshold
Fi
lm
th
ic
kn
es
s
/
nm
Fi
lm
re
si
st
iv
it
y
/
Ω
·m
Capacity / Ah/kgS
film thickness (left axis)
film resistivity (right axis)
Figure 5.27: Thickness of the Li2S film on the carbon matrix in the positive electrode
and corresponding resistivity during a C/20 discharge.
128
5.6 Multi-step model – polysulfide shuttle extension
In this section, the cycling performance and degradation of the cell are analyzed, tak-
ing into account the polysulfide shuttle as described in section 4.2.7. Implementation
and effects of the polysulfide shuttle model are discussed in detail in Ref. [P3], where
this mechanism was applied to a simplified model of the Li/S battery.
The calibration performed in Ref. [P3], however, is not valid for the multi-step
model, since the reaction mechanism for the polysulfide reduction/oxidation is dif-
ferent, as are the geometry and composition of the cell. For the cell described by
Ref. [P3], the model predicts that the solid part of the negative electrode is composed
of approximately 25 vol.-% Li2S after only three cycles. For the cells used in this work,
the amount deposited on the lithium surface is much smaller, as affirmed by SEM
images and EDX spectra of electrodes examined after cycling, see Fig. 5.28. Small
amounts of sulfur are not detected by EDX, especially at low exposures (as indicated
by the total number of counts in Fig. 5.28). Still, to keep the sulfur content in the neg-
ative electrode below the EDX detection limit of ∼ 2 % even after 200 cycles, no more
than one ppm of Li2S may be deposited during each cycle.
50 µm
0
100
200
300
0.0 1.0 2.0 3.0
C
N
O
F
S
C: 22 %
N: 25 %
O: 37 %
F: 17 %
S: 0 %C
ou
nt
s
X-ray energy / keV
Figure 5.28: Left: SEM micrograph of a lithium electrode removed from a Li/S cell af-
ter 200 cycles. Magnification: 500×. The electrode’s surface is more rough
than the pristine Li foil, but no large depositions or even dendrites are
present. Right: EDX spectrum of the same area, indicating considerable
amounts of electrolyte decomposition products, but no signs of sulfur or
Li2S. Note that Li does not show up in the EDX spectra, cf. section 2.4.1.
Therefore, the additional parameters needed for the shuttle and degradation mech-
anism are not identical to Ref. [P3]. Neither are they calibrated, but simply assumed
so that they result in a clearly visible effect, while not contradicting experimental find-
ings. To achieve this, the precipitation rate is chosen identical to the positive electrode.
This rate is rather fast, so that virtually all S2− formed in the negative electrode is
129
precipitated in place. The CTR rate was lowered subsequently until the rate of Li2S
formation was slow enough, so that the above-mentioned one ppm per cycle is not
exceeded. The values of the additional parameters are listed in Tab. 5.3.
Table 5.3: Additional parameters for the polysulfide shuttle model.
Reaction rate scaling factors ξq
ξCTR 2.5 · 10−22
ξprecip 1.0
Initial anode volume fractions εn
Lithium 0.37
Li2S 0.00
Electrolyte 0.63
With this parametrization, the effect on discharge and charge profiles is presented
in Fig. 5.29. The overall shape of the profiles is not strongly affected, except for the
position of the end of charge: The steep voltage rise that usually triggers the end of
charge is shifted to the left on the capacity axis, way into the negative range: More
charge is transferred into the cell than extracted in either the previous or the following
discharge half-cycle. This is possible because the shuttle mechanism allows the cell to
accept charge without changing the SoC.
2.0
2.4
2.8
-50 0 50 100 150 200 250 300
C
el
lv
ol
ta
ge
/
V
Capacity / Ah/kgS
without PS shuttle
with PS shuttle
Figure 5.29: Effect of the polysulfide shuttle on discharge/charge profiles. Note that
the capacity axis extends below zero, since more charge is transferred into
the cell than extracted in the previous discharge.
130
The loss in Coulombic efficiency may be tolerable for some applications, but there is
also a loss of active material. Unlike the reversible capacity decay discussed in section
5.1, this effect cannot be easily recovered by reasonably slow refresh cycles or CCCV
charging (data not shown). Instead, the deposition of Li2S on the negative electrode is
practically irreversible. While technically, the reaction is implemented reversibly, the
backward reaction does not proceed at a significant rate because it would require the
oxidation of S2−x directly above the highly reducing lithium surface. Fig. 5.30 presents
the simulated evolution of Li2S during the first ten cycles. The amount of Li2S formed
is influenced by the relative rates in Tab. 5.3 (data not shown), but the formation of Li2S
per se is inevitable. For the cell simulated here, it takes several cycles to accumulate
a few ppm of Li2S. While this does not sound like a large amount, it causes the SoH
to decrease at a slow, but steady rate. Additionally, the formation of a passivating
layer of Li2S on the lithium surface may affect the cell performance long before a
significant fraction of the total sulfur is lost. Lacking fundamental information about
other degradation modes in the cell, it could not be determined whether or not this
effect contributes significantly to the capacity fade, neither could the ultimate cause
for the experimentally observed degradation be identified.
0.0
0.4
0.8
1.2
0 10 20 30 40 50 60
2.0
2.5
Vo
lu
m
e
fa
ct
io
n
of
Li
2S
/
pp
m
C
el
l v
ol
ta
ge
/
V
Time / h
cell voltage
Li2S
Figure 5.30: Evolution of the amount of Li2S deposited at the negative electrode.
Besides analyzing one possible mode of degradation of the Li/S battery, the above
section also illustrated that additional effects can be implemented easily in the existing
modeling framework, potentially enabling the study of many interesting electrochem-
ical effects in the Li/S cell.
131
5.7 Conclusions
5.7.1 Achievements
First of all, it is a major methodological achievement of this work to enable simulations
of all relevant electrochemical techniques with just one consistent set of physically-
based governing equations and one authoritative set of parameters. This is an accom-
plishment completely novel to the domain of Li/S batteries; even many established
chemistries lack such a capable model.
Regarding the simulations, transport phenomena and charging modes of the cell
were studied in section 5.1, on the basis of charge/discharge profiles and cycling plots
obtained from the simplified global two-step model. Results of the more detailed
multi-step model were introduced in section 5.2, with a parameter set obtained pri-
marily from Ref. [196], and other pre-existing literature. The results include charge/
discharge plots, also at different rates, as well as concentration profiles of lithium ions
and polysulfide species during cycling. In addition, impedance spectra of the cell
at various SoC were presented. Based on the the evolution of solids in the positive
electrode, the two voltage plateaus during discharge could be explained conclusively.
In section 5.3, we could show that a rigorous calibration of the model in its present
form is possible and meaningful, using data available from literature as well as data
recorded experimentally. A highly selective determination of individual parameters
could be achieved, since various key features of the charge/discharge profiles and
impedance spectra are very sensitive to certain physical parameters of the model, e.g.
the hysteresis of the plateau voltages, the onset and slope of the transition region be-
tween the voltage plateaus, or the high-frequency intercept of the impedance spectra,
among others. The simulation results obtained from the calibrated multi-step model
are considered plausible and could be partly validated in section 5.4. Even though dis-
crepancies remain (see below), the model enables a thorough discussion of the charge
and discharge behavior, cyclic voltammetry profiles, and electrochemical impedance
spectra at different stages during the discharge. We could show what the performance
would look like for different formation efficiencies, i.e. different fractions of activated
Li2S. In addition to reproducing experiments, many properties of the battery can be
studied, which are not easily accessible to the experiment: for example, the volume
changes in the positive electrode were analyzed and put in relation to the concentra-
tions of dissolved polysulfides in section 5.5. The concentration of Li+ across the cell
was studied at different SoC. Finally, the model can assist with the interpretation of
reversible and irreversible cell degradation. New reactions or effects can be accom-
modated easily in order to quickly study phenomena like the polysulfide shuttle or
various degradation modes. Even though the accuracy of such studies cannot be ex-
132
pected to be very good without extensive validation, the model can promptly indicate
trends and major effects.
5.7.2 Shortcomings
As discussed above, the model does not match the experimental results perfectly. Ac-
tually, it does not match nearly as well as a typical fitted equivalent circuit, cf. for
example Refs. [182, 262, 263]. Also, as discussed in section 5.4, it falls short of explain-
ing certain phenomena even qualitatively:
• The end of charge looks very different from experimental results for typical ex-
periments, see e.g. Fig. 5.18. It also looks different from the simulations pre-
sented in section 5.2, compare e.g. Figs. 5.7a and 5.20. Since the voltage rise at
the end of charge is of purely kinetic origin, an explanation might be that there
are different reaction mechanisms in place for charge and discharge, which are
not considered separately in the model.
• The end of discharge does not depend on the discharge rate as much as it should,
see Fig. 5.18. This is a strong indication that there are actually more effects
triggering the end of discharge than accounted for in the model.
• The voltage overshoot at the beginning of the charging half-cycle is consider-
ably larger for the simulation than for the experimental reference data, see e.g.
Fig. 5.13. This is a strong indication that the precipitation and dissolution of Li2S,
which is the root cause of the high overpotential, is not represented well by the
single, symmetrical, non-electrochemical reaction in the multi-step model.
• Probably the most severe limitation of the model is that the reaction mechanism
itself is not truly validated. Quite the contrary, it is known to be incomplete
and some studies suggest that at least parts of the mechanism are implausible
[44, 142, 264]. While different reaction mechanisms could be implemented in
the model easily, these findings do question the value of the current calibration
based on the multi-step mechanism.
Despite these limitations, the model is already very useful and versatile in its present
state. Further improvements are suggested and discussed below:
133
5.7.3 Possible improvements and outlook
Improved parametrization. Of course, the values of all model parameters can be im-
proved by fitting more precise and more reproducible data – or even simply fitting
more data. Considering the large parameter space, though, the insight that can be
gained by mere statistical improvements of the data base is limited at best, cf. sec-
tion 4.3.3. A more promising approach is to determine more parameters a priori by
explicit measurements. The values of all non-fitted parameters have been researched
thoroughly, see section 5.3. However, there are many more parameters which could
have been determined by external sources, if suitable data were available. Measuring
such data would allow to reduce the number of fitted parameters and abolish the need
to calibrate groups of parameters together. Some notable examples are listed below:
• The charge transfer processes can be studied by rotating disc electrode measure-
ments [44], helping to verify or improve the reaction mechanism.
• X-ray absorption near edge structure (XANES) studies can provide a better un-
derstanding of the chemical composition of the electrodes before and after cy-
cling [148, 264, 265], especially in conjunction with X-ray photoelectron spec-
troscopy (XPS) [265].
• In situ X-ray diffraction (XRD) can be used to analyze the composition of solids
in the electrodes [155, 156].
• NMR experiments can provide diffusion coefficients of all dissolved species, see
e.g. [266].
• UV–Vis spectroscopy can be used to analyze the composition polysulfide solu-
tions semi-quantitatively [141, 142], or even to measure the speed of dispropor-
tionation reactions as well as diffusion constants [267].
• DFT calculations taking into account the effect of the electrolyte can provide
more accurate thermodynamic data of the dissolved intermediates.
• HPLC with electrolytes extracted in operando can possibly tell polysulfides apart
[268, 269].
• In operando imaging of single particles could improve the representation of the
evolution of active surface areas in the model.
More important than this input to the model, however, is the improvement of the
model itself, described in the following paragraphs:
134
Improved reaction mechanism. Even the more detailed multi-step reaction mech-
anism does not include all possible reactions in the Li/S cell. In fact, the inclusion
of every possible effect is not an end in itself – approximations, simplifications, and
exclusions are an essential part of every decent model. However, at least the addition
of disproportionation reactions as well as a more detailed description of precipitation
and dissolution are considered critical tasks. As discussed above, Refs. [44, 142, 264]
provide a good starting point for formulating a revised reaction mechanism, fully
supported by experiments.
Microstructure. While a full 3D model is beyond the scope and besides the purpose
of our modeling activities, the parametrization could be improved by simulating rep-
resentative elements of an electrode in full 3D or by analyzing the structure in other
ways in order to obtain more accurate effective parameters. An approach similar to
Refs. [270, 271] could be followed, using SEM, FIB-SEM, or TEM images to establish
a realistic microstructure layout as a base for the computation of effective transport
parameters and active surface areas as outlined in Ref. [272].
Degradation. Many more chemical degradation mechanisms may be added to the
model in a straightforward fashion, ranging from simple parasitic reactions like sol-
vent oxidation to a full SEI model in the style of Refs. [273–275]. Additionally, one
can think of non-chemical degradation mechanisms such as mechanical deformation
[276], cracking [218], agglomeration, or the build-up of insulating layers [277]. All of
these directly affect the transport in the cell, but can also result in loss of electric con-
tact in parts of the porous cathode [278]. However, lacking systematical studies on the
degradation effects, it would be difficult to determine the right reaction mechanism(s)
and parameters for even more side reactions. On the other hand, a first systematical
and methodologically sound experimental analysis of the polysulfide shuttle has been
published recently by Janek et al. [279], which might enable a better parametrization
and hence more realistic simulation of this particular parasitic reaction in the future.
Transport in the liquid electrolyte. First of all, a more detailed description of the
ionic liquid is required in order to improve the modeling of transport in the liquid
electrolyte. Another interesting improvement would be the description of locally non-
neutral concentrated electrolytes [207], potentially built on top of a thermodynamically
consistent description of the ionic transport in liquid electrolytes [203]. The latter
would also pave the way for a theoretically sound temperature-dependent model of
the Li/S battery, outlined next.
135
Non-isothermal model. Adding a description of heat sources and transport to the
model is straightforward from a theoretical point of view [280]. However, since almost
all effects in the cell depend on the temperature one way or the other, this measure
would effectively double the number of parameters required. Depending on the phe-
nomenon, entropies, activation energies, temperature coefficients, and so on would
need to be determined in addition to the present parameters. That issue put aside,
this extension is desirable, because it would enable the use of this model as a back-end
for higher level, system-scale simulations, which rely on a correct energy balance.
A full implementation of heat generation and transfer is available within the mod-
eling framework Denis, see e.g. Ref. [281]. The approach taken is to include an addi-
tional conservation equation for energy in addition to the conservation of mass and
charge:
d
dt
(wel + wchem + wheat) = div (~wheat) . (5.1)
Here wel is the electrical, wchem the chemically stored energy density, and ~wheat, the
heat flux. The total heat Q tot equals wheat times the volume of the CV; it is composed of
several contributions including, among others, heat from exothermic reactions Qelchem,
polarization heating Qpol, and resistive heating Qres:
Q tot = Qelchem + Qpol + Qres + ... . (5.2)
These sources need to be implemented one by one. In addition, one more equation is
needed, describing the transport of heat.
$cp · ∂T
∂t
= −div (~wheat) +∑
j
Qj , (5.3)
where Qj are the heat sources as defined above, $ is the density, and cp the effective
heat capacity at constant pressure. Finally, boundary conditions are needed, describing
the flux in and out of the computational domain. A detailed description of the theory
and implementation can be found in Ref. [282, chap. 3.4].
To conclude, there are many possible extensions to the model; some exist only as
concepts, some ready to use. The only limitation is the number of parameters that can
be determined faithfully and reliably.
136
6 Conclusions and outlook
6.1 Summary of achievements
This work contributes to the field of Li/S battery research in two respects: First, the
understanding of the electrochemistry of Li/S cells was improved by analyzing the
system with the assistance of a physically-based electrochemical model. Second, a
novel cathode material, carbon-coated Li2S, has been demonstrated to work well in
Li/S batteries. In the following, the achievements and major results of this work are
summarized chapter by chapter.
In the introduction, the recent history of electrochemical energy storage was laid
out, culminating in the ultimate motivation for this study – the need for better batter-
ies. Furthermore, general features of the Li/S battery were presented and remaining
challenges, which still prevent the widespread use of this promising electrochemical
system were addressed. Chapter 1 closed with an overview of this work, defining the
scope and goals for both the experimental and theoretical parts.
Materials and methods of the experimental work were described in chapter 2. Fol-
lowing an overview of the literature on Li2S based electrodes, the material and elec-
trode design was presented together with a detailed discussion of the goals and a
summary of the processing steps. A special emphasis was on the preparation and
analysis of the active material, carbon-coated Li2S particles. The active material was
prepared by ball-milling Li2S powder and applying a carbon coating by CVD in a
custom-designed tube furnace. Since Li2S is very reactive with trace water as well as
oxygen, a protective atmosphere was required during the entire preparation. Next, the
other components of the cell were described, including the lithium counter electrode,
the polypropylene separator, the ionic liquid based electrolyte with additives as well
as the additional components of the positive electrode. The latter include the binder
SBR and various forms of carbon, enhancing the specific surface area and conductiv-
ity. Also, the preparation of porous electrodes and the assembly of coin cells were
illustrated.
The second part of chapter 2 elaborated on the characterization of the materials,
electrodes, and cells. A sound characterization could only be achieved combining
different techniques such as electron microscopy (SEM and TEM), spectroscopy (EDX,
137
Raman), chemical tests, and the entire spectrum of electrochemical tests, including
galvanostatic and potentiostatic cycling, cyclic voltammetry, impedance spectroscopy,
and combinations thereof. Finally, other methods or materials that did not work well
were briefly discussed in section 2.5.
The experimental results were presented in chapter 3, which roughly followed the
chronology of this work: First, the characterization of the active material and elec-
trodes made thereof was presented in section 3.1. The effectivity of different coating
protocols was evaluated by SEM imaging and EDX spectroscopy. The best results were
obtained using acetylene as carbon precursor, deposited at 400–450 °C. Both Raman
spectra and TEM images confirm that a regular, uniform carbon coating was applied
to the particles. Next, SEM images and EDX spectra of several different electrodes
were compared to optimize composition, fabrication, and handling.
Despite numerous tweaks and improvements to the active material, the electrolyte,
and the electrode preparation, Li2S was not contained in the electrode as intended.
This could be shown by a chemical test as well as a run in a transparent beaker cell,
where the leakage of polysulfides into the electrolyte was evident.
Nevertheless, cells prepared from the novel material exhibit a promising behavior as
analyzed in section 3.2. To exploit their full potential, the activation (a.k.a. formation)
needed to be adjusted, as discussed in section 3.2.3. We could show that almost all
the capacity can be activated at voltages as low as 3.4 V. While an estimation of SoC
and SoH based on impedance spectra was intended, the reproducibility of the results
did not allow for an authoritative analysis. Next, the shape of the charge/discharge
profiles was analyzed and found to be generally very similar to that of “regular” Li/S
cells with sulfur/carbon electrodes, as analyzed in section 3.2.5. Finally, cycling results
were presented for various cells and conditions, demonstrating a slow, but steady
capacity decay. While the rate of decay, as well as the initial discharge capacity of
slightly above 1000 Ah · kg−1sulfur, leave room for improvement, the remaining capacity
of almost 500 Ah · kg−1sulfur after 200 cycles was higher than any value published for Li2S
based electrodes at that time.
The modeling part started with a general recapitulation of the means and purpose
of modeling in chapter 4, before the methodology of our physically-based continuum
model was presented. The phenomena described by the model and underlying as-
sumptions were discussed along with the equations used in the model. These include
the transport in the liquid electrolyte, the electrochemical and electrical description
of the cell as well as the representation of different phases and their microstructures
in the one-dimensional model. Next, two different reaction mechanisms were intro-
duced: the simple two-step global model and the more detailed multi-step model.
Both comprise a set of electrochemical reactions and corresponding expressions for
138
the evolution of the phases’ active surface areas. Finally, a possible mode of degrada-
tion – the formation of Li2S at the negative electrode – was integrated into the model.
The chapter continued with remarks on the steps taken to successfully implement the
model as an extension to the existing modeling framework Denis. Finally, the strategy
for model calibration and a consideration of the relevant errors were presented.
The presentation of the simulation results in chapter 5 started off with the results of
the global model in section 5.1. Here, transport in the liquid electrolyte and cycling
protocols were studied, illustrating the advantages of suppressing sulfur mobility in
the electrolyte and CCCV charging to fully recover the capacity after discharging.
Also the effect of a slow “refresh” cycle was illuminated. The more detailed multi-step
model was first presented with literature parametrization in section 5.2. As shown
by discharge profiles, impedance spectra, and the evolution of solid and dissolved
species, this model generally produces plausible results, including the well-known two
distinct voltage plateaus during discharge and the formation/dissolution of solid S8
and Li2S. The results confirm the findings previously published in Ref. [196]. Similar
to that publication, however, the end of discharge could not be explained conclusively.
Therefore, in a second step, all parameters in the multi-step model were rigorously
calibrated to match the experimental data presented in chapter 3. The calibration as
well as a partial validation were described in sections 5.3 and 5.4. A broad selection
of data was considered for the calibration, ranging from literature values to the ex-
perimentally known composition of the cell, and its electrochemical performance in
the form of discharge profiles and impedance spectra. Several important parameters
were determined by fitting this data. For each fit, a parameter variation was presented,
illustrating the effect of the parameter. After correct parametrization, simulation re-
sults reproduced experiments to a reasonably good degree and could be used to better
understand and explain the empirical findings. More simulations were presented in
section 5.5, including the evolution of the grand total and individual concentrations of
dissolved species, as well as a study of the formation of a passivating Li2S film on the
electrochemically active carbon in the positive electrode. For the calibrated model, the
resistance of this film actually triggers the end of discharge for a wide range of oper-
ating conditions. Finally, the effect of adding the polysulfide shuttle and the related
deposition of Li2S at the negative electrode to the model was studied in section 5.6.
For the highly porous cell simulated, the slightly reduced Coulombic efficiency and
the deposition of several ppm Li2S at the lithium electrode caused only minor effects
in terms of capacity and voltage profiles.
139
6.2 Limitations of this work and ideas for future research
The work presented here is certainly not immaculate and complete in every respect.
Instead, there are several known limitations and more research is needed to address
them.
Some immediate suggestions for experimentally improving the Li2S-based elec-
trodes have been expounded in section 3.3.2, including the preparation of smaller
and more uniform particles, the application of a more uniform and complete coating
as well as calendering electrodes. Still, it is anything but clear whether these measures
can help to effectively confine polysulfides within the carbon shell [165]. Interestingly,
recent research showed that well-designed cells can tolerate polysulfides dissolved in
the liquid electrolyte for some 1500 full cycles, even with a lithium metal counter elec-
trode [76]. If confinement was no longer intended, the thickness of the coating and
the carbon content could be further reduced, yielding electrodes with extremely high
sulfur content, while maintaining large internal surfaces and good conductivity at the
same time.
Regarding the model, several suggestions were made in section 5.7.3 to further im-
prove the parametrization. More importantly, however, the reaction mechanisms im-
plemented so far are known to be incomplete. Especially the precipitation and disso-
lution of Li2S (and potentially other species) needs to be represented in more detail,
e.g. by introducing asymmetric reaction pathways. Also, it was discovered that the
disproportionation reactions among the dissolved polysulfides play a vital role for the
electrochemistry of the cell [44, 142, 283]. They need to be added and calibrated in a
future version of the multi-step reaction mechanism. While the set of electrochemical
reactions in the Li/S cell is still not known in its entirety, the references cited above
provide a good starting point for formulating a revised reaction mechanism. Also,
more simulations can be run, exploiting the existing capabilities of the model, e.g.
galvanostatic intermittent titration (GITT), or scans of the overpotential, i.e. fast-sweep
cyclic voltammetry. On the other hand, the same simulations can be run with different
cathode and electrolyte compositions. Finally, more side reactions can be added eas-
ily to study additional effects, e.g. thermal decomposition, self discharge, and further
degradation modes. All in all, the model provides a flexible tool for further studies of
Li/S batteries.
140
6.3 Outlook beyond academia
While Li/S batteries are not quite ready for the mass market yet, there is steady and
inevitable progress that will virtually force this technology into the production lines
eventually – even if only for certain applications. Cardinal remaining challenges for
Li/S batteries, as often listed in review articles [37, 62, 284, 285], are the still incomplete
understanding of the complex electrochemistry, the limited cycle life, the insufficient
power density for cells with realistically high loadings, the failure to suppress side
reactions including the polysulfide shuttle, and finally the not yet mature develop-
ment of large scale, low-cost fabrication strategies. Despite a recent breakthrough
[57], arguably the negative electrode is another issue [286, 287]. An increasing num-
ber of scientists and engineers is working on all aspects of the battery to overcome
those remaining limitations; taking into account what could be achieved both in terms
of technological advancement and gain of knowledge during just the last four years,
prospects are good. It seems more than likely that the goals set by the German Bundes-
entwicklungsplan Elektromobilität [19], the U. S. DRIVE consortium [288], and other
strategic key players are not only within reach, but will be met by advanced Li/S
batteries sooner rather than later.
Unfortunately, the remarkable progress at the material or electrode level does not
translate into an equally large gain at the system or application level. A realistic es-
timate of the impact of Li/S cells on the battery system performance in the context
of electromobility can be found in Refs. [53, 289]. Because of various system necessi-
ties and restraints, the effective gain in specific energy would probably not exceed a
factor of three compared to Li-ion batteries available in 2012 and a factor of approx-
imately one compared to next-generation lithium intercalation materials. Figures for
volumetric energy density look even worse. While this analysis might look discourag-
ing at first glance, reaching the goals mentioned in the above publications would still
be a big success. Even if there is little or no benefit in terms of energy density com-
pared to future high-voltage Li-ion batteries, there are still the advantages in terms of
cost, availability, safety, and environmental sustainability. On the contrary, one might
even want to trade off some of the capacity in order to further simplify the battery:
A lithium/polysulfide battery with a liquid positive electrode was recently proposed
independently by two research groups [145, 157]. Skipping the last reduction steps
(to avoid the formation of solid Li2S) makes the battery more robust, reliable and af-
fordable at the expense of only a fraction of the capacity. Also, some of the battery
components are known to be available from renewable sources [290], improving long-
term material availability and reducing environmental impact. Finally, there is the
possibility to use the battery in a hybrid system together with another secondary bat-
tery, fuel cell, or supercapacitor, which completely changes the use profile [291–293]
141
and might alleviate some of the issues still associated with deep-cycling Li/S cells.
As prices go down – and potentially, the price of Li/S batteries could be very low
[32, 33] – this technology also becomes more attractive for large-scale grid storage ap-
plications [9]. Already today, the electrochemical storage and controlled use of wind
and solar energy is more cost-effective than immediately feeding all power into the
electricity grid as it becomes available [294, 295]. Still, the segment of grid storage
is dominated by pumped hydro-power. Other technologies such as high-temperature
Na/S batteries or flywheels are used for special purposes only, e.g. short term high-
power supply [9]. The further advancement of Li/S batteries, but also other tech-
nologies, e.g. novel flow batteries [296, 297], or improved fuel cells and electrolyzers
[298–300], might fundamentally change the game.
To conclude, more research is needed in order to advance the understanding, opti-
mization, and application of batteries in general and Li/S technology in particular. In
this regard, it is a good sign that policy makers seem to realize that the development
of next-generation systems for electrical energy storage is one of the high importance,
high impact research topics of the next decade(s). Consequently, there is an increasing
number of publicly and privately funded efforts worldwide [54]. These include the
activities of the Helmholtz Society, the German Aerospace Center, and the Helmholtz
Institute Ulm, promoting this dissertation.
142
A Appendix
A.1 List of symbols
Table A.1: List of symbols. Except for the indices, nomenclature follows Ref. [59].
symbol unit description
an – activity of dissolved species n
AVm m2 ·m−3 volume-specific surface area of interface m
cn, cn, ref mol ·m−3 (reference) concentration of dissolved species n
Cdl, m F ·m−2 area-specific double-layer capacity of interface m
(ds) - label for dissolved species
Dn, Dn, eff m2 · s−1 (effective) transport coefficient of species n
E V cell voltage
E0 V open-circuit voltage
Eact, q J ·mol−1 activation energy of reaction q,set to 0 J ·mol−1 for all reactions
F As ·mol−1 Faraday’s constant, 9.6485 · 104 As ·mol−1
gn J ·mol−1 molar Gibbs free energy of species n
∆Gq J ·mol−1 Gibbs free energy of reaction q
iVdl A ·m−3 double-layer current source term
iVF A ·m−3 Faradaic current source term
itot A ·m−2 current density
i, j – general purpose indices
Jn mol ·m−2 · s−1 flux of species n
143
symbol unit description
kq, fwd, kq, rev mol ·m−2 · s−1 forward and reverse rate of reaction q
k0, q, fwd, k0, q, rev mol ·m−2 · s−1
forward and reverse rate constants
(pre-exponential factors) of reaction q
LCA, LAN m thickness of cathode and anode, respectively
l m thickness of a film or coating
m – index of interfaces
m kg mass
M kg ·mol−1 molar mass
(for phases, M denotes the average molar mass)
~nCV – normal vector of the CV surface
n – index of species
p0 Pa atmospheric pressure, 1.0133 · 105 Pa
p – index of phases
q – index of chemical reactions
qmax Ah · kg−1 specific charge
Q J heat
r m particle radius
R J ·mol−1 ·K−1 universal gas constant, 8.3145 J ·mol−1 ·K−1
Rp mol ·m−3 net production rate of phase p
(s) - label for solid species
s˙n, s˙n
mol ·m−3 · s−1,
mol ·m−2 · s−1
volume and area specific net rate of production of
species n, respectively
t s time
T K temperature
VCV, ∂VCV m3, m2 volume and surface of a CV, respectively
w J ·m−3 specific energy or specific heat
144
symbol unit description
W J work or energy
y m
spatial position across the MEA
(i.e. from cathode to anode)
y – solution vector of the DAE system
z – number of electrons in a charge-transfer step
Z Ω·m2 complex impedance
αq –
symmetry factor of charge transfer reaction q,
set to 0.5 for all reactions
β –
temperature exponent in Arrhenius rate equation,
set to 0 for all reactions
γn – activity coefficient of species n
εp – volume fraction of phase p
er – dielectric constant (relative permittivity)
η V overpotential
µn J ·mol−1 chemical potential of species n
νn,q – stoichiometric coefficient of species n in reaction q
ξ variable phenomenologically motivated fit parameter
$ kg ·m−3 density
ρ Ω·m resistivity
σ, σ2 variable uncertainty and variance of a statistical quantity
τp – tortuosity of phase p
φCA, elde, φAN, elde V potential of the cathode and anode, respectively
∆φ V potential difference
χ2 – goodness-of-fit parameter
145
A.2 Calculation of specific capacities, energy densities and
state of charge
One can find many slightly different values for the theoretical volumetric and gravi-
metric capacities of various cathode materials in the literature. However, since the
specific charge is a fundamental material property, there is only one correct value for
each material, which can be derived as follows
qmax = z · F ·M−1 , (A.1)
where z is the change in oxidation state, equal to the number of electrons transferred,
M is the molar mass of the species, and F is Faraday’s constant. Values of qmax are
reported in Tab. A.2 for several systems. The specific capacity qmax can be calculated
for either the charged or discharged state; for a fair comparison of different materials
the heavier state should be used, as is reported in Tab. A.2. Contrary to this postula-
tion, the specific capacity is often reported as Ah/kgsulfur, not Ah/kgLi2S, in this work.
This is a minor compromise to improve comparability with other works; the factor for
conversion of Ah/kgsulfur to Ah/kgLi2S is 0.698.
The specific charge contained in a battery electrode q˜max equals
q˜max = qmax ·mactive material ·m−1electrode , (A.2)
where mactive material and melectrode are the total mass of the active material and the
entire electrode, respectively. Because qmax is a fundamental property of each material,
it is useful to compare different materials, especially from a theoretical point of view.
On the other hand, q˜max takes the composition of the electrode into account and is
therefore more popular among experimentalists, because of its technological relevance.
Finally, there is ˜˜qmax, which is based on the system’s weight – a figure popular among
engineers. Of course, ˜˜qmax < q˜max ≤ qmax always holds.
While the capacity of a cell depends on the oxidation state of the active material, the
specific energy depends on the standard Gibbs free energy of reaction ∆G:
wmax = ∆G ·M−1 . (A.3)
Again, the value may also be expressed in units of electrode weight w˜max or system
weight ˜˜wmax. In contrast, the specific energy is determined experimentally by integrat-
ing the charge density times the voltage at which it is extracted:
wexp =
ˆ q=0
q=qmax
E (q) dq . (A.4)
146
However, the cell voltage E(q) and even the open-circuit voltage E0 (q) are not intrin-
sic functions of the cathode active material. Instead, they depend on the electrode’s
microstructure as well as the electrolyte and counter electrode used. Therefore, it
is reasonable for different authors to report significantly different (practical) energy
densities wexp < wmax.
That leaves two options for the definition of the SoC of a battery. According to the
experimental definition, 100 % corresponds to the state at the end of charge and 0 %
to the end of discharge. According to the theoretical definition, 100 % corresponds
to the situation where all accessible sulfur in the cell is present in the form of S8,
whereas 0 % corresponds to the situation where all accessible sulfur is present as Li2S.
Here, accessible means that the sulfur is present in a form which can be transformed
into S8 by means of (electro-)chemical reactions in the cell. While the latter is a more
fundamental definition, it implies that the SoC will never be exactly 100 % or 0 %,
since, for example, some sulfur will always be dissolved in the electrolyte. To avoid
confusion, the latter definition is chosen in this work throughout.
Finally, the cell’s SoH is unanimously defined as the fraction of the initial capacity
that is still available at a given time.
Table A.2: Maximum theoretical specific capacity and specific energy of various mate-
rials. See also Ref. [1], Tab. V. Annotations: aestimate, actual ∆G varies with
SoC, see Ref. [301]; bexcl. product water.
chemistry M z ∆G qmax wmax
g ·mol−1 MJ ·mol−1 Ah · kg−1 kWh · kg−1
Be+ 12 O2  BeO 25.01 2 0.609 2143 6.76
2 Li+O2  Li2O2 45.88 2 0.644 1168 3.9
2 Li+ 18 S8  Li2S 45.95 2 0.441 1166 2.67
Al+ 34 O2 +
3
2 H2O+
Al(OH)3
78.00 3 1.272 1030 4.5
Li+CoO2  LiCoO2 97.87 1 0.201a 273 0.57a
Li+ LiFePO4 
LiFePO4
157.76 1 0.352a 170 0.62a
1
2 Pb+
1
2 PbO2 +
H2SO4  PbSO4 +H2O
303.26b 2 0.088 177 0.08
147
A.3 List of chemicals
Table A.3: List of chemicals used.
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a
A
ld
ri
ch
Te
tr
ae
th
yl
en
e
gl
yc
ol
di
m
et
hy
l
et
he
r
TE
G
D
M
E
22
2.
28
10
01
l
≥
99
%
14
3-
24
-8
Si
gm
a
A
ld
ri
ch
Po
ly
et
hy
le
ne
gl
yc
ol
di
m
et
hy
l
et
he
r
PE
G
D
M
E
∼2
40
∼1
08
0
l
?
24
99
1-
55
-7
Si
gm
a
A
ld
ri
ch
C
ar
bo
n
bl
ac
k
Su
pe
r
C
65
–
19
00
s
≥
99
.8
%
13
33
-8
6-
4
Ti
m
ca
l
1,
3-
D
io
xo
la
ne
D
O
L
74
.0
8
10
60
l
99
.8
%
64
6-
06
-0
Si
gm
a
A
ld
ri
ch
1,
2-
D
im
et
ho
xy
et
ha
ne
(D
im
et
hy
lg
ly
co
l)
D
M
E
90
.1
2
86
7
l
99
.5
%
11
0-
71
-4
Si
gm
a
A
ld
ri
ch
N
-M
et
hy
l-
2-
py
rr
ol
id
on
e
N
M
P
99
.1
3
10
28
l
an
hy
dr
ou
s,
99
.5
%
87
2-
50
-4
Si
gm
a
A
ld
ri
ch
M
ul
ti
-w
al
le
d
ca
rb
on
na
no
tu
be
s
M
W
C
N
T
–
–
s
>
95
%
30
80
68
-5
6-
6
C
he
ap
Tu
be
s
A
rg
on
A
r
39
.9
5
–
g
99
.9
99
%
74
40
-3
7-
1
Pr
ax
ai
r
A
ce
ty
le
ne
C
2H
2
26
.0
4
–
g
?
74
-8
6-
2
Pr
ax
ai
r
Et
hy
le
ne
C
2H
4
28
.0
5
–
g
?
74
-8
5-
1
Pr
ax
ai
r
148
A.4 List of equipment
Table A.4: List of equipment
device name/model manufacturer
high energy ball mill 8000M Mixer/Mill
Thomas Scientific, Swedesboro, NJ,
USA
YSZ grinding balls
4039GM, diameter = 10, 5,
and 3 mm
Inframat Advanced Materials,
Manchester, CT, USA
CVD furnace
Lindberg Blue M
Mini-Mite
Thermo Fisher Scientific, Waltham,
MA, USA
glove box 1 UNILab MBraun, Garching, Germany
glove box 2 LAB master 130 MBraun, Garching, Germany
TEM F20 UT Tecnai FEI, Hillsboro, OR, USA
Raman LabRAM (confocal) Horiba Jobin Yvon, Edison, NJ, USA
Karl Fischer titrator DL39 Coulometer
Mettler Toledo, Greifensee,
Switzerland
Ultrasonic tip Digital Sonifier 450
Branson Ultrasonics Corporation,
Danbury, CT, USA
Ultrasonic bath 2510 Ultrasonic Bath
Branson Ultrasonics Corporation,
Danbury, CT, USA
Molecular sieves Zeolite type 4Å TRICAT, Hunt Valley, MD, USA
Arbin cycler MSTAT
Arbin Instruments, College Station,
TX, USA
BioLogic cycler (EIS) VSP BioLogic, Claix, France
BioLogic cycler (CVs) VMP3 BioLogic, Claix, France
temperature chamber Model TEC1 TestEquity LLC, Moorpark, CA, USA
SEM microscope JSM-7000F JEOL, Akishima, Tokyo, Japan
coin cell parts Belleville Stack
National Research Council Canada,
Montreal, QC, Canada
hot plate/magnetic stirrer 120V Stirrer/hot plate Corning Inc., Corning, NY, USA
149
A.5 Parameter sets for the models
Table A.5: Parameters used for the global model. Values are given for T = 303.15 K
and p = 101325 Pa.
cathode
T
hi
ck
ne
ss
:4
0
µm
C
on
tr
ol
vo
lu
m
es
:8
Bu
lk
ph
as
es
Vo
lu
m
e
fr
ac
ti
on
ε 0
To
rt
uo
si
ty
τ 0
Sp
ec
ie
s
D
en
si
ty
/
In
it
ia
lc
on
ce
nt
ra
ti
on
D
iff
us
io
n
co
ef
fic
ie
nt
/
m
2
·s
−1
Su
lf
ur
0.
5
1.
0
S 8
(s
)
2.
07
·1
03
kg
·m
−3
–
C
ar
bo
n
0.
2
1.
0
C
2.
26
·1
03
kg
·m
−3
–
El
ec
tr
ol
yt
e
0.
3
1.
2
C
4H
6O
3
1.
20
·1
03
kg
·m
−3
/1
.1
8
·1
04
m
ol
·m
−3
–
Li
+
1.
00
·1
03
m
ol
·m
−3
1
·1
0−
10
TF
SI
−
7.
67
·1
02
m
ol
·m
−3
1
·1
0−
10
S2
−
2.
36
·1
01
m
ol
·m
−3
1
·1
0−
10
S2
− 2
2.
36
·1
01
m
ol
·m
−3
1
·1
0−
10
S2
− 4
2.
36
·1
01
m
ol
·m
−3
1
·1
0−
10
S2
− 6
2.
36
·1
01
m
ol
·m
−3
1
·1
0−
10
S2
− 8
2.
36
·1
01
m
ol
·m
−3
1
·1
0−
10
S 8
(d
s)
3.
54
·1
01
m
ol
·m
−3
1
·1
0−
10
Li
th
iu
m
su
lfi
de
(L
i 2
S)
1
·1
0−
4
1.
0
Li
2S
(s
)
1.
64
·1
03
kg
·m
−3
–
In
te
rf
ac
es
Sp
ec
ifi
c
ar
ea
A
V 0
A
re
a-
sp
ec
ifi
c
ca
pa
ci
ty
C
dl
R
ea
ct
io
n(
s)
Fo
rw
ar
d
ra
te
co
ns
ta
nt
M
ol
ar
G
ib
bs
fr
ee
en
er
gy
su
lf
ur
su
rf
ac
e
1
·1
06
m
2
·m
−3
0
F
·m
−2
2
Li
+
+
1 /
8
S 8
(d
s)
+
2
e−

Li
2S
(s
)
1.
0
·1
01
7
m
ol
·m
−2
·s
−1
−4
41
.4
kJ
·m
ol
−1
S 8
(d
s)

S 8
(s
)
1.
0
·1
03
m
ol
·m
−2
·s
−1
15
.0
kJ
·m
ol
−1
separator
T
hi
ck
ne
ss
:1
00
µm
C
on
tr
ol
vo
lu
m
es
:5
Bu
lk
ph
as
es
Vo
lu
m
e
fr
ac
ti
on
ε 0
To
rt
uo
si
ty
τ 0
Sp
ec
ie
s
El
ec
tr
ol
yt
e
1.
0
1.
2
se
e
ca
th
od
e
In
te
rf
ac
es
:n
on
e
anode
T
hi
ck
ne
ss
:2
00
µm
C
on
tr
ol
vo
lu
m
es
:4
Bu
lk
ph
as
es
Vo
lu
m
e
fr
ac
ti
on
ε 0
To
rt
uo
si
ty
τ 0
Sp
ec
ie
s
D
en
si
ty
/
In
it
ia
lc
on
ce
nt
ra
ti
on
Li
th
iu
m
0.
7
1.
0
Li
(s
)
5.
34
·1
02
kg
·m
−3
El
ec
tr
ol
yt
e
0.
3
1.
2
se
e
ca
th
od
e
In
te
rf
ac
es
Sp
ec
ifi
c
ar
ea
A
V 0
A
re
a-
sp
ec
ifi
c
ca
pa
ci
ty
C
dl
R
ea
ct
io
n(
s)
Fo
rw
ar
d
ra
te
co
ns
ta
nt
M
ol
ar
G
ib
bs
fr
ee
en
er
gy
an
od
e
su
rf
ac
e
2
·1
06
m
2
·m
−3
0
F
·m
−2
Li
(s
)

Li
+
+
e−
1.
0
·1
0−
3
0.
0
J·
m
ol
−1
150
Table A.6: Initial set of parameters for the multi-step model (for T = 298 K and p =
101325 Pa).
cathode
Th
ic
kn
es
s:
41
µm
C
on
tr
ol
vo
lu
m
es
:5
Bu
lk
ph
as
es
Vo
lu
m
e
fr
ac
ti
on
ε 0
To
rt
uo
si
ty
τ 0
Sp
ec
ie
s
D
en
si
ty
/
In
it
ia
lc
on
ce
nt
ra
ti
on
D
iff
us
io
n
co
ef
fic
ie
nt
/
m
2
·s
−1
Su
lf
ur
0.
16
1.
0
S 8
(s
)
2.
07
·1
03
kg
·m
−3
–
C
ar
bo
n
0.
06
19
1.
0
C
2.
26
·1
03
kg
·m
−3
–
El
ec
tr
ol
yt
e
0.
77
8
1.
0
C
4H
6O
3
1.
20
·1
03
kg
·m
−3
–
Li
+
1.
02
4
·1
03
m
ol
·m
−3
1
·1
0−
10
PF
− 6
1.
02
3
·1
03
m
ol
·m
−3
4
·1
0−
10
S2
−
8.
46
·1
0−
10
m
ol
·m
−3
4
·1
0−
10
S2
− 2
5.
35
·1
0−
7
m
ol
·m
−3
1
·1
0−
10
S2
− 4
2.
05
·1
0−
2
m
ol
·m
−3
1
·1
0−
10
S2
− 6
3.
31
·1
0−
1
m
ol
·m
−3
6
·1
0−
10
S2
− 8
1.
82
·1
0−
1
m
ol
·m
−3
6
·1
0−
10
S 8
(d
s)
1.
94
m
ol
·m
−3
1
·1
0−
9
Li
th
iu
m
su
lfi
de
(L
i 2
S)
1
·1
0−
4
1.
0
Li
2S
(s
)
1.
64
·1
03
kg
·m
−3
–
In
te
rf
ac
es
Sp
ec
ifi
c
ar
ea
A
V 0
A
re
a-
sp
ec
ifi
c
ca
pa
ci
ty
C
dl
R
ea
ct
io
ns
Fo
rw
ar
d
ra
te
co
ns
ta
nt
R
ev
er
se
ra
te
co
ns
ta
nt
Su
lf
ur
|E
le
ct
ro
ly
te
1
·1
05
m
2
·m
−3
0
F
·m
−2
S 8
(s
)

S 8
(d
s)
1.
90
0
·1
02
m
ol
·m
−2
·s
−1
1.
00
0
·1
03
m
ol
·m
−2
·s
−1
C
ar
bo
n|
El
ec
tr
ol
yt
e
1
·1
05
m
2
·m
−3
1.
36
F
·m
−2
1 /
2
S 8
(d
s)
+
e−

1 /
2
S2
− 8
7.
72
5
·1
01
6
m
ol
·m
−2
·s
−1
2.
94
0
·1
0−
24
m
ol
·m
−2
·s
−1
3 /
2
S2
− 8
+
e−

2
S2
− 6
4.
33
1
·1
01
9
m
ol
·m
−2
·s
−1
1.
19
0
·1
0−
20
m
ol
·m
−2
·s
−1
S2
− 6
+
e−

3 /
2
S2
− 4
3.
19
3
·1
01
7
m
ol
·m
−2
·s
−1
4.
19
1
·1
0−
17
m
ol
·m
−2
·s
−1
1 /
2
S2
− 4
+
e−

S2
− 2
2.
37
5
·1
01
4
m
ol
·m
−2
·s
−1
7.
50
5
·1
0−
21
m
ol
·m
−2
·s
−1
1 /
2
S2
− 2
+
e−

S2
−
4.
65
5
·1
01
5
m
ol
·m
−2
·s
−1
4.
73
8
·1
0−
19
m
ol
·m
−2
·s
−1
Li
2S
|E
le
ct
ro
ly
te
1
·1
05
m
2
·m
−3
0
F
·m
−2
2
Li
+
+
S2
−

Li
2S
(s
)
2.
75
0
·1
0−
2
m
ol
·m
−2
·s
−1
5.
00
0
·1
0−
11
m
ol
·m
−2
·s
−1
separator
Th
ic
kn
es
s:
9
µm
C
on
tr
ol
vo
lu
m
es
:5
Bu
lk
ph
as
es
Vo
lu
m
e
fr
ac
ti
on
ε 0
To
rt
uo
si
ty
τ 0
Sp
ec
ie
s
D
en
si
ty
Se
pa
ra
to
r
0.
37
1.
0
no
ne
El
ec
tr
ol
yt
e
0.
63
1.
0
se
e
ca
th
od
e
1.
20
·1
03
kg
·m
−3
In
te
rf
ac
es
:n
on
e
anode
Th
ic
kn
es
s:
10
0
µm
C
on
tr
ol
vo
lu
m
es
:5
Bu
lk
ph
as
es
Vo
lu
m
e
fr
ac
ti
on
ε 0
To
rt
uo
si
ty
τ 0
Sp
ec
ie
s
D
en
si
ty
Li
th
iu
m
0.
37
1.
0
Li
(s
)
5.
34
·1
02
kg
·m
−3
El
ec
tr
ol
yt
e
0.
63
1.
0
se
e
ca
th
od
e
1.
20
·1
03
kg
·m
−3
In
te
rf
ac
e
Sp
ec
ifi
c
ar
ea
A
V 0
A
re
a-
sp
ec
ifi
c
ca
pa
ci
ty
C
dl
R
ea
ct
io
n
Fo
rw
ar
d
ra
te
co
ns
ta
nt
M
ol
ar
G
ib
bs
fr
ee
en
er
gy
Li
th
iu
m
|E
le
ct
ro
ly
te
1
·1
06
m
2
·m
−3
1.
80
F
·m
−2
Li
(s
)

Li
+
+
e−
4.
09
·1
0−
9
0.
0
J·
m
ol
−1
151
Table A.7: Calibrated set of parameters for the multi-step model (for T = 298 K and
p = 101325 Pa). Notes: ?(elyte) is a placeholder species, representing the liq-
uid electrolyte. Its density is chosen to match the composition of electrolyte
#5 in Tab. 2.2 on page 25.
cathode
T
hi
ck
ne
ss
:2
5
µm
C
on
tr
ol
vo
lu
m
es
:5
Bu
lk
ph
as
es
Vo
lu
m
e
fr
ac
ti
on
ε 0
To
rt
uo
si
ty
τ 0
Sp
ec
ie
s
D
en
si
ty
/
In
it
ia
lc
on
ce
nt
ra
ti
on
D
iff
us
io
n
co
ef
fic
ie
nt
/
m
2
·s
−1
Su
lf
ur
0.
04
14
0
1.
00
S 8
(s
)
2.
07
·1
03
kg
·m
−3
–
C
ar
bo
n
0.
03
24
8
1.
00
C
2.
26
·1
03
kg
·m
−3
–
St
yr
en
e-
bu
ta
di
en
e
ru
bb
er
0.
02
80
0
1.
00
[C
58
H
69
] n
9.
65
·1
02
kg
·m
−3
–
El
ec
tr
ol
yt
e
0.
69
14
3
1.
12
(e
ly
te
)?
1.
29
·1
03
kg
·m
−3
–
Li
+
1.
51
·1
03
m
ol
·m
−3
1
·1
0−
11
TF
SI
−
1.
00
·1
03
m
ol
·m
−3
1
·1
0−
11
N
O
− 3
5.
00
·1
02
m
ol
·m
−3
1
·1
0−
11
S2
−
6.
19
·1
0−
25
m
ol
·l−
1
1
·1
0−
11
S2
− 2
3.
65
·1
0−
19
m
ol
·l−
1
1
·1
0−
11
S2
− 4
2.
12
·1
0−
7
m
ol
·l−
1
1
·1
0−
11
S2
− 6
7.
16
·1
0−
4
m
ol
·l−
1
1
·1
0−
11
S2
− 8
2.
92
·1
0−
1
m
ol
·l−
1
1
·1
0−
11
S 8
(d
s)
9.
38
m
ol
·l−
1
1
·1
0−
11
Li
th
iu
m
su
lfi
de
(L
i 2
S)
0.
00
12
0
1.
00
Li
2S
(s
)
1.
64
·1
03
kg
·m
−3
–
Pa
ss
iv
e
lit
hi
um
su
lfi
de
(p
-L
i 2
S)
0.
10
54
9
1.
00
Li
2S
(s
)
1.
64
·1
03
kg
·m
−3
–
A
rg
on
0.
10
00
0
1.
00
A
r
1.
78
kg
·m
−3
–
In
te
rf
ac
es
Sp
ec
ifi
c
ar
ea
A
V 0
A
re
a-
sp
ec
ifi
c
ca
pa
ci
ty
C
dl
R
ea
ct
io
ns
Fo
rw
ar
d
ra
te
co
ns
ta
nt
M
ol
ar
G
ib
bs
fr
ee
en
er
gy
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Own publications
Peer-reviewed publications
[P1] J. P. Neidhardt, D. N. Fronczek, T. Jahnke, T. Danner, B. Horstmann, and W. G.
Bessler. A flexible framework for modeling multiple solid, liquid and gaseous phases in
batteries and fuel cells. Journal of The Electrochemical Society 159, A1528–A1542
(2012).
[P2] D. N. Fronczek and W. G. Bessler. Insight into lithium-sulfur batteries: Elementary
kinetic modeling and impedance simulation. Journal of Power Sources 244, 183–188
(2013). 16th International Meeting on Lithium Batteries.
[P3] A. F. Hofmann, D. N. Fronczek, and W. G. Bessler. Mechanistic modeling of poly-
sulfide shuttle and capacity loss in lithium-sulfur batteries. Journal of Power Sources
259, 300–310 (2014).
[P4] N. A. Cañas, D. N. Fronczek, N. Wagner, A. Latz, and K. A. Friedrich. Experimen-
tal and Theoretical Analysis of Products and Reaction Intermediates of Lithium-Sulfur
Batteries. The Journal of Physical Chemistry C 118, 12106–12114 (2014).
An updated list of my publications can
be found on my Orcid profile page:
http://orcid.org/0000-0003-4924-5415
180
Presentations
[C1] C. Hellwig, D. N. Fronczek, S¸. Sörgel, and W. G. Bessler. Physically based modeling
and simulation of a LiFePO4-based lithium-ion battery (poster). 8th Symposium on
Fuel Cell and Battery Modeling and Experimental Validation. March 8–9, 2011,
Jülich, Germany.
[C2] D. N. Fronczek. Modeling of electrochemistry and transport in the lithium-sulfur bat-
tery (oral presentation). HIU meeting series. September 7, 2011. Stuttgart/Ulm/
Karlsruhe, Germany.
[C3] D. N. Fronczek and W. G. Bessler. Lithium/sulfur batteries: global and elementary
modeling approaches (oral presentation). 9th Symposium on Fuel Cell and Battery
Modeling and Experimental Validation. April 2–4, 2012, Sursee, Switzerland.
[C4] D. N. Fronczek and W. G. Bessler. Insight Into Li/S Batteries: Elementary Kinetic
Modeling and Impedance Simulation (poster). 16th International Meeting on Lithium
Batteries. June 17–22, 2012, Jeju, South Korea.
[C5] D. N. Fronczek and W. G. Bessler. A virtual Li/S battery: Modeling, simulation and
computer-aided development (oral presentation). Next Generation Batteries 2012.
July 18–20, 2012, Boston, MA, USA.
[C6] D. N. Fronczek. Electrochemical energy storage – an overview (lecture). Diablo Valley
College. December 15, 2012, Concord, CA, USA.
[C7] D. N. Fronczek, S. E. Choi, A. Latz, W. G. Bessler, and E. J. Cairns. Next Genera-
tion Battery Electrochemistry Analyzed: Towards a Validated Computational Model of
the Lithium/Sulfur Cell (oral presentation). 224th Meeting of the Electrochemical
Society. October 28–November 2, 2013, San Francisco, CA, USA.
[C8] D. N. Fronczek, A. Latz, E. J. Cairns, and W. G. Bessler. Design and Validation of a
Computational Model of the Lithium/Sulfur Cell (poster). 17th International Meeting
on Lithium Batteries. June 10–14, 2014, Como, Italy.
181
Contributions to miscellaneous works
[X1] D. N. Fronczek, T. Danner et al. Batterien im Einsatz. tecTV (Verein Deutscher In-
genieure e. V.), March 7, 2012. http://www.youtube.com/watch?v=WtJafStGkas .
[X2] D. N. Fronczek, T. Danner, B. Horstmann, and W. G. Bessler. Modeling the next
battery generation: Lithium-sulfur and lithium-air cells. Kraftwerk Batterie. March
6–7, 2012, Münster, Germany.
[X3] J. P. Neidhardt, T. Jahnke, D. N. Fronczek, T. Danner, B. Horstmann, and W. G.
Bessler. A flexible framework for modeling multiple solid, liquid and gaseous phases in
batteries and fuel cells. 9th Symposium on Fuel Cell and Battery Modeling and
Experimental Validation (ModVal 9). April 2–4, 2012, Sursee, Switzerland.
[X4] E. J. Cairns et al. Beyond Lithium Ion: The Lithium/Sulfur Cell. Beyond Lithium Ion
V. June 5–7, 2012, Berkeley, CA, USA.
[X5] J. P. Neidhardt, D. N. Fronczek, T. Jahnke, T. Danner, B. Horstmann, and W. G.
Bessler. A flexible modeling framework for multi-phase management in SOFCs and
other electrochemical cells. 10th European SOFC Forum 2012. June 26–29, 2012,
Luzern, Switzerland.
[X6] A. F. Hofmann, D. N. Fronczek, A. Latz, and W. G. Bessler. Simulating the poly-
sulfide shuttle effect: a thermodynamically consistent, fully reversible, numerical Li/S-
battery model. 10th Symposium for Fuel Cell and Battery Modeling and Experi-
mental Validation (ModVal 10). March 18–20, 2013, Münster, Germany.
[X7] Z. Lin, D. N. Fronczek, C. Nan, S. E. Choi, Y. Zhang, E. J. Cairns, and M.-K.
Song. Improved Sulfur/Lithium Sulfide Nano-composite Electrodes for Next-Generation
Lithium Cells. CalCharge Kickoff Event. May 3, 2013, Berkeley, CA, USA.
[X8] A. F. Hofmann, A. Latz, D. N. Fronczek, and W. G. Bessler. Simulating the poly-
sulfide shuttle effect: a thermodynamically consistent, fully reversible, numerical Li/S-
battery model. 2nd International Conference on Materials for Energy (EnMat II).
May 12–16, 2013, Karlsruhe, Germany.
[X9] N. A. Cañas, B. Pascucci, N. Wagner, and K. A. Friedrich. On the understanding
and improvement of high capacity lithium/sulfur (Li/S) batteries. 14th Ulm Electro-
Chemical Talks (UECT2014). June 23–26, 2012, Ulm, Germany.
182
Events and workshops without original contribution
[E1] Next-generation battery activities at DLR. Hannover Messe. April 4–6, 2011, Han-
nover, Germany.
[E2] HE-Lion Battery school. July 20–21, 2011, Rauischholzhausen, Germany.
[E3] First convention of the Helmholtz Institute Ulm for Electrochemical Energy Stor-
age. October 20–22, 2011, Ulm, Germany.
[E4] X-Raptor business game: Setup and commissioning of an intelligent factory.
February 16–17, 2012, Düsseldorf, Germany.
[E5] Beyond Lithium Ion V conference. June 5–7, 2012, Berkeley, CA, USA.
[E6] MIC/ZAE Workshop: Charakterisierung poröser Materialien. April 8–9, 2014, Würzburg,
Germany.
[E7] 14th Ulm ElectroChemical Talks (UECT2014). June 23–26, 2014, Ulm, Germany.
183
Academic résumé
of Dipl.-Phys. David Norman Fronczek
born on November 23rd, 1983
in Würzburg, Germany
Education and research
01/2011–12/2014 Graduate student at the University of Stuttgart,
supervisor: Prof. Dr. Wolfgang G. Bessler
01/2011–12/2014 Researcher at the German Aerospace Center (DLR), Institute of
Engineering Thermodynamics, Stuttgart, Germany
04/2012–05/2013 Visiting research scholar at Lawrence Berkeley National Labora-
tory (LBNL), Berkeley, CA, USA
06/2013–12/2014 Associated researcher at the Helmholtz Institute Ulm for Electro-
chemical Energy Storage (HIU), Ulm, Germany
10/2003–10/2010 Undergraduate student at the University of Würzburg,
degrees earned: Physik-Diplom, 1. Staatsexamen Biologie/Physik
11/2008–10/2009 Diplom student at the Rudolf-Virchow-Center for Experimental
Biomedicine (RVZ), Würzburg, Germany, supervisors: Dr. Ingrid
Teßmer & Prof. Dr. Bert Hecht
11/2007–07/2008 Intern student at ZAE Bayern, Würzburg, Germany
09/1994–07/2003 Franz-Miltenberger-Gymnasium Bad Brückenau
Honors and awards
• Graduate fellow of the Foundation of the German Industry (sdw)
• Awardee of grants by the German Academic Exchange Service (DAAD) and the
Italian Elettrochimica ed Energia (e&e)
Trivia
• You are reading the 154th revision of this document.
• total number of Denis simulation runs: 3494
• total number of successful Denis runs: 1824
• data recorded and generated: 61.7 GB
• coin cells assembled: 237
• coin cells tested: 178
• parts, chemicals, and services ordered: 76
• number of e-mails sent (received): 2690 (9976)
• lines of code written or modified by the author: 14 628
(approx. 23 % of the simulation framework in 2014)
• number of commits to the code and model repository: 209
• total distance traveled for this thesis: 87 934 km
(not including the daily commute)
• different domiciles 2011–2014: 6
• LinkedIn contacts collected: 137
• cups of coffee consumed at work: 1022
• Scientific wordles (like the one on page ) prepared and designed: 12
• This document was produced using free software: LyX/LaTeX for typesetting
and gnuplot, Inkscape, chemfig, and the GIMP for illustrations.
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Acknowledgments
First and foremost I would like to thank my supervisor Wolfgang Bessler. Already in
2011 you gave me a great start into this project. You did so not only by choosing a great
topic to work on and teaching me what I needed to know, but also by trusting and
believing in me right from the start. During the entire past four years you have been
both demanding and promoting, determined yet open-minded, and always, always
very cordial and encouraging.
Next, I would like to thank Elton Cairns, not only for hosting me at LBNL, but
also for letting me tap from his enormous well of experience and expertise in long
discussions, e-mails, and phone calls. You taught me to always keep the next goal in
mind, but also look at the big picture, making me a better scientist.
Also, I would like to thank Arnulf Latz and Andreas Friedrich for hosting and fund-
ing me during my entire time at DLR and HIU, but particularly for your continued
support and advice.
Finally, many thanks go to Stefano Passerini for joining the board of examiners.
Of course many more people contributed to the success of this work. In my group at
DLR I am most thankful to (in no particular order) Timo Danner, Jonathan Neidhardt,
Birger Horstmann, Christian Hellwig, Andreas Hofmann, and Klaus Göckelmann,
who all directly supported my modeling work one way or the other. In my group
at LBNL, I am most indebted to Chris Choi who has been working for me for nine
months and made valuable contributions to almost all aspects of the experimental
part of this work, to Kunpeng Cai who voluntarily and most comradely helped me
to get started in the U.S. in general and at LBNL in particular, and to Min-Kyu Song
from whom I could learn so much, but also to Caiyun Nan, Wujun Fu, and Zhan Lin
who all have been working closely with me.
For many a good discussion, great ideas, and contributions I would like to thank
Natalia Cañas, Vitaliy Yurkiv, Moritz Henke, Nico Hörmann, Valene Cruz, Bruno
Scrosati, Ulrich Sauter, Jake Christensen, Michel Fouré, Burçak Ebin, Anthony Fer-
rese, John Newman, Venkat Srinivasan, Peter Therkelsen, Rutooj Desphande, Zhihui
Wang, Charles Delacourt, Ulrike Vogl, and Simon Lux, the latter also for providing
Raman spectroscopy data, and Xiangyun Song for the TEM images.
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Also, I would like to thank all the people in the research groups of Arnulf Latz,
Andreas Friedrich and Norbert Wagner in Stuttgart; the groups of Vince Battaglia,
Gao Liu, Venkat Srinivasan, Jeff Urban, and John Kerr in Berkeley and the groups
of Arnulf Latz, Max Fichtner, and Stefano Passerini in Ulm. Without the support and
companionship of all of you, my time as a Ph.D. student would have been considerably
less productive and most certainly less fun.
Thankfully, equipment and materials required for my experimental activities were
provided by UC Berkeley and Lawrence Berkeley National Laboratory. In addition, I
would like to acknowledge the funding I received during the past years by the Ger-
man Academic Exchange Service (DAAD), the Elettrochimica ed Energia (e&e), and
especially the Foundation of the German Industry (sdw). I am most thankful for
the support of the sdw without which this work might still have been possible – but
spending a year in the U.S. and raising two children at the same time would not.
In the end, I owe it all to my family – and especially my wife Stefanie – that I
could do what I did and learn what I have learned to become who I am today. My
appreciation is beyond words.
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