APPROVED: Jincheng Du, Major Professor Witold Brostow, Committee Member Nigel Shepherd, Committee Member

and Chair of the Department Materials Science and Engineering

Costas Tsatsoulis, Dean of the College of Engineering

Mark Wardell, Dean of the Toulouse Graduate School

ATOMISTIC COMPUTER SIMULATIONS OF DIFFUSION MECHANISMS IN

LITHIUM LANTHANUM TITANATE SOLID STATE ELECTROLYTES

FOR LITHIUM ION BATTERIES

Chao-Hsu Chen

Thesis Prepared for the Degree of

MASTER OF SCIENCE

UNIVERSITY OF NORTH TEXAS

August 2014

Chen, Chao-Hsu. Atomistic Computer Simulations of Diffusion Mechanisms in

Lithium Lanthanum Titanate Solid State Electrolytes for Lithium Ion Batteries. Master

of Science (Materials Science and Engineering), August 2014, 82 pp., 4 tables, 28

figures, 76 numbered references.

Solid state lithium ion electrolytes are important to the development of next

generation safer and high power density lithium ion batteries. Perovskite-structured

LLT (La2/3-xLi3xTiO3, 0 < x < 0.16) is a promising solid electrolyte with high lithium ion

conductivity. LLT also serves as a good model system to understand lithium ion

diffusion behaviors in solids. In this thesis, molecular dynamics and related atomistic

computer simulations were used to study the diffusion behavior and diffusion

mechanism in bulk crystal and grain boundary in lithium lanthanum titanate (LLT)

solid state electrolytes. The effects of defect concentration on the structure and

lithium ion diffusion behaviors in LLT were systematically studied and the lithium ion

self-diffusion and diffusion energy barrier were investigated by both dynamic

simulations and static calculations using the nudged elastic band (NEB) method. The

simulation results show that there exist an optimal vacancy concentration at around

x=0.067 at which lithium ions have the highest diffusion coefficient and the lowest

diffusion energy barrier. The lowest energy barrier from dynamics simulations was

found to be around 0.22 eV, which compared favorably with 0.19 eV from static NEB

calculations. It was also found that lithium ions diffuse through bottleneck structures

made of oxygen ions, which expand in dimension by 8-10% when lithium ions pass

through. By designing perovskite structures with large bottleneck sizes can lead to

materials with higher lithium ion conductivities. The structure and diffusion behavior of

lithium silicate glasses and their interfaces, due to their importance as a grain

boundary phase, with LLT crystals were also investigated by using molecular

dynamics simulations. The short and medium range structures of the lithium silicate

glasses were characterized and the ceramic/glass interface models were obtained

using MD simulations. Lithium ion diffusion behaviors in the glass and across the

glass/ceramic interfaces were investigated. It was found that there existed a minor

segregation of lithium ions at the glass/crystal interface. Lithium ion diffusion energy

barrier at the interface was found to be dominated by the glass phase.

ii

Copyright 2014

by

Chao-Hsu Chen

iii

ACKNOWLEDGEMENTS

I would like to thank all people who have helped and encouraged me during

my study, and made the completion of this thesis possible.

During my research life at UNT, I want to offer my deepest appreciation to my

advisor, Dr. Jincheng Du. He motivated and encouraged me to keep passionate on

my research work. In addition, he was always willing to help me deal with the issues

on my research. Therefore, my research work and this thesis have become smooth

and successful. Also, I would like to convey my hearty grateful to all my thesis

committee members, Dr. Nigel Shepherd and Dr. Witold Brostow, for their support

and suggestions on my thesis.

Last but not least, I dedicate my deepest gratitude to my wife for her sacrifice.

She always showed kindness and patience to me during my research work.

iv

TABLE OF CONTENTS

Page ACKNOWLEDGEMENTS ......................................................................................... iii LIST OF TABLES ...................................................................................................... vi LIST OF FIGURES ....................................................................................................vii CHAPTER 1 INTRODUCTION ................................................................................... 1

1.1 Brief History .......................................................................................... 1 1.2 Principle of Lithium Ion Battery ............................................................. 2 1.3 Materials of Electrolytes Solid State Lithium Ion Batteries .................... 3 1.4 Motivation ............................................................................................. 5 1.5 Thesis Layout ....................................................................................... 8

CHAPTER 2 MOLECULAR DYNAMICS SIMULATION DETAILS AND METHODOLOGY ..................................................................................................... 11

2.1 Introduction ......................................................................................... 11 2.2 Theory ................................................................................................. 12

2.2.1 Verlet Algorithm ........................................................................ 12 2.2.2 The Velocity Verlet Algorithm ................................................... 13 2.2.3 Leap Frog Algorithm ................................................................. 13

2.3 Molecular Dynamics Simulation .......................................................... 14 2.3.1 Ensembles ............................................................................... 14 2.3.2 Potentials ................................................................................. 15

2.4 Methodology of Data Analysis ............................................................ 16 2.4.1 Mean Square Displacement and Diffusion Coefficient ............. 16 2.4.2 Neutron Structure Factor .......................................................... 17

CHAPTER 3 DEFECT CONCENTRATION EFFECT ON LITHIUM ION DIFFUSION IN LITHIUM LANTHANUM TITANATE SOLID STATE ELECTROLYTES ............... 19

3.1 Abstract ............................................................................................... 19 3.2 Introduction ......................................................................................... 19 3.3 Methodology ....................................................................................... 21

3.3.1 Introduction .............................................................................. 22 3.3.2 Initiation of Crystal .................................................................... 22

3.4 Bulk Crystal Structure and Effect of Temperature on Lattice Parameter ............................................................................................................ 25

v

3.5 Effect of Vacancy Concentration on Lithium Ion Diffusion .................. 26 3.6 Diffusion Energy Barrier: Static Calculations ...................................... 33 3.7 Lithium Ion Diffusion Mechanism ........................................................ 33 3.8 Total Ionic Conductivity Calculation .................................................... 36

CHAPTER 4 STRUCTURE AND LITHIUM ION DIFFUSION IN SILICATE GLASSES AND AT THEIR INTERFACES WITH LITHIUM LANTHANUM TITANATE CRYSTALS .............................................................................................................. 39

4.1 Abstract ............................................................................................... 39 4.2 Introduction ......................................................................................... 39 4.3 Methodology ....................................................................................... 41

4.3.1 Introduction .............................................................................. 41 4.3.2 Initiation of Glass System ......................................................... 43 4.3.3 Build the Glass/Crystal Interface Structure .............................. 45

4.4 The Structure of Lithium Silicate Glasses ........................................... 48 4.5 The Structure of the Lithium Silicate Glass/LLT Crystal Interface ....... 53 4.6 Diffusion Coefficients in Lithium Silicate Glasses and at the Boundary

............................................................................................................ 54 4.7 Lithium Ion Diffusion Behavior in Interface System ............................. 58

CHAPTER 5 DIFFUSION ANISOTROPY AND CATION RADIUS EFFECT IN LITHIUM LANTHANUM TITANATE ......................................................................... 61

5.1 Abstract ............................................................................................... 61 5.2 Introduction ......................................................................................... 61 5.3 Methodology ....................................................................................... 62 5.4 Diffusion Anisotropy of Lithium Ions in LLT ......................................... 63 5.5 Effect of a Site Cation Substitution on Lithium Ion Diffusion ............... 66

CHAPTER 6 SUMMARY .......................................................................................... 71 CHAPTER 7 FUTURE RESEARCH ......................................................................... 74 REFERENCES ......................................................................................................... 75

vi

LIST OF TABLES

Page

Table 1.1 Materials of solid state electrolytes for lithium ion batteries ........................ 5

Table 2.1 Ionic charges and Buckingham potential parameters ............................... 16

Table 3.1 Comparison of calculated and experimental structure of LLT (Li0.3La0.567TiO3)

................................................................................................................................. 23

Table 4.1 Glass composition parameters for MD simulation ..................................... 44

Figure 3.9: The steps of Li ion which migrates through the bottleneck. The left picture

of each step exhibits the size of bottleneck structure and the right one shows a view

vii

LIST OF FIGURES

Page

Figure 1.1: The ideal perovskite crystal structure of lithium lanthanum titanate (LLT).

Lithium and lanthanum are distributed over A-site. TiO6 octahedral are exclusively

connected through corner sharing. In defected crystal, lithium ions are replaced by

lanthanum ions with each substitution creates two lithium vacancies. Grey octahedral:

TiO6 octahedral, purple ball: lithium, blue ball: lanthanum, red ball: oxygen................ 8

Figure 3.1: (a) The initial structure of lithium lanthanum titanate (LLT). (b) The defect

structure of LLT at 600K (Purple: lithium; Blue: lanthanum) ...................................... 24

Figure 3.2: The lattice parameter as function of x value in LLT vacancy structure

(La2/3-xLi3xTiO3) .......................................................................................................... 26

Figure 3.3: (a) Mean square displacement of lithium lanthanum titanate (LLT) (b)

MSD in logarithm ....................................................................................................... 29

Figure 3.4: Lithium ion diffusion coefficients as a function of x value in La2/3-xLi3xTiO3

……………………………………………………………………………………………….30

Figure 3.5: Lithium ion diffusion energy barrier as a function of x value in

La2/3-xLi3xTiO3 ............................................................................................................. 30

Figure 3.6: Static energy barrier from NEB calculations. It shows the energy barrier

and the associated structure of of A-site lithium ions diffuse in LLT. (Purple ball: Li,

Red ball: O, Grey ball: Ti) .......................................................................................... 32

Figure 3.7: Trajectories of lithium ions for composition Li0.2La0.6TiO3. MD simulation is

at 600 K for 160 ps .................................................................................................... 34

Figure 3.8: The bottleneck structure of lithium lanthanum titanate (LLT) (Grey ball: Ti;

Red ball: oxygen; Blue ball: lithium; and the bottleneck structure is schematically

shown in yellow bonds.) ............................................................................................ 34

viii

perpendicular to the diffusion pathway. The black numbers (Å ) are the distance

between oxygen ions of bottleneck structure. It is obvious that the largest bottleneck

size is at step C. (Red: oxygen. Purple: Lithium. Grey: Titanium) ........................... 35

Figure 3.10: Charge carrier concentration as a function of x value in La2/3-xLi3xTiO3

………………………………………………………………………………………….……38

Figure 3.11: Ionic conductivity as a function of x value in La2/3-xLi3xTiO3 ................... 38

Figure 4.1: The structure of the lithium silicate oxide Li2O-2SiO2. Golden pyramids:

silicon oxygen tetrahedrons, red ball: oxygen, blue ball: lithium ions ........................ 45

Figure 4.2: Atomic structure of the glass/crystal interface (a) and zoom in view of the

interface (b). Yellow ball: silicon, red ball: oxygen, green ball: titanium, light blue ball:

lanthanum; purple ball: lithium ................................................................................... 47

Figure 4.3: Comparison of calculated and experimental neutron structure factor

function of lithium disilicate glass. Solid line: MD simulations; circles: experiment data

.................................................................................................................................. 49

Figure 4.4: Li-O pair distribution function as a function of Li2O concentration in lithium

silicate glasses. Arrow points to increase of Li2O concentration ................................ 49

Figure 4.5: Change of lithium ion coordination number as a function of Li2O

concentration ............................................................................................................. 51

Figure 4.6: Distribution of lithium ion coordination number of 30 Li2O-70SiO2

compositions ............................................................................................................. 52

Figure 4.7: Qn distribution as a function of Li2O concentration .................................. 52

Figure 4.8: Z-Density profile across the interface ...................................................... 54

Figure 4.9: (a) Linear (a) and logarithm (b) mean square displacement of lithium ions

in lithium disilicate (LS33) glass ................................................................................ 56

Figure 4.10: Diffusion coefficients of lithium ions in glass–crystal interface and

different composition of glasses for different temperatures (The unit D is cm2/s) ...... 57

ix

Figure 4.11: Diffusion energy barrier for lithium disilicate glass, LLTO crystal, and the

glass-crystal interfacial structure ............................................................................... 57

Figure 5.1: Lithium ion diffusion coefficient under different external electrical field

along X-axis and Z-axis for composition Li0.2La0.6TiO3 (MD simulations at 600 K) .... 64

Figure 5.2: Trajectories of lithium ions with 43MV/m external electrical field for

composition Li0.2La0.6TiO3. MD simulations at 600 K for 160 ps with electrical field

applied along z-axis................................................................................................... 65

Figure 5.3: The diffusion coefficient of lithium lanthanum titanate (LLT), lithium

gadolinium titanate (LGT), and lithium Ytterbium titanate (LYT) at 600K ................... 67

Figure 5.4: The free volume calculation (a) schematically shows the free volume in

grey color (Blue color: surface area). (b)The comparisons of system free volume and

Li+ diffusion energy barrier among La (1.032Å ), Gd (0.938 Å ), and Yb (0.868 Å ) in LLT,

LGT, and LYT, respectively ........................................................................................ 69

Figure 5.5: The pair distribution functions of La-O, Gd-O, and Yb-O at 600K ........... 70

1

CHAPTER 1

INTRODUCTION

1.1 Brief History

With the development of lighter and thinner portable electronic products,

electronic components have become smaller and smaller. Indeed, applications such

as cameras, mobile phones, and laptops computers are wireless, portable, and

multi-functional but all of them require portable power sources. For these portable

power supply system, high energy storage capacity, light weight, and high stability are

desired. In 21st century, the portable electronic devices enrich our life and lithium

batteries have become the most common portable energy source. The common

batteries such as carbon-zinc battery or alkaline battery which can’t be recharged

after usage are called primary battery. For the batteries used in mobile phones and

laptops are secondary batteries. They can be recharged and reused many times. For

environmental and economic considerations, the secondary batteries dominate our

daily life.

Secondary batteries have been dominated by nickel-cadmium batteries for

decades. In 1991, the new generation of nickel-metal hydride batteries and lithium

secondary batteries were commercialized. They not only meet the requirement of

electronic products but also are characterized by their high energy density,

2

rechargeable, and environmental friendly. Therefore, the global yield of batteries is

nickel-cadmium batteries mainly, followed by nickel-metal hydride batteries and

lithium ion batteries are the least.

The initial development of lithium batteries was primary battery. In 1991, lithium

ion secondary batteries were released by Sony [1]. They have characteristic of high

energy density and voltage operation, stable charge and discharge, wide-ranged

operating temperature, long storage life with more than 500 charge and discharge

cycles. Therefore, they are currently the most important secondary battery.

1.2 Principles of Lithium Ion Battery

The conventional lithium ion batteries such as those used in comment

electronic devices use lithium carbonaceous materials as anode and intercalating

compound such as LiMO2 (M=Co, Ni, Mn) as cathode, which are separated by a

lithium-ion conducting electrolyte layer that is usually made of solution of LiPF6 in

organic solution such as ethylene carbonate-diethylcarbonate. The reactions of

anode and cathode are shown below:

Cathode:

LiCoO2

charge

↔

discharge

𝐿𝑖(1−𝑥)𝐶𝑜𝑂2 + 𝑥 𝐿𝑖+ + x 𝑒− (1.1)

3

Anode:

C6 + 𝑥 𝐿𝑖 + 𝑥 𝑒−

charge

↔

discharge

𝐶6𝐿𝑖𝑥 (1.2)

Total:

LiCoO2 + 𝐶6

charge

↔

discharge

𝐿𝑖(1−𝑥)𝐶𝑜𝑂2 + 𝐶6𝐿𝑖𝑥 (1.3)

In addition to electronic devices, lithium ion batteries have been actively

pursued in hybrid and electrical cars and as stationary energy storage devices to

compensate the intermittent nature of other renewable energy sources such as solar

and wind energies due to their high energy density and high voltage [2, 3]. In these

applications, the cost, safety, stored energy density, charge/discharge rates, and

service life of the batteries are critical parameters. These parameters are closely

linked to the electrode and electrolyte materials used in lithium ion batteries [2-4].

1.3 Materials of Electrolytes of Solid State Lithium Ion Batteries

Currently there are several types of solid state lithium electrolytes that show

promising properties and behaviors (Table 1.1). Among the solid state electrolytes,

perovskite structured lithium lanthanum titanate (LLT) ceramics have one of the

highest ionic conductivity (10-3 S/cm) [5-8], which is the focus of study of this paper. In

LLT, perovskite structure units are separated by lithium layer and lanthanum layers.

4

Lanthanum substitution of lithium site leads to lithium ion vacancy formation and

lithium ion diffuse through the vacancy mechanism through the bottle-neck structure

leads to high ionic conductivity in these materials [5-8]. Garnet structured lithium

lanthanum zirconate also has relatively high ionic conductivity (10-4 S/cm). Lithium

ions occupied 3 different crystalline sites in the Li7La3Zr2O12 framework structure, and

lithium ion conduction pathway is through face-sharing tetrahedral and octahedral

lithium sites [8-11]. Li-analogues of NASICON structures containing Ti4+ ions have

also been found to have high ionic conductivities (up to 10-3 at room temperature).

However, due to the reduction of Ti4+, the Ti-free Li-analogues of NASICONs have

been investigated [12]. Another type of solid lithium electrolyte is LAGP

glass-ceramics. Li1+xAlxGe2-x(PO4)3 (LAGP with x=0.5) crystals were formed from heat

treatment of amorphous powers and total conductivity as high as 2 x 10-4 S/cm at

room temperature was obtained [12]. LIPON is another type solid state electrolyte

that has been used in thin film batteries. LIPON thin films were made by sputtering

Li3PO4 powder in nitrogen gas. Inaba et all have found that three-coordinated N atom

are dominated in LIPON structure. It leads to relatively high ionic conductivity (3.1 x

10-6 S/cm at 25℃) [13-20]. The three-coordinated N atom in the LIPON film may

create higher cross-link density which can facilitate the lithium ions migration between

P-O chains to improve the ionic conductivity [21]. Inorganic sulfide and other glass

5

solid electrolytes such as Li2S-P2S5 have been shown to have higher ionic

conductivity (10-3 to 10-2 S/cm) than oxide materials. In particular, sulfide crystalline

Li10GeP2S12 has the highest known ambient temperature conductivity (1.2 x 10-2 S/cm)

which is close to organic liquid electrolytes. The major disadvantage of sulfide based

electrolytes is their hygroscopic nature in ambient environment [22].

Table 1.1 Materials of solid state electrolytes for lithium ion batteries

1.4 Motivations

Lithium ion solid state electrolytes have important technological applications in

various electrochemical devices such as all solid state lithium ion batteries,

electrochromic devices, and sensors [2, 3]. Compared to conventional lithium ion

batteries that use liquid or polymeric electrolytes, all solid state lithium ion batteries

have higher thermal stability, free of leakage issues, and resistance to shock and

vibration [23, 24]. In combination of their high voltage and high power density, all solid

Materials Ionic Conductivity Features

La2/3−xLi3xTiO3 10-3 S/cm Highest ionic conductivity in ceramics

Li7La3Zr2O12 10-4 S/cm 3-D framework structure,

short Li-Li migration pathway

Li1+xAlxGe2-x(PO4)3 2 x 10-4 S/cm Ti-free NASICON structure

LIPON 3.1 x 10-6 S/cm Higher cross-link density, Li ions can

migrate in P-O chains

Li2S-P2S5 10-3 to 10-2 S/cm Inorganic sulfide material.

Conductivity is close to liquid electrolytes Li10GeP2S12 1.2 x 10-2 S/cm

6

state lithium ion batteries are very promising next generation batteries for stationary

power in renewable energies production and in all electrical or hybrid transportation

systems. One of the key issues in developing all solid state lithium ion batteries and

related devices is the development of solid state lithium ion electrolyte with high

lithium ion conductivity and interfacial stability, especially at the interface with the

anode that usually contains highly reactive lithium metals.

The requirements of electrolytes for lithium ion batteries include high ionic

conductivity and low electronic conductivity, retention of electrode/electrolyte

interface during cycling, chemical and thermal stability, safety and cost consideration

[4]. The commonly used lithium/graphite electrode (anode) operates near the full

potential of lithium metal that can result in lithium dendrite formation and lead to

potential electrical shortening, that can cause heat generation, thermal run away, and

even fire when the electrode is in contact with organic flammable electrolytes [2].

Applications in stationary energy storage and automobiles put even more stringent

requirements of safety and reliability of these batteries [25]. Lithium ion batteries with

solid state electrolytes have the advantages of high thermo and electrical stability,

resistance to shocks and vibrations that are suitable for applications such as

transportation and stationary power storage [23, 24].

One common issue of current solid state lithium ion electrolytes is their

7

relatively low lithium ion conductivity [7, 24]. How to improve the ionic conductivity of

solid state electrolytes is thus a critical issue with great technological importance.

Solid electrolytes that have high ionic conductivities have been actively investigated.

Bates and co-workers have discovered nitrogen doped lithium phosphate glasses

with high lithium ion conductivity as solid state electrolytes, which have been used in

thin film lithium ion batteries [26-28]. Perovskite structured lithium lanthanum titanate

(LLT), La2/3−xLi3xTiO3, has also drawn considerable attention as a promising solid

lithium ion electrolyte (Figure 1.1) [6, 7]. The high conductivity is due to the lithium ion

vacancies introduced as a result of substituting the A site lithium ions with lanthanum

ions.

In LLT, lithium ions diffuse through the vacancy mechanism [7]. According to

previous studies, the conduction of lithium ions depends on the lithium ion vacancy

concentration [29]. A maximum of conductivity was observed at around 40%

vacancies [6, 7]. Polycrystalline ceramics or thin films are the common forms of the

solid state electrolytes. In these electrolytes, the grain boundaries and intergranular

phases that separate each of the highly conductive crystal grains are usually critical

to the overall lithium ion conductivities.

8

Figure 1.1: The ideal perovskite crystal structure of lithium lanthanum

titanate (LLT). Lithium and lanthanum are distributed over A-site. TiO6

octahedral are exclusively connected through corner sharing. In defected

crystal, lithium ions are replaced by lanthanum ions with each substitution

creates two lithium vacancies. Grey octahedral: TiO6 octahedral, purple ball:

lithium, blue ball: lanthanum, red ball: oxygen.

1.5 Thesis Layout

Chapter 1 gives the brief history and the principles of lithium ion batteries. The

common materials which are used to be electrolytes in lithium ion batteries are

mentioned. The advantages and disadvantages of liquid and solid state are

compared, and the previous studies not only on computational but also experimental

research are also presented in this chapter

Chapter 2 presents the theory of molecular dynamics. The algorithm, statistic

ensemble, and potential functions are included in this section. The methodologies of

9

structural analyze and property calculations are also discussed.

Chapter 3 presents the results on lithium lanthanum titanate simulation. The

effects of defect concentration of lithium ion diffusion are shown. The dynamic and

static simulations on lithium ion diffusion energy barrier are compared. The lithium ion

diffusion mechanisms are schematically presented. Total ionic conductivities are

calculated by Nernst-Einstein equation with appropriate charge carrier concentration

equation presented by previous studies.

Chapter 4 presents the structure analyze of the lithium silicate glass and

lithium silicate glass/lithium lanthanum titanate crystal interface. Lithium ion diffusion

coefficients and energy barriers are calculated and compared among crystal, glass,

and interface. Two linear range behavior of lithium ion diffusion in lithium silicate is

studied, and it is found that diffusion coefficients of interface are dominated by glass

system. Some experimental studies between glass and crystal system are also

discussed.

Chapter 5 presents the LLT crystal system with applying the external electrical

field. 2D or 3D lithium ion diffusion behavior is analyzed. Cation radius effect is

studied by substituting lanthanum ions with gadolinium and ytterbium ions,

respectively. Free volume is calculated and compared with lithium ion diffusion

coefficient and energy barrier in lithium lanthanum titanate (LLT), lithium gadolinium

10

titanate (LGT), and lithium ytterbium titanate (LYT).

Chapter 6 summarizes the results of lithium ions diffusion in crystal, glass, and

interface systems, and chapter 7 presents the future research on solid state

electrolytes.

11

CHAPTER 2

MOLECULAR DYNAMICS SIMULATION DETAILS AND MOTHODOLOGY

2.1 Introduction

The concept of molecular dynamics (MD) simulations was firstly mentioned in

1950 by Irving Kirkwood [30] and was further developed and has become a versatile

and very powerful molecular level simulation method that are widely used in physics,

chemistry, biology, and material science. MD mainly utilizes empirical potential

functions to describe the interactions between molecules in the system. By integration

of the equation of motion iteratively at a constant time step, the position, velocity, and

acceleration of each atom or molecule can be recorded in every step of simulations,

based on which thermodynamic properties and other physical properties can be

calculated.

The earliest molecular dynamics simulation was used by Alder and Wainwright

in 1957 and 1959 [31, 32]. They successfully simulated the force between two hard

spheres. In 1964, Rahman utilized the real potential for liquid argon simulation;

subsequently [33], Rahman and Stillinger have completed the simulation of liquid

water [34]. This is the first study of the real system simulation by using molecular

dynamics theory.

12

2.2 Theory

Molecular dynamics simulation is based on Newton’s second law to determine

the position and velocity of molecule in next time step:

F = ma = m𝑑𝑣

𝑑𝑡= m

𝑑2𝑟

𝑑𝑡2 (2.1)

While the forces were calculated by taking derivatives of the potential energies,

majority of the calculations were on the integration of the equation of motions (EOM).

There are several common algorithms for integration of EOM:

2.2.1 Verlet Algorithm

r(t + ∆t) = 2r(t) − r(t − ∆t) + a(t)∆t2 + 𝑂(∆𝑡4) (2.2)

With this method, we only need to consider the positions. Due to the fact that

the function doesn’t include the velocity, we need to utilize finite difference method to

calculate the kinetic energy and temperature:

v(t) =𝑟(𝑡+∆𝑡)−𝑟(𝑡−∆𝑡)

2∆𝑡+ 𝑂(∆𝑡2) (2.3)

The velocity error is 𝑂(∆𝑡2), and the truncation error is 𝑂(∆𝑡4). The Verlet Algorithm

reveals the problem to precision of velocity and positions despite it is accurate and

stable, and it is calculated one step behind atoms positions [35].

13

2.2.2 The Velocity Verlet Algorithm

r(t + ∆t) = r(t) + v(t)∆t +1

2𝑎(𝑡)∆𝑡2 (2.4)

𝑣(t + ∆t) = v(t) +1

2[𝑎(𝑡) + 𝑎(𝑡 + ∆𝑡)]∆𝑡 (2.5)

The velocity Verlet algorithm is faster and more stable than simple Verlet

algorithm. It computes velocities, accelerations, and positions at t + Δt. As we can see

the equation (2.5), only one set of velocities, accelerations, and positions need to be

stored at time t. That’s why it doesn’t require much memory to compute.

2.2.3 Leap Frog Algorithm

v (t +1

2∆t) = v (t −

1

2∆t) + a(t)∆t (2.6)

r(t + ∆t) = r(t) + v(t +1

2∆t)∆t (2.7)

v(t) =𝑣(𝑡+

1

2∆𝑡)+𝑣(𝑡−

1

2∆𝑡)

2 (2.8)

Velocity are firstly calculated at time equal to t + Δt/2, and it is used to calculate

the positions r(t) at time equal to t +Δt. It requires less storage when we do the large

scale simulations because this algorithm makes positions and velocities leap one

over the other. Also, the velocities can be clearly calculated even if the velocities and

position are not calculated at the same time.

14

2.3 Molecular Dynamics Simulation

MD simulations were performed using the parallel general purpose molecular

dynamics code DL_POLY developed at Daresbury Laboratory UK [36]. Long range

Coulombic interactions are calculated using the Ewald sum method. The Verlet

Leapfrog algorithm with a time step of 1*10-15 second was used in the integration of

equation of motion.

2.3.1 Ensembles

The ensemble concept is proposed by Gibbs in 1878. From the microscopic

point of view, some limitations will be introduced in order to maintain the stability of

the simulation system. We can classify the ensembles into 2 parts:

Isobaric isothermal ensemble (NPT): this is characterized by fixing number of

atoms (N), pressure (P), and temperature (T). We use this ensemble to study

the initial structural change, and have the whole system relaxed.

Micro canonical ensemble (NVE): this is characterized by fixing number of

atoms (N), volume (V), and energy (E). The temperature will fluctuate in this

ensemble because of constant energy. We use NVE ensemble after each NPT

ensemble simulation.

We use Verlet leapfrog algorithm along with NPT and NVE ensemble in this work.

15

2.3.2 Potentials

Partial charge pair-wise potentials were used in all simulations of our work.

The set of potentials was similar to the widely used BKS and TTAM potentials [37]

which utilize partial charges from quantum mechanical calculations and obtain other

parameters by fitting the structure and properties of relevant crystals [38, 39].

Short-range potentials acting between pairs of atoms include both repulsive (due to

electron cloud overlap) and attractive (due to Van der Waals or dispersion interaction)

terms [38]. Short‐range interactions of the potentials have the Buckingham form:

V(r)=Aexp(-r/ρ)-C/r6 (2.9)

where r is the distance between two atoms and A, C, and ρ are parameters. The

charges of O, Si, Li, La, Ti, were assigned to −1.2, 2.4, 0.6, 1.8, and 2.4, respectively.

This potential set has been successfully used to study silica and a number of silicate

glasses [39]. The potential parameters are listed in Table 2.1. It is know that the

original Buckingham potential has issues where the energy diverges to negative

infinity when r is small. In order to correct the original Buckingham potential, a

repulsion function V(r) was used to replace the original Buckingham potential. Here rc

is defined as r value to be between the first maximum and first minimum of the

Buckingham potential and where the third derivative of Buckingham potential is zero.

16

V(r) has the function form of

V(r)=B/rn+Dr2 (2.10)

where n, D, and B are fit to make the potential, force, and first derivative of the force

continuous from both functions at rc.

Table 2.1 Ionic charges and Buckingham potential parameters

Pairs A (eV) ρ (Å) C (eV·Å 6)

O-1.2-O-1.2 2029.2204 0.343645 192.58

Si2.4-O-1.2 13702.905 0.193817 54.681

Li0.6-O-1.2 41051.938 0.151160 0.0

La1.8-O-1.2 4369.393 0.278603 60.278

Ti2.4-O-1.2 23707.909 0.185580 14.513

2.4 Methodology of Structural Analysis and Property Calculations

2.4.1 Mean Square Displacement and Diffusion Coefficient

Mean square displacements (MSD) are calculated from NVE trajectories. After

initial steps of equilibrium, the remaining steps were recorded every 10 steps under

the microcanonical ensemble (NVE). The diffusion coefficient D can be calculated

17

from MSD according to Einstein diffusion equation:

D =1

6limt→∞

d

dt⟨|ri(0) − ri(t)|

2⟩ (2.11)

Mean square displacement (MSD) is defined as,

MSD = ⟨|ri(0) − ri(t)|2⟩ (2.12)

where ri is the position of particle i, ri(0) and ri(t) are the positions of the particle at time

0 and time t, respectively. To ensure statistical meaningful results, MSD calculations

are usually averaged over the same type of particles and over large number of origins.

In this work, we average over all the lithium ions and 400 origins during MSD

calculations. With such a large number of configuration recording, the diffusion

pathways of ions can be generated and visualized. By utilizing the visualization

method, the preferred diffusion directions of ions in crystalline LLT can be identified:

either the diffusion is preferred along the a-b plane or along the c-axis direction.

2.4.2 Neutron Structure Factor Calculations

In order to validate the simulated glass structures, the neutron structure factors

were calculated from the simulated glasses and compared with available

experimental data. The partial structure factor can be obtained by Fourier

transforming the pair distribution function gij(r) through

Sij(Q) = 1 + ρ0 ∫ 4πr2[gij(r) − 1]R

0

sin(Qr)

Qr

sin(πr R⁄ )

πr R⁄dr (2.13)

18

in which ρ0 is the average atom number density, Q is the scattering factor and R is the

maximum value of the integration in real space which is set to half of the size of one

side of the simulation cell. The sin(πr R⁄ )

πr R⁄ part is the Lorch type window function [40]

which reduces the effect of finite simulation cell size during the Fourier transformation.

The total neutron structure factor was calculated by

SN(Q) = (∑ cibini=1 )−2∑ cicjbibjSij(Q)

nij=1 (2.14)

where ci and cj are the fractions of atoms; bi and bj are neutron scattering lengths.

The neutron scattering used are 5.803, 4.1491, and -1.90 fm for oxygen, silicate, and

lithium respectively [41].

19

CHAPTER 3

DEFECT CONCENTRATION EFFECT ON LITHIUM ION DIFFUSION IN LITHIUM

LANTHANUM TITANATE SOLID STATE ELECTROLYTES

3.1 Abstract

Solid state electrolytes with high lithium ion conductivity are critical to the

development of next generation safer and more efficient lithium ion batteries.

Perovskite structured lithium lanthanum titanium oxide (LLT, La2/3-xLi3xTiO3) with

introduced lithium ion vacancies through lanthanum/lithium substitution, has been

shown to be a promising solid electrolyte. In this chapter, we have investigated the

effect of defects on the diffusion behaviors in LLT using molecular dynamics

simulations with the goal to obtain fundamental understanding of the diffusion

mechanism and the effect of crystallography orientation, a site atom size on the

diffusion with lithium vacancy concentration, and crystalline lithium solid state

electrolyte. Lithium ion diffusion energy barriers are obtained by dynamic and static

calculations using the nudge elastic band (NEB) method. The total ionic conductivity

is calculated by Nernst-Einstein equation and compare with the experimental data.

3.2 Introduction

Among the most promising glass and ceramics lithium ion solid electrolytes,

20

lithium lanthanum titanate (LLT) ceramics have attracted considerable attention since

its first report of bulk ionic conductivity of 1x10-3 S/cm at ambient temperature in the

early 1990s [5, 42]. Subsequent work has contributed to the understanding of the

conduction mechanism and the effect of partial or total substitution of La and Ti and

synthesis or sintering condition on the crystal structure and electrical conductivity. It is

generally believed that the high ionic conductivity of LLT is due to A-site vacancies

which are caused by La/Li substitutions. The defect reaction can be written as,

OTiLaLiLi

TiOLiLaOTiLaVLaOLa 3612642 ')(

3223

(3.1)

The compositions of LLT with lithium ion vacancies due to La/Li substitution are

usually represented as La2/3-xLi3xTiO3 with x ranging from 0 to 0.16. In

perovskite-structured LLT, lithium ions occupy A-site and each site is surrounded by

twelve oxygen ions. It is generally believed that lithium ions diffuse through the

vacancy mechanism by crossing a bottle neck structure formed by four oxygen ions to

an adjacent vacant site. A dilation of the lattice as measured by the positive activation

volume of 1.6-1.7 cm3/mol was observed when lithium ion jump to an adjacent

vacancy [24]. It was also suggested that TiO6 units have different level of tilting that

results in non-uniform distribution of bottleneck dimensions in the crystal structures

and consequently a distribution of diffusion energy barriers. It was found that

substituting La with smaller lanthanide elements led to cubic to orthorhombic lattice

21

change of LLT structures. The conductivity decreases and activation energy increase

as a result of this substitution [24]. The exact diffusion dimensionality (2D or 3D) of

lithium ions is still controversial [24]. It is proposed that at low temperature the

diffusion is 2D while at high temperature it becomes 3D [24]. Although considerable

understanding has been achieved in LLT, detailed mechanistic understanding of

lithium ion diffusion and the effect of composition and associated local structural

changes on diffusion is still lacking.

3.3 Methodology

3.3.1 Introduction

Molecular dynamics (MD) simulations have been widely used to investigate ion

migration in crystalline solids due to their ability to provide atomic level details of ionic

diffusion and to study the temperature and pressure effect on diffusion behaviors.

Lithium ion diffusion in LLT was studied by Katsumata et. al. Both fully ionic model

(FIM) and partially ionic model (PIM) pair potentials were used to study the diffusion

behaviors of lithium ions in LLT with x=0.67 [6, 7]. It was found that lithium ions diffuse

through the vacancy mechanism by crossing the “bottleneck” formed by oxygen ions

when they diffuse to adjacent A-site vacancies. The size of the bottleneck and their

relation to lithium ion diffusion coefficient and energy barrier were also studied. Pair

22

distribution function analysis was used to study the diffusion path and it was found

that the first peak intensity of Ti-Li pair distribution gTi-Li(r) increased when r approach

the bottlenecks. The result showed Li ions exist at various positions between A-site

and bottleneck so that the migration of Li ions in the cell was explained [6, 7]. The

diffusion coefficient of lithium ion was calculated to be 1.9-6.4x10-7 cm2/s at 500K with

different A-site ions arrangements and compared with the experimental data [6, 7].

3.3.2 Initiation of Crystal System

To ensure the accuracy of our potential, we I initially have our LLT perfect

structure relaxed. The lattice parameter and atom position are calculated from the

initial relaxed LLT perfect structure in Table 3.1. Our LLT structure shows good

agreement with experimental data [43]. The total volume is within 1.5% difference

compared to the experimental values. After the first relaxation, we randomly produce

the lithium ion vacancies and replace with lanthanum ions in LLT perovskite structure.

According to the defect equation, each replacing can form 2 sites of lithium vacancies

because of the charge balance of the system. In the perfect LiLa(TiO3)2 structure,

titanium ions occupy the B-site of the ABO3 pervorskite structure while Li and La ions

occupy the A-site, alternating layer by layer. After introducing the lithium vacancies,

the lithium layer and lanthanum layer become fully mixed up. Thus, the structure

23

becomes isotropic along (100), (010) and (001) directions (Figure 3.1). To ensure the

accuracy of diffusion studies, we used 10x10x10 super cells with over 8000 atoms in

each model. Nine configurations have been generated with nine different vacancy

concentrations of lithium ions, x= 0.157, 0.147, 0.137, 0.117, 0.097, 0.087, 0.067,

0.057, and 0.037. The isothermal and isobaric ensemble (constant number of atoms,

pressure, and temperature (NPT)) with a Hoover thermostat and barostat relaxation

times (ps) are used in the simulations. At each temperature, after NPT runs for

200,000 steps, a MD run with microcanonical ensemble (constant number of atoms,

volume, and energy (NVE)) is used for another 200,000 steps advance equilibrium

the system.

Table 3.1 Comparison of calculated and experimental structure of LLT

(Li0.3La0.567TiO3)[43]

Occupancy Exp. at 25 °C

(P4/nbm)

Simulation

(this work)

a (Å ) - 5.482 5.518

c (Å ) - 7.746 7.752

Volume - 232.78 236.09

alpha/beta/gamma (°) - 90.0/90.0/90.0 90.0/90.0/90.0

La1 (2c) 0.357 0.25/0.75/0.00 0.250/0.750/0.0032

La2 (2d) 0.778 0.25/0.75/0.50 0.250/0.50/0.4968

Ti (4g) 1 0.25/0.25/0.2452 0.250/0.250/0.2452

O1 (8m) 1 0.5180/0.4820/0.2640 0.4999/0.4999/0.2519

O2 (2a) 1 0.25/0.25/0. 0.250/0.250/0.0032

O2 (2b) 1 0.25/0.25/0.5 0.250/0.250/0.4968

Li (8m) 0.15 0.108/-0.108/0.036 0.000/0.000/0.0032

24

(a)

(b)

Figure 3.1: (a) The initial structure of lithium lanthanum titanate (LLT).

(b) The defect structure of LLT at 600K (Purple: lithium; Blue: lanthanum)

25

3.4 Bulk Crystal Structure and Effect of Temperature on Lattice Parameter

The perovskite-structured LLT have been reported to demonstrate volume

change after introducing lithium ion vacancies [24]. In perfect LLT, i.e. no lithium ion

vacancy is introduced; lithium and lanthanum ions occupy alternating A-site layers

(Figure 3.1(a)). Lithium vacancies were introduced by replacing lithium ions with

lanthanum ions and simultaneously creating two lithium ion vacancies (Figure 3.1(b)).

After initial random replacing lithium ions with lanthanum and introduction of lithium

ion vacancies, the vacancy sites were only distributed on the lithium layers. However,

after relaxation at 600K for 200 ps MD runs, lithium ion vacancies were also found to

exist in the lanthanum layers (Figure 3.1(b)). This would impact the diffusion

anisotropy of lithium ions, i.e. lithium ions can now also diffuse through direction

perpendicular to the alternating layers in additions to parallel to the layers. As the

ionic radius for lanthanum ion (1.032Å ) is considerably larger than lithium ion (0.76 Å )

[44], the substitution would lead to expansion of the lattice. It was indeed found that

the lattice parameters gradually increase with more lanthanum ion substitutions. This

trend is confirmed by studying the lattice parameter change as a function of

composition. The lattice parameter a, b, and c are found to decrease with almost

linearly with increasing x from 0.04 to 0.12 (decreasing the percentage of Li ions

vacancy) (Figure 3.2). This is in good agreement with the experimental observations

26

[24]. Further increase of x values beyond 0.12, the lattice parameters remain constant.

Experimental data also showed a decrease of slope of lattice parameter change for x

larger than 0.13 [24].

5.84

5.85

5.86

5.87

5.88

5.89

0.04 0.06 0.08 0.1 0.12 0.14 0.16

a,b

c

La

ttic

e P

ara

me

ter

(An

gstr

om

)

X value

Figure 3.2: The lattice parameter as function of x value in LLT vacancy

structure (La2/3-xLi3xTiO3)

3.5 Effect of Vacancy Concentration on Lithium Ion Diffusion

To understand the dynamic behavior and lithium ion transport in the crystal

systems, we performed mean square displacement (MSD) calculations based on the

trajectories from MD simulations. To obtain statistically meaningful results of MSD,

relatively large number atoms and averaging over large number of origins are needed

27

in the calculations. Figure 3.3(a) and 3.3(b) show the MSDs of LLT with x=0.067 at

different temperature. It is usually calculated from NVE trajectories after NPT

equilibrium.

In the logarithm of MSD (Figure 3.3(b)), it is classified into three regions. The

initial region is the ballistic region which MSD is proportional to t2. The following

region is the crossover region which between ballistic region and diffusion region. The

last region is called diffusion region, and the MSD is proportional to t. The linear range

of at long time was used for the calculation of diffusion coefficients. Figure 3.4 shows

the diffusion coefficients of different percentage of lithium ion vacancies at 600 K. In

order to increase the accuracy of our simulation, we generated five independent

structures each with random La/Li substitution and vacancy distributions for each

composition. The highest diffusion coefficient is found to be at x=0.067 which gives

1.59x10-5 cm2/s. This result has good agreement with the experimental data [5, 24, 42,

45]. In addition, according to Arrhenius equation

D = D0exp (−Ea

RT) (3.2)

where Ea is diffusion energy barrier, T is temperature and R is gas constant. If we take

the logarithm respectively,

logD = logD0 −Ea

R

1

T (3.3)

where D0 is pre-exponential factor. The slope of logD over 1/T gives activation energy

28

barrier of diffusion. Figure 3.5 is the energy barrier as a function of composition (x

value) obtained for temperature range of 400 to 800 K. The linear fit of Arrhenius

equation has high quality with R2 values higher than 0.98. A higher temperature range

as compared to the usual room temperature that the electrolyte is usually used was

adopted because higher temperature facilitates diffusion and improves the statistics

of diffusion coefficients. The trend of diffusion energy barriers look shows a minimum

at around 40% lithium ion vacancy (x= 0.067) with the energy barrier being around

0.216 eV. This is in good agreement with recent first principles DFT NEB calculations

that found the energy barrier being 0.23 eV [46]. Experimentally, it was found that

activation energy barrier for lithium ion conduction is 0.4 eV for room temperature and

0.15 eV for high temperature (100-400 °C) for composition Li0.34La0.5lTiO2.94 [5-8, 29].

The simulated diffusion energy barrier fell well in the range of the two values.

29

0

5

10

15

20

0 10 20 30 40 50

800K

686K

600K

480K

400KM

ea

n s

qu

are

dis

pla

cem

en

t (A

2)

Time (ps)

(a)

0.01

0.1

1

10

0.01 0.1 1 10

800K

686K

600K

480K

400K

Mea

n s

qu

are

dis

pla

cem

ent

(A2)

Time (ps)

(b)

Figure 3.3: (a) Mean square displacement of lithium lanthanum titanate (LLT)

(b) MSD in logarithm.

30

0.8

1

1.2

1.4

1.6

1.8

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

X value

Diffu

sio

n c

oe

ffic

ient(

10

-5cm

2/s

)

Figure 3.4: Lithium ion diffusion coefficients as a function of x value in

La2/3-xLi3xTiO3.

0.2

0.25

0.3

0.35

0.4

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

X Value

Activa

tion E

nerg

y(e

V)

Figure 3.5: Lithium ion diffusion energy barrier as a function of x value in

La2/3-xLi3xTiO3.

31

3.6 Diffusion Energy Barrier—Static Calculations

In order to calculate lithium ions diffusion energy barrier in LLT, we also utilized

nudged elastic band (NEB) [47] method for the static calculation. The minimum

energy pathway for diffusion and the energy barrier can be efficiently found by NEB

method [47]. In this work, we try to move a lithium ion from A-site through the

bottleneck to adjacent A-site vacancy. The total distance is 4 Å . We firstly generated

the initial and final structure, and constructed a set of images (replicas) between them.

Total of 100 images were used to obtain accurate energy path. The lattice energies of

all the images were relaxed simultaneously with a spring acting along the reaction

pathway to avoid them collapsing to each other. Figure 3.6 shows the minimum

energy path of lithium ion diffusion in LLT obtained from NEB calculations. The saddle

point is located at 1.64 Å with energy barrier of 0.19 eV, which corresponds to lithium

ion being in the bottleneck structure (inset of Figure 3.6). A shoulder of the minimum

energy path was observed on the longer distance side. This was found to be related

to the final lowest energy state not being located at the center of the cell. As lithium

ions are small relative to the vacancy site, it might take a position away from the cell

center. The energy barrier from static NEB calculations is in good agreement with the

dynamic calculation (0.22 eV) reported in chapter 3.2. NEB calculations based on first

principles DFT found the energy barrier to be 0.23 eV [46]. Because of the mixing of

32

lithium and lanthanum ions after lithium vacancy introduction, the local environments

of lithium ions are different. The slightly energy difference (0.036 eV) between initial

and final structure confirms this observation and was found to be caused by the local

environments around lithium ion and the vacancy especially the number of lanthanum

ions in the next nearest neighbors.

-4039.75

-4039.7

-4039.65

-4039.6

-4039.55

-4039.5

0 0.5 1 1.5 2 2.5 3 3.5 4

Latt

ice

En

erg

y (

eV

)

Distance (Angstrom)

Figure 3.6: Static energy barrier from NEB calculations. It shows the energy

barrier and the associated structure of of A-site lithium ions diffuse in LLT.

(Purple ball: Li, Red ball: O, Grey ball: Ti)

33

3.7 Lithium Ion Diffusion Mechanism

As we mentioned above, each lanthanum substitution can form 2 lithium ions

vacancies. Lithium ions are able to diffuse within these vacancy sites. Figure 3.7

shows the diffusion pathway of lithium ions. We can see that lithium ions are likely to

migrate in A-site through the bottleneck structure (Figure 3.8) to adjacent A-site. The

bottleneck structure is surrounded by 4 oxygen atoms. In our simulation work, we

found out that when lithium ions try to migrate though the bottleneck, the bottleneck

will become broadened, and the time spend of lithium ions within the bottleneck is

less than the time spend in A-site. Figure 3.9 shows the change of bottleneck

structure based on the difference of lithium position. The bottleneck size increased

through A to C and decreased through C to E. In addition, the lithium ion only takes

5ps to migrate through the bottleneck (B to D). It can clearly explain why the diffusion

path way is rich in A-site. Inaguma et all also point out that for the ideal perovskite

structure, the size of bottleneck is smaller than a lithium ion diameter. Thus, the

dilation of the bottleneck must occur, when lithium ions try to jump to the adjacent

A-site. They also mentioned that the activation volume of bottleneck has positive

value (the available for the migration of lithium ion = the initial volume at the

bottleneck volume + the activation volume). In other words, the positive activation

volume means that the dilation of bottleneck took place [45, 48-50].

34

Figure 3.7: Trajectories of lithium ions for composition Li0.2La0.6TiO3. MD

simulation is at 600 K for 160 ps.

Figure 3.8: The bottleneck structure of lithium lanthanum titanate (LLT)

(Grey ball: Ti; Red ball: oxygen; Blue ball: lithium; and the bottleneck

structure is schematically shown in yellow bonds.)

35

Step A Step B

Li ion tries to migrate through bottleneck. Li ion is close to bottleneck.

Step C Step D

Li ion is within the bottleneck. Li ion migrates through bottleneck

Step E

Li ion is away from the bottleneck and ready to migrate to another vacancy site

Figure 3.9: The steps of Li ion which migrates through the bottleneck. The

left picture of each step exhibits the size of bottleneck structure and the right

one shows a view perpendicular to the diffusion pathway. The black

numbers (Å ) are the distance between oxygen ions of bottleneck structure.

It is obvious that the largest bottleneck size is at step C. (Red: oxygen.

Purple: Lithium. Grey: Titanium)

36

3.8 Total Ionic Conductivity Calculation

The total ionic conductivity of a solid can be expressed as

𝜎 = ∑ 𝑛𝑖𝑍𝑖𝜇𝑖𝑖 (3.4)

where ni is the charge carrier concentration, Zi is charge of lithium ions(+1), and µ i is

mobility of ionic charge carrier i. The main factors that affect ionic conductivity thus

include the charge carrier concentration and the mobility, which can be correlated to

the self-diffusion coefficient (D) through the Einstein’s equation. Kawai et al.

suggested that the mobility of lithium ions remained constant at ambient temperature

[24] due to the observation that the conduction energy barriers remain constant of

around 0.35 eV in the range in wide composition ranges [51]. The ionic conductivity

was thus dominated by charge carrier concentrations, which involved both the lithium

ion and vacancy concentrations. The estimation of charge carrier concentration

showed a dome shape behavior as a function of x value in La2/3-xLi3xTiO3 (Figure 3.10)

with the maximum at x= 0.067. The available lithium ion concentration in

La2/3-xLi3xTiO3 can be expressed as NLi: 3x/Vs, in which Vs is the unit cell volume, and

lithium ion vacancy concentration expressed a (0.33-2x)/Vs. Assuming the total A site

concentration N=Nv+NLi to be identical in terms of symmetry and energies, the total

charge carrier concentration can be expressed as [51].

𝑛𝐿𝑖 =𝑁𝐿𝑖𝑁𝑉

𝑁=

(𝑥−6𝑥2)

(0.33+𝑥)𝑉𝑠 (3.5)

37

Based on the charge carrier concentration estimation above, a dome-shaped

conductivity as a function to x was obtained with the maximum at x=0.067 [24]. This

charge carrier concentration was combined with diffusion coefficient to calculate the

lithium ion conductivity through the linkage of Nernst-Einstein equation:

𝜎 =𝑧2𝑒2𝑛𝐷

𝑘𝑇 (3.6)

where z is the charge value of charge carrier, which is +1 in the case of lithium ion, e

is electron charge, n is charge carrier concentration and D is diffusion coefficient. The

calculated electrical conductivity at 600 K is shown in Figure 3.11. The maximum ionic

conductivity was obtained to be 0.043 S/cm at x= 0.067. Experimental conductivity

measurements also showed a similar shape although the exact maximum position

was slightly different: x equals 0.11 vs 0.067. Lithium ion diffusion coefficients also

showed a dome shape behavior as a function of x value in LLT (Figure 3.4) from our

simulations. The maximum of diffusion coefficient happens at x equals 0.067 (around

40% lithium ion vacancy). It is interesting to point out that the activation energy barrier

of lithium ions from simulations was found not to be constant but instead ranged from

0.22 to 0.35 eV and showed an extreme (minimum) at around x equals 0.067 (Figure

3.5). The coincidence of the highest lithium ion diffusion coefficient and lowest

activation energy barrier at a composition with x= 0.067 (around 40% lithium ion

vacancy) from simulations is in good agreement with the maximum behavior of

38

experimental conductivity data.

1

2

3

4

5

6

7

8

9

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Cha

rge

Carr

ier

Co

nce

ntr

ation

(1

02

0cm

-3)

X Value

Figure 3.10: Charge carrier concentration as a function of x value in

La2/3-xLi3xTiO3

0

0.01

0.02

0.03

0.04

0.05

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Ion

ic C

on

du

ctivity (

S/c

m)

X value

Figure 3.11: Ionic conductivity as a function of x value in La2/3-xLi3xTiO3

39

CHAPTER 4

STRUCTURE AND LITHIUM ION DIFFUSION IN LITHIUM SILICATE GLASSES

AND AT THEIR INTERFACES WITH LITHIUM LANTHANUM TITANATE CRYSTALS

4.1 Abstract

Solid state lithium ion electrolytes are important to the development of next

generation safer and higher power density lithium ion batteries. Lithium lanthanum

titanate ceramics is a promising solid state electrolyte with high lithium ion

conductivity. In this chapter, we present investigations of the structure and diffusion

behavior of lithium silicate glasses and their interfaces with crystalline lithium

lanthanum titanates using molecular dynamics simulations. The atomic structure at

the ceramic/glass interface will be examined. Lithium ion diffusion behavior in the

glass and across the interface will investigated and correlated to the electrical

conductivities of these materials.

4.2 Introduction

Two main issues remain for LLT as a solid electrolyte for battery applications.

The first one is the reduction of lithium conductivity by 1-2 orders of magnitude in

sintered ceramics as compared to the bulk conductivity. This was explained by

diffusion barriers caused by grain boundaries. The second one is the reduction of Ti

40

from Ti4+ to Ti3+, and associated increase of electronic conductivity, at the

electrolyte/anode interface. Recent studies showed that the total conductivity can be

improved by introducing highly conductive lithium silicate glassy grain boundaries [52,

53] or intergranular thin films. To address the second issue, a separation layer has

been used to separate the LLT electrolyte and the electrolyte to alleviate the reduction

of titanium ions. LLT ceramics have recently been investigated as coatings to

electrode materials to enhance both ionic and electronic conductivity. Meng et al have

observed that Li ion diffusion is higher in coating samples than in the uncoated

samples [46]. Also, the impedances of Li ions transportation in the

solid-electrolyte-interphase (SEI) layer and interfacial charge transfer and are

reduced up to 50% in the coated samples [46]. Despite these known limitations of LLT

as a solid electrolyte in lithium ion batteries, it remains a promising solid electrolyte

material and, more importantly, serves as a unique model system to investigate

fundamental diffusion mechanism and structure-mobility relationships, which can

pave way to the development of future generation solid state electrolytes.

Nan and coworkers have recently discovered experimentally that by using

lithium silicate glasses as the intergranular thin films, the ionic conductivity of LLT

ceramic system can be greatly improved [52]. Using the lithium silicate glass as the

grain boundary phase was found to enhance the conductivity of polycrystalline

41

materials. It has been proposed that the homogeneous glass intergranular phase can

decrease the anisotropic effect of lithium ion diffusion thus improve lithium ion

conductivity in these solid state electrolytes [53]. However, detailed understanding of

the diffusion mechanism across the glasses and across the glass/crystal interface is

still poorly understood. One main obstacle is the lack of understanding of the complex

structures of the glasses and especially at the interfaces.

4.3 Methodology

4.3.1 Introduction

Molecular dynamics (MD) simulations have been widely used to study the

structure and diffusion behaviors in lithium and other alkali containing glasses [38, 54,

55]. Cormack et al. have investigated the migration of sodium silicate glass by

molecular dynamics simulation, and observed a few sequence jumps between

selected sites [38]. Habasaki et al. have investigated the mechanism of the ion

conduction in glass by MD simulation. The diffusion coefficient conductivity tends to

increase logarithmically with increase of alkali contents [56]. Pedone et al. and Du et

al. have obtained the bond length of Li–O which is from 1.95 Å to 1.98 Å with

increasing the Li2O mole percentage in lithium silicate oxide [41, 57]. The

coordination numbers are also increased from 3.5 to 3.9 and approach 4 for disilicate

42

glass [41, 57]. The activation energy for lithium silicate glass of previous studies is

from 0.75 eV to 0.85 eV [58-60]. Lammert et al. studied the sequence of a lithium ion

which left one cluster and moved into a different one and this step is recorded as a

jump [61]. The trajectories of lithium silicate diffusion pathway were also studied in

several previous simulation works to understand the diffusion mechanisms in the

amorphous matrix [55, 57, 58, 61, 62].

MD simulations have also been utilized to study the interface of amorphous

and crystalline materials. Rushton et al. have studied the interface of sodium, lithium

alkali-barosilicate glass in contact with MgO, CaO, and SrO crystals, respectively [63].

The interfaces were formed between the stable (100) and (110) surfaces of the

rocksalt crystals. The number of alkali species (Na and Li) within the interface was

investigated and they concluded that the change of alkali content at the interface

depends on the crystal phase and crystallographic orientation with respect to the

glass [63]. In addition, Garofalini and Shadwell studied the behavior of lithium silicate

glass/V2O5 crystal interface which is similar to our system [60]. They created different

surface terminations (vanadium and oxygen) of (001) and (010) surfaces of V2O5

crystals and used them to build interface models with lithium silicate glasses. The

(010) surface was found to form better interface [60]. The reason is that the energy

barrier of lithium ion diffusion along <010> direction is similar to those of the glasses,

43

but the energy barrier of lithium ion diffusion along <001> direction in the crystals was

very different from the glass, which resulted in a pile up of lithium ions at the interface.

The glass/crystal interface formation thus created a barrier of lithium ion diffusion.

Lithium build-up was found at the (001)-oriented interface but not at (010)-oriented

interface [60].

4.3.2 Initiation of Glass System

Wide composition range in the Li2O-SiO2 glass system has been studied to

provide systematic study of structure and property variations. Experimentally it was

found that phase separation exists in certain compositions in the glass formation

range of binary Li2O-SiO2 glasses [64]. In our simulations, however, we only consider

homogeneous glasses. The glass compositions simulated are xLi2O-(1-x)SiO2 with

x=0.1, 0.2, 0.3, 0.33, 0.4, and 0.46. These glasses are named LS10, LS20, LS30,

LS33, LS40 and LS46, respectively. The total atoms in the cubic simulation cell are

3000, and the lattice parameter is 33.8 Å . The detailed glass composition parameters

are listed in Table 4.1. The isothermal and isobaric ensemble (constant number of

atoms, pressure, and temperature (NPT)) with a Hoover thermostat and barostat

relaxation times (ps) were used in the simulations. At each temperature, after NPT

runs for 60,000 steps, a MD run with microcanonical ensemble (constant number of

44

atoms, volume, and energy (NVE)) is used for another 60,000 steps to advance the

equilibrium of the system. The initial structure was generated by randomly put atoms,

with proper composition and density, in cubic simulation boxes, with initial constraints

of shortest interatomic distance to avoid atoms being too close to each other. The

glass structures are generated by melting and quenching process. After initial

relaxation at 0 K, the systems are heated up through 300 K, 1000 K, and 3000 K to

4000 K to melt the glass. The systems are gradually cooled down to 300 K through

steps of 3500 K, 3000 K, 2500 K, 2000 K, 1500 K, 1000 K, and 300 K with a nominal

cooling rate of 0.5 K/ps. Structure analyses of the glasses were averaged based on

the trajectories recorded every 50 steps under NVE runs at 300 K. Figure 4.1 shows

the glass structure.

Table 4.1 Glass composition parameters for MD simulation

Percentage (mol%)

Density

(g/cc)*

Atom number

Li2O SiO2 O Si Li

LS10 10 90 2.235 1900 900 200

LS20 20 80 2.283 1800 800 400

LS30 30 70 2.330 1700 700 600

LS33 33.3 66.7 2.345 1667 667 666

LS40 40 60 2.346 1600 600 800

LS46 46 54 2.343 1540 540 920

* Density data from ref. [65]

45

Figure 4.1: The structure of the lithium silicate oxide Li2O-2SiO2. Golden

pyramids: silicon oxygen tetrahedrons, red ball: oxygen, blue ball: lithium

ions.

4.3.3 Build the Glass/Crystal Interface Structure

As we mentioned above, the LLT defect structure is isotropic along (100), (010),

and (001) after the lithium vacancies are introduced. We chose the (001) surface of

the La2/3-xLi3xTiO3 structure to build the interface with lithium silicate glasses. The

crystal/glass interface was generated by first generating and relaxing the (001) crystal

surface using NPT ensemble. Subsequently, the glass phase was generated by

perfectly matching the lateral dimension of the crystal surface while maintaining the

glass density and total cell volume. After the glass and crystal are generated, the two

were put together with a vacuum gap of 3–4 Å . The size of the simulation cell with the

46

interface is 23×23×99Å . The whole system was relaxed under constant pressure at

1500K to give sufficient thermal energy for interface relaxation, while avoiding melting

of the interface, and then gradually cooled down to 300K. Similar procedures were

used to generate the interfaces of titanium oxides [66]. The simulation was performed

under constant pressure (NPT) ensemble. At 300K, the final 40,000 steps during NVE

run, configurations were recorded every 50 steps, and the structural analyses were

averaged over these last 801 configurations. Figure 4.2 shows the snapshot of atomic

structure of glass/crystal interface.

47

(a)

(b)

Figure 4.2: Atomic structure of the glass/crystal interface (a) and zoom in

view of the interface (b). Yellow ball: silicon, red ball: oxygen, green ball:

titanium, light blue ball: lanthanum; purple ball: lithium.

48

4.4 The Structure of Lithium Silicate Glasses

Figure 4.3 shows the comparison of the neutron structure factors which are

calculated from simulated structures and experimental data [67]. The calculated

structure factor from MD simulations is generally in good agreement with the

experimental data. There are some noticeable differences: the intensity is slightly

higher and the valley is slightly deeper in the structure factor from simulations than

those from experiment. The good agreement of the structure factors indicates that the

potential models used can well reproduce the structure of the lithium silicate glasses.

The Li–O pair distribution functions as a function of Li2O concentration in lithium

silicate glasses are shown in Figure 4.4. The Li–O bond length increases from 1.94 Å

to 1.97 Å as Li2O concentration increases from 10 to 46 mol% which is the same as in

previous studies [41, 57]. The peak intensity also increases with increasing lithium

oxide concentrations, suggesting an increase of coordination number.

49

-0.5

0

0.5

1

1.5

2

0 5 10 15 20

SN (

Q)

Q (Angstrom-1

)

Figure 4.3: Comparison of calculated and experimental neutron structure

factor function of lithium disilicate glass. Solid line: MD simulations; circles:

experiment data [67].

0

1

2

3

4

5

6

1.5 2 2.5 3 3.5 4 4.5 5

Li2O 10 mol%

Li2O 20 mol%

Li2O 30 mol%

Li2O 40 mol%

Li2O 46 mol%

g (

r)

r (Angstrom)

Figure 4.4: Li-O pair distribution function as a function of Li2O concentration

in lithium silicate glasses. Arrow points to increase of Li2O concentration.

50

The average coordination number of lithium ions indeed increases from 3.4 to

3.8, as Li2O concentration increases from 10 to 46 mol%. This is shown in Figure 4.5.

The lithium ion coordination numbers can be partitioned into 3 contributions: bridging

oxygen (BO), non-bridging oxygen (NBO) and free oxygen (FO), which were

classified based on the number of silicon around each oxygen being two, one or zero,

respectively. It can be seen in figure 4.5 that with increasing lithium oxide

concentration the NBO contribution gradually increases. The FO contribution is very

small and remains almost constant with Li2O concentration. Figure 4.6 shows the

distribution of lithium ion coordination numbers (calculated using a cutoff obtained

from the first minimum of Li–O pair distribution functions (around 2.58 Å ). Lithium ions

have coordination numbers ranging from 2 to 6 with majority of them having 3, 4, and

5 coordination. For the 30 Li2O-70SiO2 composition (shown in Figure 4.6), lithium ion

coordination number is around 3.6. Lithium coordination numbers found in this work

are in good agreement with earlier MD simulations [41].

Qn (meaning silicon oxygen tetrahedron with n BO) distribution is a measure

of the medium range structure of silicate glasses. Very importantly, Qn distribution

can be measured from solid state NMR studies or Raman spectroscopy [68].

Comparing the Qn distribution from simulation with those from experiments is another

important validation of the simulated structure models. The silicate glass distributions

51

of our work and experimental results from NMR studies [68] are compared in figure

4.7. With increasing lithium oxide concentration, Q1 and Q2 increase, and Q4

decreases, monotonically. The percentage of Q3 however, shows a maximum at

around the disilicate concentration. This is in excellent agreement with experimental

data obtained by Maekawa et al. from NMR studies of lithium silicate glasses that are

also shown in Figure 4.7. Similar maximum was observed in simulations of lithium

disilicate glasses using a different set of potential models [55] and the simulations of

sodium silicate glasses [39].

0

1

2

3

4

0 10 20 30 40 50

Li Cood. #BO#NBO#FO#

Li C

oo

rd.

Nu

mb

er

Li2O (mol %)

Figure 4.5: Change of lithium ion coordination number as a function of Li2O

concentration.

52

0

10

20

30

40

50

60

2 3 4 5 6

Pe

rce

nta

ge

Li Coord. Number

30Li2O-70SiO

2

Average Li Coord. 3.6

Figure 4.6: Distribution of lithium ion coordination number of 30 Li2O-70SiO2

compositions.

Figure 4.7: Qn distribution as a function of Li2O concentration (Experimental

data from Ref [68]).

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50

Perc

enta

ge

Li2O(mol%)

Q1

Q2

Q3

Q4

Q2 exp

Q3 exp

Q4 exp

53

4.5 The Structure of Lithium Silicate Glass/LLT Crystal Interface

The structure of simulated lithium lanthanum titanate crystal with lithium ion

vacancies after NPT MD simulations was compared and found to be in good

agreement with experimental data. The cell parameters of LLT with 60% lithium ion

vacancy have an average cell parameter of 3.896 Å and 7.828 Å for a and c,

respectively, in the tetragonal unit cells of LLT. This compares well with experimental

cell parameters 3.872 Å and 7.785 Å of LLT with similar lithium vacancy concentration

[69]. This suggests that the partial charge potentials used in this work give good

description of the defected structure of the lithium lanthanum titanate system.

The Z-density profile analysis was used to determine the distribution of atoms

along z-direction. Fig. 4.8 shows the atom density along the z-direction. We can see

that the interface is located approximately between 40 Å and 50 Å (relative distance

along the Z-direction). Lithium lanthanum titanate crystal face is below around 40 Å

as it is shown that there is no silicon detected in this range. On the other hand, no

density of lanthanum or titanium is found above around 50 Å where the lithium silicate

glass is. The Z-density profile also shows a local maximum of the lithium ion density

near the interface. This suggests that there is a certain level of lithium ion segregation

at the glass/crystal interface. This segregation can be related to the high mobility of

lithium ions and relatively large number of defected sites and free volume at the

54

interface.

0

20

40

60

80

100

20 30 40 50 60

SiLiLaOTi

Ato

m n

um

ber

density (

num

be

r/nm

3)

Z-distance (Angstrom)

Figure 4.8: Z-Density profile across the interface.

4.6 Diffusion Coefficients in Lithium Silicate Glasses and At The Boundary

Mean square displacement is also utilized in glass and interface system. It is

calculated from the NVE trajectories after NPT equilibrium which we mentioned

above. Figure 4.9(a) and 4.9(b) present the MSD of lithium silicate glass (LS33). We

calculate diffusion coefficients by utilizing the linear range of long time. The diffusion

coefficients of lithium ions in glass-crystal interface and different compositions of

glasses for different temperatures (800 K-3500 K) are shown in Figure 4.10. We can

obviously see that there is a change of slope for two different temperature ranges:

800 K to 2000 K and 2500 K to 3500 K. The higher temperature range has a steeper

slope, suggesting a higher diffusion energy barrier. Temperature ranges used in MSD

55

calculations in the literature varied greatly: Pedone et al. used 1000-2600 K

temperature range in the calculations of sodium diffusion in sodium silicate glasses

[70]. Kob et al, on the other hand, observed non-linear behaviors of diffusion

coefficient versus 1/T for the diffusion in silica and alumina silicate glasses. These

were explained by the mode coupling theory [71]. In this work, we clearly see a two

linear range behavior of lithium ion diffusions. The linear trend is generally good for

both temperature ranges. However, the quality of fitting is slightly better for glasses

with higher Li2O concentrations, for example the R2 values for linear fitting are 0.996

for LS40 and 0.992 for LS10 for the high temperature range, while the R2 values are

0.999 for LS40 and 0.982 for LS10 for the low temperature. By using the equations

(3.1) and (3.2), the diffusion energy barrier can be obtained. Energy barriers of the

glasses, the crystal phase (with 40% of lithium ion vacancies), and that of the

interface are shown in Figure 4.11. The energy barrier of the lithium silicate glasses

decreases with increasing Li2O concentration from 0.39 eV to 0.32 eV at 800 K to

2000 K and from 0.77 eV to 0.74 eV at 2000 K to 3500 K. The value of lithium ion

diffusion energy barriers obtained from the high temperature range are in agreement

with experimental data and previous MD simulation (ranging from 0.75 eV to 0.85 eV)

[58-60]. The energy barrier for the interface (lithium disilicate glass with lithium

lanthanum titatnate crystal) is 0.32 eV. For the defected crystal phase, the barrier is

56

lower with a value of around 0.22 eV at 800 K to 2000 K, which is close to the

experimental value 0.33 eV obtained from lithium ionic conductivity measurements

[72].

(a)

(b)

Figure 4.9: (a) Linear (a) and logarithm (b) mean square displacement of

lithium ions in lithium disilicate (LS33) glass.

0

100

200

300

400

500

600

700

800

0 50 100

MSD

(nm

2)

time(ps)

2000

1450

1140

940

800

0.01

0.1

1

10

100

1000

0.01 0.1 1 10 100

MSD

(nm

2)

time(ps)

2000

1450

1140

940

800

57

Figure 4.10: Diffusion coefficients of lithium ions in glass–crystal interface

and different composition of glasses for different temperatures (The unit D is

cm2/s).

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50

Glass 800-2000

Glass 2000-3500

Interface

Crystal

Ac

tiva

tio

n E

nerg

y(e

V)

Li2O(mol%)

Figure 4.11: Diffusion energy barrier for lithium disilicate glass, LLT crystal,

and the glass-crystal interfacial structure.

58

4.7 Lithium Ion Diffusion Behavior in Interface System

Liquid electrolytes consisting lithium salt in an organic solvent are widely used

in current lithium ion batteries. In addition to the safety concern we mentioned above,

liquid electrolytes are also prone to decomposition at the anode during the charge

process. If proper organic solvents are used, the decomposition can be controlled on

the initial charge process [73]. With the usage of solid electrolytes, both the safety

and decomposition issues can be avoided. In addition, the solid state electrolyte

provides higher thermal and mechanical stability as compared to liquid electrolytes

hence are better for future generation lithium ion batteries, especially for

transportation and energy storage applications. Development of crystalline/glass

hybrid structure can be a very promising approach to obtain high ionic conductivity

solid electrolytes.

From the structural point of view, Nan’s group reported that in LLT structure,

lanthanum ions are layered by La3+-rich and La3+-deficit layer. The lithium ions can

only migrate two-dimensionally within the La3+-deficit layer. They introduced lithium

silicate into the LLT grain boundary to remove the anisotropy of the grain. Therefore,

the migration of lithium ions becomes three-dimensional, and the inserted lithium ions

can provide lithium ions in various sites for conduction. The potential barrier for

lithium ions across the grain boundary can be reduced [52]. However, as we

59

mentioned some results above, the lanthanum layer (La3+-rich) and lithium layer

(La3+-deficit) are fully mixed up when heat is applied (Figure 3.1). The system

becomes isotropy at all directions. For this reason, we can say that lithium ions can

migrate three-dimensionally in a normal polycrystalline LLT structure. We also

investigated the diffusion behavior of lithium ions in amorphous lithium silicate glass

and crystalline LLT interface. We found that the diffusion energy barrier at the

interface is dominated by the glass phase. Higher lithium oxide concentration is

preferred for lithium silicate glass in order to lower the barrier. In addition, higher

lithium oxide concentration means higher lithium ion density in glass which can

improve the chance of lithium ions to migrate across the interface.

Lithium ion diffusion energy barriers show differences among the glass, the

crystal and at the interface. The barrier in the glass decreases slightly with increasing

lithium oxide content but is higher than that in the lithium lanthanum titanate crystal

with introduced vacancy defects. The barrier at the interface is obviously dominated

by the glass phase, with a value close to the disilicate glass composition. This means

that the intergranular thin films play a critical role in determining the total ionic

conductivity of the polycrystalline system. In order to improve the total ionic

conductivity, lithium silicate glasses with high lithium oxide concentration is preferred

since the barrier decreases with increasing lithium oxide concentration. In addition,

60

higher lithium oxide concentration in the glass also means higher density of lithium

ions at the interface that can increase the preexponentional factor (higher number of

available sites and higher frequency of jumping) for lithium ion diffusion. The

observed segregation of lithium ions at the interface can also help improve the

chance of lithium ion diffusion across the interface. Experimental (such as high

resolution TEM) investigations and determination of the lithium ion concentration at

the crystal/glass interface would be useful to validate the observed structures of the

interface from simulations.

61

CHAPTER 5

DIFFUSION ANISOTROPY AND CATION RADIUS EFFECT IN LITHIUM

LANTHANUM TITANATE

5.1 Abstract

Lithium ion self-diffusion under electrical field is studied and diffusion energy

barriers and diffusion heterogeneity in different crystallographic directions are

investigated in this chapter. It is found that lithium ion diffusion shows 3D behavior

because of the mixture of lithium and lanthanum layer when heat is applied. The size

effects of the rare earth ions on the diffusion behaviors have also been studied. The

free volume of lithium lanthanum titanium oxide (LLT), lithium gadolinium titanium

oxide (LGT), and lithium ytterbium titanium oxide (LYT) are calculated, and the

diffusion energy barriers were compared. It is found that the size of bottleneck

structure that lithium diffuse through plays an important role in determining the

diffusion energy barriers, with the larger rare earth cations on the A site of the

perovskite structure favoring higher lithium ion diffusion and lower the diffusion

energy barriers.

5.2 Introduction

Molecular dynamic (MD) simulations with applying external electrical field have

62

also been studied in both amorphous and crystalline systems. Heuer et al. pointed out

that the current density and lithium ion diffusivity of lithium silicate glass are increased

with increasing the strength of electrical field. The relation between current density

and the field strength is close to linear for fields around E=5x107 V/m, but shows

non-linear behavior above E=5x107 V/m [74]. Soolo et al. have studied the diffusion

coefficient of lithium ions in Li+-Nafion with electrical field. The diffusion coefficient

rises with increasing the field strength, and the tendency is even more pronounced at

higher field strength [75]. The conductivity of poly(ethylene oxide)10:LiClO4 with

adjustment electrical field was also studied by Wang et al.. They have found that the

conductivity of the poly(ethylene oxide)10:LiClO4 electrolyte is sensitive to the

adjustment of electrical field and temperature loop. Furthermore, it can also be

enhanced after a compound treatment of both a primary electrical field and

heating-cooling loop, due to the formation of more ordered crystalline structures [76].

5.3 Methodology

In order to study the diffusion anisotropy and to answer whether the diffusion is

2D or 3D behavior, electrical field was applied in the simulation cell and along certain

directions to observe the diffusion behavior. The homogenous electrical field was

applied in our 40% lithium ion vacancy system. We utilize the same condition which

63

we set in chapter 2. Four forces of electrical field were generated separately along

x-direction and z-direction, 13, 22, 30, 43MV/m. The MD runs with microcanonical

ensemble (NVE) for 200,000 steps in each forces of external electrical field. MSD are

also calculated from NVE trajectories with configuration records every 10 steps in

remain 160,000 steps. Diffusion coefficients and energy barrier can also be

calculated by Einstein equation (2.11) and Arrhenius equation (3.2) (3.3).

5.4 Diffusion Anisotropy of Lithium Ions in LLT

Figure 5.1 shows the diffusion coefficient as the function of force field along

x-axis and z-axis. The diffusion coefficient increases with increasing electrical field

strength at 600K. The result shows good agreement with the previous studies [74-76].

Moreover, we can see that the diffusion tendency of x-axis and z-axis are similar.

Firstly, we expect that the diffusion along z-axis is harder than x-axis. According to the

LLT structure (Figure 1.1), lanthanum ions and lithium ions are separated into layers

along z-axis. For this reason, lanthanum layers might be the obstacle for the diffusion

of lithium ions. The lithium ions should obtain enough energy or other external forces

in order to diffuse across the lanthanum layers. However, the trajectory of our

simulation without electrical field (Figure 3.7) shows that the diffusion along the z-axis

is obvious. Figure 5.2 also shows the trajectory of lithium ions with external electrical

64

field. The vertical direction is z-direction, and horizontal direction is x-direction.

0.5

1

1.5

2

2.5

3

10 15 20 25 30 35 40 45

X-axis

Z-axis

Diffu

sio

n C

oe

ficie

nt (1

0-5

cm

2/s

)

Electrical Field (MV/m)

Figure 5.1: Lithium ion diffusion coefficient under different external electrical

field along X-axis and Z-axis for composition Li0.2La0.6TiO3 (MD simulations

at 600 K).

Figure 5.2: Trajectories of lithium ions with 43MV/m external electrical field

for composition Li0.2La0.6TiO3. MD simulations at 600 K for 160 ps with

electrical field applied along z-axis.

65

Compare Figure 3.7 and Figure 5.2, even though we applied the electrical field

on z-direction, the diffusion behavior along z-axis was not obviously increased or

decreased. In addition, no matter we apply the external electrical field along z-axis or

x-axis, the diffusion behaviors tend to be the same (Figure 5.1). In conclusion, we can

say that lithium ions diffusion tendency is similar along x-direction and z-direction. As

we mentioned above, when we increase the temperature during the diffusion

simulation process, the lanthanum layers and lithium layers will randomly mix (Figure

3.1). Moreover, when introducing the vacancies in LLT, we replace lithium ions with

lanthanum ions by random substitution, so that lithium layers will have some

lanthanum ions because of substitution, and then (001), (100), and (010) become

isotropic. Therefore, it can explain why the diffusion behaviors along x-axis and z-axis

are similar.

5.5 Effect of A Site Cation Substitution on Lithium Ion Diffusion

In the initial LLT structure with 40% lithium ions vacancy, we substitute

lanthanum with gadolinium and ytterbium, namely, lithium gadolinium titanate (LGT)

and lithium ytterbium titanate (LYT). Gadolinium and ytterbium are lanthanoid

elements, and both of them have 3+ valences like lanthanum. The ionic radius of

lanthanum, gadolinium, and ytterbium are 1.032Å , 0.938Å , and 0.868Å , respectively

66

[44]. At first, we have an assumption that if we substitute smaller atoms (Gd, Yb),

there will be more spaces within the unit cell, and then lithium ions are easier to

diffuse so that the diffusion coefficient can be improved.

By utilizing the same simulation process we mentioned above, we obtain the

diffusion coefficient of LLT, LGT, and LYT in Figure 5.3. We can obvious see that the

diffusion coefficient of LLT is the highest and LYT is the lowest. It decreases with

decreasing the atomic radius of lanthanum, gadolinium, and ytterbium. Figure 5.3

also shows the slope differences among LLT, LGT, and LYT. The slope of the curve fit

of LLT is the smallest and the slope of the curve fit of LYT is the largest. Thus, the

energy barrier is LYT(0.456eV)>LGT(0.289eV)>LLT(0.216eV). It can be calculated by

the Arrhenius equation, higher diffusion coefficient should come with lower diffusion

energy barrier. Due to the fact that lithium ions diffusion in LLT is dominated by the

bottleneck expansion, therefore, we can say that the larger atom in the structure will

make the bottleneck expand. The concept is the same as we need a bigger box if we

want to place a larger basketball.

67

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

-4

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

LGT

LYT

LLTD

iffu

sio

n C

oe

ffic

ien

t (L

og

D)

1000/T

Figure 5.3: The diffusion coefficient of lithium lanthanum titanate (LLT),

lithium gadolinium titanate (LGT), and lithium Ytterbium titanate (LYT) at

600K.

Lithium ions diffusion shows difference among LLT, LGT, and LYT. The results

above show that the larger cation (La) will not block the lithium ions diffusion,

whereas the smaller cation (Yb) substitution does not have any advance for lithium

ions diffusion. It indicates that the lithium ions diffusion is not dominated by the size of

cation. It is dominated by the size of bottleneck structure. Another way to determine

the size of bottleneck is to calculate the free volume in our structure. According to

Figure 3.8, we can see that bottleneck forms a cubic-like structure. Therefore, much

free volume within the structure means bottleneck structure expands more. Figure

68

5.4(a) schematically shows the free volume in grey color and figure 5.4(b) is the

comparisons of system free volume and lithium ion diffusion energy barrier in LLT,

LGT, and LYT. The total free volume is 3422.24 Å 3 for LLT, 2434.58 Å 3 for LGT, and

1665.51 Å 3for LYT. In addition, the interatomic distance is La-O>Gd-O>Yb-O which

shows in their pair distribution function (Figure 5.5). Thus, we can explain that why

the diffusion of lithium ions is easier in LLT than in LGT and LYT.

69

(a)

0

500

1000

1500

2000

2500

3000

3500

0

0.1

0.2

0.3

0.4

0.5

1.032 0.938 0.868

Free Volume

Energy barrier

Fre

e V

olu

me

(A

ng

str

om

3)

En

erg

y b

arrie

r (eV

)

Ionic Radius (Angstrom)

(b)

Figure 5.4: The free volume calculation (a) schematically shows the free

volume in grey color (Blue color: surface area). (b)The comparisons of

system free volume and Li+ diffusion energy barrier among La (1.032Å ), Gd

(0.938 Å ), and Yb (0.868 Å ) in LLT, LGT, and LYT, respectively [44].

70

0

1

2

3

4

5

6

0 5 10 15 20

LaOGdOYbO

g(r

)

r (Angstrom)

Figure 5.5: The pair distribution functions of La-O, Gd-O, and Yb-O at 600K.

71

CHAPTER 6

SUMMARY

Molecular dynamics simulations have been utilized to study the temperature

and composition effect on the structure, diffusion and dynamic properties of lithium

lanthanum titanate, lithium silicate glasses and lithium silicate glass/lithium lanthanum

titanate crystal interfaces. The diffusion behaviors of lithium ions were investigated by

calculating the mean square displacements at different temperatures. Defects and

lithium ion diffusion behaviors of A-site deficient lithium lanthanum titanate structures

in a wide composition range have been systematically studied using molecular

dynamic simulations with effective partial charge pairwise potentials. A maximum of

lithium ion diffusion coefficient and minimum of lithium diffusion activation energy

barrier at x= 0.067 was obtained, in good agreement with the experimental results.

The lowest lithium ion diffusion energy barrier of around 0.22 eV from dynamic

simulations agrees with static NEB minimum energy path calculations of around 0.19

eV using the same force field. The diffusion mechanism was discussed with

observation of dynamic diffusion and static minimum energy pathways. The preferred

diffusion direction was proved by applying external electrical field. The diffusion

coefficients of both x-direction and z-direction are increased due to the electrical field.

The free volume of LLT, LGT, and LYT is calculated, and we found that larger atom

72

(lanthanum) will make the bottleneck become larger so that the free volume increases.

It was found that the lithium ions migrate through A-site vacancy site with about

8-10% lattice expansion when they across the bottleneck. It is suggested that

enlarging bottleneck size by substituting larger A-site cation might be a mechanism to

increase lithium ion conductivities of these solid state electrolytes.

The simulated lithium silicate glass structures were compared with

experimental data, including neutron structure factor and Qn distributions, and found

the two are in good agreement. The Li-O bond distance is around 1.9 Å and the

average coordination number of lithium ions is around 3.6. Both the Li-O bond length

and lithium coordination number increase with increasing lithium oxide concentration.

Lithium silicate glass/lithium lanthanum titanate crystal interfaces were constructed

and analyzed. It was observed that there exists certain level of lithium ion segregation

at the interface. The energy barriers show two temperature range linear behaviors.

The barriers decrease with increasing Li2O concentration in lithium silicate glasses.

This was explained by the increase percentage of NBO and average lithium ion

coordination in glasses with higher Li2O concentration. The interface diffusion energy

barrier is found to be dominated by the glass phase, which has a higher diffusion

energy barrier than the crystal phase. Increasing the diffusion coefficients and

lowering the diffusion energy barrier in the glass phase, by using glasses with higher

73

lithium oxide concentration, can thus improve the total lithium ionic conductivity of the

interface and, consequently, the polycrystalline ceramics.

74

CHAPTER 7

FUTURE RESEARCH

This thesis has so far studied the structure of lithium lanthanum titanate,

lithium silicate, and their interface. Lithium ion diffusion coefficients and energy

barriers are simulated by molecular dynamics simulation. The lithium ion diffusion

mechanisms are also investigated and compared between dynamics and static

simulation.

One of the major issue of LLT ceramics as solid state electrolyte in lithium ion

batteries is due to titanium reduction that increases the electronic conductivity, which

is detrimental for electrolytes, at the electrolyte/anode interface. Therefore, we have

done some initial investigation on Li(1+x)AlxGe(2-x)(PO4)3 solid state electrolyte which is

titanium-free NASICON (acronym of Na Super Ionic Conductor) type structure. The

total ionic conductivity of Li(1+x)AlxGe(2-x)(PO4)3 glass-ceramic with x=0.5 (10-4 S/cm) is

comparable to polycrystalline LLT ceramics but does not have the issue of interfacial

reduction [12]. This system can be systematically studied by using MD simulations to

identify the best solid state electrolytes for future generation all solid state lithium ion

batteries.

75

REFERENCES

[1] H. Shimotake, Progress in Batteries and Solar Cells, JEC Press, 1989, .

[2] Z. Yang, J. Liu, S. Baskaran, C.H. Imhoff and J.D. Holladay, "Enabling renewable

energy-and the future grid-with advanced electricity storage," JOM, vol. 62, pp.

14-23, 2010.

[3] A. Manthiram, "Materials challenges and opportunities of lithium ion

batteries," Journal of Physical Chemistry Letters, vol. 2, pp. 176-184, 2011.

[4] J.B. Goodenough and Y. Kim, "Challenges for rechargeable Li

batteries," Chemistry of Materials, vol. 22, pp. 587-603, 2010.

[5] Y. Inaguma, C. Liquan, M. Itoh, T. Nakamura, T. Uchida, H. Ikuta and M. Wakihara,

"High ionic conductivity in lithium lanthanum titanate," Solid State Commun., vol.

86, pp. 689-93, 06. 1993.

[6] T. Katsumata, Y. Inaguma, M. Itoh and K. Kawamura, "Molecular dynamics

simulation of the high lithium ion conductor, La0.6Li0.2TiO3," Journal of the

Ceramic Society of Japan, vol. 107, pp. 615-21, 07. 1999.

[7] T. Katsumata, Y. Inaguma, M. Itoh and K. Kawamura, "Influence of covalent

character on high Li ion conductivity in a perovskite-type Li ion conductor:

prediction from a molecular dynamics simulation of La0.6Li0.2TiO3," Chemistry

of Materials, vol. 14, pp. 3930-6, 2002.

[8] Y. Li, J.T. Han, C.A. Wang, S.C. Vogel, H. Xie, M. Xu and J.B. Goodenough, "Ionic

distribution and conductivity in lithium garnet Li7La3Zr2O12," J.Power Sources,

vol. 209, pp. 278-281, 7/1. 2012.

[9] J. Awaka, N. Kijima, H. Hayakawa and J. Akimoto, "Synthesis and structure

analysis of tetragonal Li7La3Zr2O12 with the garnet-related type

structure," Journal of Solid State Chemistry, vol. 182, pp. 2046-2052, 8. 2009.

[10] R. Murugan, V. Thangadurai and W. Weppner, "Fast lithium Ion conduction in

garnet-type Li7La3Zr2O12," Angewandte Chemie International Edition, vol. 46,

pp. 7778-81, 10/15. 2007.

76

[11] M. Huang, T. Liu, Y. Deng, H. Geng, Y. Shen, Y. Lin and C. Nan, "Effect of

sintering temperature on structure and ionic conductivity of Li7 - XLa3Zr2O12 -

0.5x (x = 0.5 0.7) ceramics," Solid State Ionics, vol. 204-205, pp. 41-45, 2011.

[12] C.R. Mariappan, C. Yada, F. Rosciano and B. Roling, "Correlation between

micro-structural properties and ionic conductivity of Li1.5Al0.5Ge1.5(PO4)3

ceramics," J.Power Sources, vol. 196, pp. 6456-6464, 8/1. 2011.

[13] N. Suzuki, S. Shirai, N. Takahashi, T. Inaba and T. Shiga, "A lithium phosphorous

oxynitride (LiPON) film sputtered from unsintered Li3PO4 powder target," Solid

State Ionics, vol. 191, pp. 49-54, 6/2. 2011.

[14] B. Wang, B.C. Chakoumakos, B.C. Sales, B.S. Kwak and J.B. Bates, "Synthesis,

Crystal Structure, and Ionic Conductivity of a Polycrystalline Lithium Phosphorus

Oxynitride with the γ-Li3PO4 Structure," Journal of Solid State Chemistry, vol.

115, pp. 313-323, 3. 1995.

[15] B. Wang, B.S. Kwak, B.C. Sales and J.B. Bates, "Ionic conductivities and

structure of lithium phosphorus oxynitride glasses," J.Non Cryst.Solids, vol. 183,

pp. 297-306, 04. 1995.

[16] W.C. West, J.F. Whitacre and J.R. Lim, "Chemical stability enhancement of

lithium conducting solid electrolyte plates using sputtered LiPON thin

films," J.Power Sources, vol. 126, pp. 134-8, 02/16. 2004.

[17] X. Yu, J.B. Bates and G.E. Jellison J., "Characterization of lithium phosphorous

oxynitride thin films," in Proceedings of the Symposium on Thin Film Solid Ionic

Devices and Materials, pp. 23-30, 1996.

[18] X. Yu, J.B. Bates, G.E. Jellison J. and F.X. Hart, "A stable thin-film lithium

electrolyte: lithium phosphorus oxynitride," J.Electrochem.Soc., vol. 144, pp.

524-32, 02. 1997.

[19] Y.A. Du and N.A. Holzwarth, "Li ion Migration in Li3PO4 Electrolytes: Effects of O

Vacancies and N Substitutions," ECS Transactions, vol. 13, pp. 75-82,

December 18. 2008.

[20] A. Hayashi and M. Tatsumisago, "Invited paper: Recent development of

bulk-type solid-state rechargeable lithium batteries with sulfide glass-ceramic

electrolytes," Electronic Materials Letters, vol. 8, pp. 199-207, 2012.

77

[21] F. Muñoz, A. Durán, L. Pascual, L. Montagne, B. Revel and A.C.M. Rodrigues,

"Increased electrical conductivity of LiPON glasses produced by

ammonolysis," Solid State Ionics, vol. 179, pp. 574-579, 6/30. 2008.

[22] A. Hayashi and M. Tatsumisago, "Invited paper: Recent development of

bulk-type solid-state rechargeable lithium batteries with sulfide glass-ceramic

electrolytes," Electronic Materials Letters, vol. 8, pp. 199-207, 2012.

[23] Y. Maruyama, H. Ogawa, M. Kamimura, S. Ono and M. Kobayashi, "Dynamical

properties and electronic structure of (LaLi)TiO3 conductors," Ionics, vol. 14, pp.

357-361, 2008.

[24] S. Stramare, V. Thangadurai and W. Weppner, "Lithium lanthanum titanates: a

review," Chemistry of Materials, vol. 15, pp. 3974-90, 10/21. 2003.

[25] A. Ritchie and W. Howard, "Recent developments and likely advances in

lithium-ion batteries," J.Power Sources, vol. 162, pp. 809-12, 11/22. 2006.

[26] J.B. Bates, N.J. Dudney, G.R. Gruzalski, R.A. Zuhr, A. Choudhury, C.F. Luck and

J.D. Robertson, "Electrical properties of amorphous lithium electrolyte thin

films," Solid State Ionics, vol. 53-56, pp. 647-654, 1992.

[27] J.B. Bates, G.R. Gruzalski, N.J. Dudney, C.F. Luck and X. Yu, "Rechargeable

thin-film lithium batteries," Solid State Ionics, vol. 70-71, pp. 619-628, 1994.

[28] J.B. Bates, N.J. Dudney, D.C. Lubben, G.R. Gruzalski, B.S. Kwak, X. Yu and R.A.

Zuhr, "Thin-film rechargeable lithium batteries," J.Power Sources, vol. 54, pp.

58-62, 1995.

[29] Y. Inaguma and M. Itoh, "Influences of carrier concentration and site percolation

on lithium ion conductivity in perovskite-type oxides," Solid State Ionics, vol.

86-88, pp. 257-260, 1996.

[30] J.H. Irving and J.G. Kirkwood, "The Statistical Mechanical Theory of Transport

Processes. IV. The Equations of Hydrodynamics," J.Chem.Phys., vol. 18, pp.

817-829, 1950.

[31] B.J. Alder and T.E. Wainwright, "Phase Transition for a Hard Sphere

System," J.Chem.Phys., vol. 27, pp. 1208-1209, 1957.

78

[32] B.J. Alder and T.E. Wainwright, "Studies in Molecular Dynamics. I. General

Method," J.Chem.Phys., vol. 31, pp. 459-466, 1959.

[33] A. Rahman, "Correlations in the Motion of Atoms in Liquid Argon," Phys.Rev.,

vol. 136, pp. A405-A411, Oct. 1964.

[34] F.H. Stillinger and A. Rahman, "Improved simulation of liquid water by molecular

dynamics," J.Chem.Phys., vol. 60, pp. 1545-1557, 1974.

[35] R.J. Sadus, Molecular Simulation of Fluids: Theory, Algorithms and

Object-Orientation, New York, NY, USA: Elsevier Science Inc, 1999, .

[36] W. Smith, M. Lesile, T.R. Forester., "The DL_POLY_2 User Manual," Daresbury

Laboratory, 2003.

[37] B. Van B.W.H., G.J. Kramer and S. Van R.A., "Force fields for silicas and

aluminophosphates based on ab initio calculations," Phys.Rev.Lett., vol. 64, pp.

1955-1958, 1990.

[38] A.N. Cormack, J. Du and T.R. Zeitler, "Alkali ion migration mechanisms in silicate

glasses probed by molecular dynamics simulations," in 79th International Bunsen

Discussion Meeting, pp. 3193-7, 2002.

[39] J. Du and A.N. Cormack, "The medium range structure of sodium silicate glasses:

A molecular dynamics simulation," in 16th University Conference, August 13,

2004 - August 15, pp. 66-79, 2004.

[40] E. Lorch, "Neutron diffraction by germania, silica and radiation-damaged silica

glasses," Journal of Physics C: Solid State Physics, vol. 2, pp. 229, 1969.

[41] J. Du and L.R. Corrales, "Compositional dependence of the first sharp diffraction

peaks in alkali silicate glasses: A molecular dynamics study," J.Non Cryst.Solids,

vol. 352, pp. 3255-3269, 2006.

[42] A.G. Belous, G.N. Novitskaya, S.V. Polyanetskaya and Y.I. Gornikov, "Study of

Complex Oxides of Composition La2/3 - xLi3xTiO3; ISSLEDOVANIE

SLOZHNYKH OKSIDOV SOSTAVA La//2/////3// minus

//xLi//3//xTiO//3," Neorganiceskie Materialy, vol. 23, pp. 470-472, 1987.

79

[43] M. Catti, M. Sommariva and R.M. Ibberson, "Tetragonal superstructure and

thermal history of Li0.3La0.567TiO3 (LLTO) solid electrolyte by neutron

diffraction," Journal of Materials Chemistry, vol. 17, pp. 1300-7, 04/07. 2007.

[44] R.D. Shannon, "Revised effective ionic radii and systematic studies of

interatomic distances in halides and chalcogenides," Acta Crystallographica,

Section A (Crystal Physics, Diffraction, Theoretical and General Crystallography),

vol. A32, pp. 751-67, 1976.

[45] M. Yashima, M. Itoh, Y. Inaguma and Y. Morii, "Crystal structure and diffusion

path in the fast lithium-ion conductor La0.62Li0.16TiO3," J.Am.Chem.Soc., vol.

127, pp. 3491-3495, 2005.

[46] D. Qian, B. Xu, H. Cho, T. Hatsukade, K.J. Carroll and Y.S. Meng, "Lithium

lanthanum titanium oxides: A fast ionic conductive coating for lithium-ion battery

cathodes," Chemistry of Materials, vol. 24, pp. 2744-2751, 2012.

[47] G. Henkelman and H. Jónsson, "Improved tangent estimate in the nudged elastic

band method for finding minimum energy paths and saddle

points," J.Chem.Phys., vol. 113, pp. 9978-9985, 2000.

[48] Y. Inaguma, J. Yu, Y. Shan, M. Itoh and T. Nakamuraa, "The Effect of the

Hydrostatic Pressure on the Ionic Conductivity in a Perovskite Lanthanum

Lithium Titanate," J.Electrochem.Soc., vol. 142, pp. L8-L11, 01/00. 1995.

[49] Y. Inaguma, "Fast percolative diffusion in lithium ion-conducting perovskite-type

oxides," J Ceram Soc Jpn, vol. 114, pp. 1103-1110, 2006.

[50] Y. Inaguma, T. Katsumata, M. Itoh, Y. Morii and T. Tsurui, "Structural

investigations of migration pathways in lithium ion-conducting La2/3-xLi3xTiO3

perovskites," Solid State Ionics, Diffusion & Reactions, vol. 177, pp. 3037-44,

11/30. 2006.

[51] H. Kawai and J. Kuwano, "Lithium Ion Conductivity of A‐Site Deficient Perovskite

Solid Solution La0.67 − x Li3x TiO3," Journal of the Electrochemical Society, vol.

141, pp. L78-L79, July 01. 1994.

[52] A. Mei, X. Wang, J. Lan, Y. Feng, H. Geng, Y. Lin and C. Nan, "Role of

amorphous boundary layer in enhancing ionic conductivity of

80

lithium-lanthanum-titanate electrolyte," Electrochim.Acta, vol. 55, pp. 2958-2963,

2010.

[53] A. Mei, X. Wang, Y. Feng, S. Zhao, G. Li, H. Geng, Y. Lin and C. Nan, "Enhanced

ionic transport in lithium lanthanum titanium oxide solid state electrolyte by

introducing silica," Solid State Ionics, vol. 179, pp. 2255-2259, 2008.

[54] Y. Hiwatari and J. Habasaki, "Dynamical significance in alkali metasilicate

glasses," Journal of Molecular Liquids, vol. 120, pp. 135-138, 6. 2005.

[55] R. Prasada Rao, T.D. Tho and S. Adams, "Ion transport pathways in molecular

dynamics simulated lithium silicate glasses," Solid State Ionics, vol. 181, pp. 1-6,

2010.

[56] J. Habasaki and Y. Hiwatari, "Molecular dynamics study of the mechanism of ion

transport in lithium silicate glasses: Characteristics of the potential energy

surface and structures," Physical Review B (Condensed Matter and Materials

Physics), vol. 69, pp. 144207-1, 04/01. 2004.

[57] A. Pedone, G. Malavasi, A.N. Cormack, U. Segre and M.C. Menziani, "Insight

into elastic properties of binary alkali silicate glasses; prediction and

interpretation through atomistic simulation techniques," Chemistry of Materials,

vol. 19, pp. 3144-54, 06/26. 2007.

[58] A. Heuer, M. Kunow, M. Vogel and R.D. Banhatti, "Characterization of the

complex ion dynamics in lithium silicate glasses via computer simulations," in

79th International Bunsen Discussion Meeting, pp. 3185-92, 2002.

[59] W. Li and S.H. Garofalini, "Molecular dynamics simulation of lithium diffusion in

Li2O-Al2O3-SiO2 glasses," Solid State Ionics, Diffusion & Reactions, vol. 166,

pp. 365-73, 01. 2004.

[60] S.H. Garofalini and P. Shadwell, "Molecular dynamics simulations of

cathode/glass interface behavior: Effect of orientation on phase transformation, Li

migration, and interface relaxation," J.Power Sources, vol. 89, pp. 190-200,

2000.

[61] H. Lammert, M. Kunow and A. Heuer, "Complete identification of alkali sites in

ion conducting lithium silicate glasses: a computer study of ion

dynamics," Phys.Rev.Lett., vol. 90, pp. 215901-1, 05/30. 2003.

81

[62] J. Habasaki and Y. Hiwatari, "Characteristics of slow and fast ion dynamics in a

lithium metasilicate glass," Phys Rev E., vol. 59, pp. 6962-6966, Jun. 1999.

[63] M.J.D. Rushton, R.W. Grimes and S.L. Owens, "Predicted changes to alkali

concentration adjacent to glass-crystal interfaces," J Am Ceram Soc, vol. 91, pp.

1659-1664, 2008.

[64] P.W.M. P.F. James, "Quantitative measurements of phase separation in glasses

using transmission electron microscopy. II. A study of lithia-silica glasses and

the influence of phosphorus pentoxide," Physics and Chemistry of Glasses, vol.

11, pp. 64-70, 1970.

[65] N.P. Bansal and R.H. Doremus, Handbook of glass properties, Orlando:

Academic Press, 1986, pp. 680.

[66] N.A. Deskins, S. Kerisit, K.M. Rosso and M. Dupuis, "Molecular dynamics

characterization of rutile-anatase interfaces," Journal of Physical Chemistry C,

vol. 111, pp. 9290-9298, 2007.

[67] M. Misawa, D.L. Price and K. Suzuki, "The short-range structure of alkali

disilicate glasses by pulsed neutron total scattering," J.Non Cryst.Solids, vol. 37,

pp. 85-97, 3. 1980.

[68] H. Maekawa, T. Maekawa, K. Kawamura and T. Yokokawa, "The structural

groups of alkali silicate glasses determined from 29Si MAS-NMR," J.Non

Cryst.Solids, vol. 127, pp. 53-64, 01. 1991.

[69] A.I. Ruiz, M.L. Lopez, M.L. Veiga and C. Pico, "Electrical properties of

La1.33-xLi3xTi2O6 (0.1x0.3)," Solid State Ionics, vol. 112, pp. 291-297, 1998.

[70] A. Pedone, G. Malavasi, A. Cormack, U. Segre and M. Menziani, "Elastic and

dynamical properties of alkali-silicate glasses from computer simulations

techniques," Theoretical Chemistry Accounts: Theory, Computation, and

Modeling (Theoretica Chimica Acta), vol. 120, pp. 557-564, 2008.

[71] A. Winkler, J. Horbach, W. Kob and K. Binder, "Structure and diffusion in

amorphous aluminum silicate: A molecular dynamics computer

simulation," J.Chem.Phys., vol. 120, pp. 384-393, 2004.

82

[72] Y. Inaguma, J. Yu, T. Katsumata and M. Itoh, "Lithium ion conductivity in a

perovskite lanthanum lithium titanate single crystal," J Ceram Soc Jpn, vol. 105,

pp. 548-550, 1997.

[73] R. Ruffo, S.S. Hong, C.K. Chan, R.A. Huggins and Y. Cui, "Impedance analysis

of silicon nanowire lithium ion battery anodes," Journal of Physical Chemistry C,

vol. 113, pp. 11390-11398, 2009.

[74] M. Kunow and A. Heuer, "Nonlinear ionic conductivity of lithium silicate glass

studied via molecular dynamics simulations," J.Chem.Phys., vol. 124, pp.

214703-1, 06/07. 2006.

[75] E. Soolo, D. Brandell, A. Liivat, H. Kasemagi, T. Tamm and A. Aabloo, "Force

field generation and molecular dynamics simulations of

Li+-Nafion," Electrochim.Acta, vol. 55, pp. 2587-2591, 2010.

[76] J. Wang and J. Lei, "Influence of DC electric field on both crystalline structure

and conductivity of poly(ethylene oxide)10:LiClO4 electrolyte," J.Polym.Sci.Part

B, vol. 50, pp. 656-667, 2012.