ATOMISTIC COMPUTER SIMULATIONS OF DIFFUSION MECHANISMS IN LITHIUM LANTHANUM TITANATE SOLID STATE ELECTROLYTES FOR LITHIUM ION BATTERIES.

Transcription

1 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 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

2 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 (La 2/3-x Li 3x TiO 3, 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

3 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.

4 Copyright 2014 by Chao-Hsu Chen ii

5 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. iii

6 TABLE OF CONTENTS Page ACKNOWLEDGEMENTS... iii LIST OF TABLES... vi LIST OF FIGURES...vii CHAPTER 1 INTRODUCTION Brief History Principle of Lithium Ion Battery Materials of Electrolytes Solid State Lithium Ion Batteries Motivation Thesis Layout... 8 CHAPTER 2 MOLECULAR DYNAMICS SIMULATION DETAILS AND METHODOLOGY Introduction Theory Verlet Algorithm The Velocity Verlet Algorithm Leap Frog Algorithm Molecular Dynamics Simulation Ensembles Potentials Methodology of Data Analysis Mean Square Displacement and Diffusion Coefficient Neutron Structure Factor CHAPTER 3 DEFECT CONCENTRATION EFFECT ON LITHIUM ION DIFFUSION IN LITHIUM LANTHANUM TITANATE SOLID STATE ELECTROLYTES Abstract Introduction Methodology Introduction Initiation of Crystal Bulk Crystal Structure and Effect of Temperature on Lattice Parameter iv

7 3.5 Effect of Vacancy Concentration on Lithium Ion Diffusion Diffusion Energy Barrier: Static Calculations Lithium Ion Diffusion Mechanism Total Ionic Conductivity Calculation CHAPTER 4 STRUCTURE AND LITHIUM ION DIFFUSION IN SILICATE GLASSES AND AT THEIR INTERFACES WITH LITHIUM LANTHANUM TITANATE CRYSTALS Abstract Introduction Methodology Introduction Initiation of Glass System Build the Glass/Crystal Interface Structure The Structure of Lithium Silicate Glasses The Structure of the Lithium Silicate Glass/LLT Crystal Interface Diffusion Coefficients in Lithium Silicate Glasses and at the Boundary Lithium Ion Diffusion Behavior in Interface System CHAPTER 5 DIFFUSION ANISOTROPY AND CATION RADIUS EFFECT IN LITHIUM LANTHANUM TITANATE Abstract Introduction Methodology Diffusion Anisotropy of Lithium Ions in LLT Effect of a Site Cation Substitution on Lithium Ion Diffusion CHAPTER 6 SUMMARY CHAPTER 7 FUTURE RESEARCH REFERENCES v

8 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 Table 3.1 Comparison of calculated and experimental structure of LLT (Li 0.3 La TiO 3 ) Table 4.1 Glass composition parameters for MD simulation vi

9 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. TiO 6 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: TiO 6 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) Figure 3.2: The lattice parameter as function of x value in LLT vacancy structure (La 2/3-x Li 3x TiO 3 ) Figure 3.3: (a) Mean square displacement of lithium lanthanum titanate (LLT) (b) MSD in logarithm Figure 3.4: Lithium ion diffusion coefficients as a function of x value in La 2/3-x Li 3x TiO 3.30 Figure 3.5: Lithium ion diffusion energy barrier as a function of x value in La 2/3-x Li 3x TiO 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) Figure 3.7: Trajectories of lithium ions for composition Li 0.2 La 0.6 TiO 3. 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.) 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

10 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) Figure 3.10: Charge carrier concentration as a function of x value in La 2/3-x Li 3x TiO Figure 3.11: Ionic conductivity as a function of x value in La 2/3-x Li 3x TiO Figure 4.1: The structure of the lithium silicate oxide Li 2 O-2SiO 2. Golden pyramids: silicon oxygen tetrahedrons, red ball: oxygen, blue ball: lithium ions 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 Figure 4.3: Comparison of calculated and experimental neutron structure factor function of lithium disilicate glass. Solid line: MD simulations; circles: experiment data Figure 4.4: Li-O pair distribution function as a function of Li 2 O concentration in lithium silicate glasses. Arrow points to increase of Li 2 O concentration Figure 4.5: Change of lithium ion coordination number as a function of Li 2 O concentration Figure 4.6: Distribution of lithium ion coordination number of 30 Li 2 O-70SiO 2 compositions Figure 4.7: Q n distribution as a function of Li 2 O concentration Figure 4.8: Z-Density profile across the interface Figure 4.9: (a) Linear (a) and logarithm (b) mean square displacement of lithium ions in lithium disilicate (LS33) glass Figure 4.10: Diffusion coefficients of lithium ions in glass crystal interface and different composition of glasses for different temperatures (The unit D is cm 2 /s) viii

11 Figure 4.11: Diffusion energy barrier for lithium disilicate glass, LLTO crystal, and the glass-crystal interfacial structure Figure 5.1: Lithium ion diffusion coefficient under different external electrical field along X-axis and Z-axis for composition Li 0.2 La 0.6 TiO 3 (MD simulations at 600 K) Figure 5.2: Trajectories of lithium ions with 43MV/m external electrical field for composition Li 0.2 La 0.6 TiO 3. MD simulations at 600 K for 160 ps with electrical field applied along z-axis Figure 5.3: The diffusion coefficient of lithium lanthanum titanate (LLT), lithium gadolinium titanate (LGT), and lithium Ytterbium titanate (LYT) at 600K 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 Figure 5.5: The pair distribution functions of La-O, Gd-O, and Yb-O at 600K ix

12 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 21 st 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, 1

13 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 LiMO 2 (M=Co, Ni, Mn) as cathode, which are separated by a lithium-ion conducting electrolyte layer that is usually made of solution of LiPF 6 in organic solution such as ethylene carbonate-diethylcarbonate. The reactions of anode and cathode are shown below: Cathode: charge LiCoO 2 Li (1 x) CoO 2 + x Li + + x e (1.1) discharge 2

14 Anode: charge C 6 + x Li + x e C 6 Li x (1.2) discharge Total: charge LiCoO 2 + C 6 Li (1 x) CoO 2 + C 6 Li x (1.3) discharge 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. 3

15 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 Li 7 La 3 Zr 2 O 12 framework structure, and lithium ion conduction pathway is through face-sharing tetrahedral and octahedral lithium sites [8-11]. Li-analogues of NASICON structures containing Ti 4+ ions have also been found to have high ionic conductivities (up to 10-3 at room temperature). However, due to the reduction of Ti 4+, the Ti-free Li-analogues of NASICONs have been investigated [12]. Another type of solid lithium electrolyte is LAGP glass-ceramics. Li 1+x Al x Ge 2-x (PO 4 ) 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 Li 3 PO 4 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 4

16 solid electrolytes such as Li 2 S-P 2 S 5 have been shown to have higher ionic conductivity (10-3 to 10-2 S/cm) than oxide materials. In particular, sulfide crystalline Li 10 GeP 2 S 12 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 Materials Ionic Conductivity Features La 2/3 x Li 3x TiO S/cm Highest ionic conductivity in ceramics Li 7 La 3 Zr 2 O S/cm 3-D framework structure, short Li-Li migration pathway Li 1+x Al x Ge 2-x (PO 4 ) 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 Li 2 S-P 2 S to 10-2 S/cm Inorganic sulfide material. Li 10 GeP 2 S x 10-2 S/cm Conductivity is close to liquid electrolytes 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 5

17 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 6

18 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), La 2/3 x Li 3x TiO 3, 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. 7

19 Figure 1.1: The ideal perovskite crystal structure of lithium lanthanum titanate (LLT). Lithium and lanthanum are distributed over A-site. TiO 6 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: TiO 6 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 8

20 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 9

21 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. 10

22 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. 11

23 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 = m dv dt = m d2 r dt 2 (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: Verlet Algorithm r(t + t) = 2r(t) r(t t) + a(t) t 2 + O( t 4 ) (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( t 2 ) (2.3) The velocity error is O( t 2 ), and the truncation error is O( t 4 ). 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]. 12

24 2.2.2 The Velocity Verlet Algorithm r(t + t) = r(t) + v(t) t a(t) t2 (2.4) v(t + t) = v(t) + 1 [a(t) + a(t + t)] t (2.5) 2 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 Leap Frog Algorithm v (t + 1 t) = v (t 1 t) + a(t) t (2.6) 2 2 r(t + t) = r(t) + v(t + 1 t) t (2.7) 2 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. 13

25 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 Ensembles The ensemble concept is proposed by Gibbs in 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. 14

26 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/r 6 (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 r c 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. 15

27 V(r) has the function form of V(r)=B/r n +Dr 2 (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 r c. Table 2.1 Ionic charges and Buckingham potential parameters Pairs A (ev) ρ (Å) C (ev Å 6 ) O O Si 2.4 -O Li 0.6 -O La 1.8 -O Ti 2.4 -O Methodology of Structural Analysis and Property Calculations 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 16

28 from MSD according to Einstein diffusion equation: D = 1 6 lim t d dt r i(0) r i (t) 2 (2.11) Mean square displacement (MSD) is defined as, MSD = r i (0) r i (t) 2 (2.12) where r i is the position of particle i, r i (0) and r i (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 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 g ij (r) through R S ij (Q) = 1 + ρ 0 4πr 2 [g ij (r) 1] 0 sin(qr) sin(πr R) Qr πr R 17 dr (2.13)

29 in which ρ 0 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) part is the Lorch type window function [40] πr R which reduces the effect of finite simulation cell size during the Fourier transformation. The total neutron structure factor was calculated by n n S N (Q) = ( i=1 c i b i ) 2 ij=1 c i c j b i b j S ij (Q) (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, , and fm for oxygen, silicate, and lithium respectively [41]. 18

30 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, La 2/3-x Li 3x TiO 3 ) 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, 19

31 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, LiLa( TiO3 ) 2 ' La O 2La 4V 6La 12Ti 36O 2 3 Li Li La Ti O (3.1) The compositions of LLT with lithium ion vacancies due to La/Li substitution are usually represented as La 2/3-x Li 3x TiO 3 with x ranging from 0 to 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 cm 3 /mol was observed when lithium ion jump to an adjacent vacancy [24]. It was also suggested that TiO 6 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 20

32 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 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 21

33 distribution function analysis was used to study the diffusion path and it was found that the first peak intensity of Ti-Li pair distribution g Ti-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 x10-7 cm 2 /s at 500K with different A-site ions arrangements and compared with the experimental data [6, 7] 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(TiO 3 ) 2 structure, titanium ions occupy the B-site of the ABO 3 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 22

34 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 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 (Li 0.3 La TiO 3 )[43] Occupancy Exp. at 25 C Simulation (P4/nbm) (this work) a (Å ) c (Å ) Volume alpha/beta/gamma ( ) /90.0/ /90.0/90.0 La1 (2c) /0.75/ /0.750/ La2 (2d) /0.75/ /0.50/ Ti (4g) /0.25/ /0.250/ O1 (8m) /0.4820/ /0.4999/ O2 (2a) /0.25/ /0.250/ O2 (2b) /0.25/ /0.250/ Li (8m) /-0.108/ /0.000/

35 (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) 24

36 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 25

37 Lattice Parameter (Angstrom) [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] a,b c X value Figure 3.2: The lattice parameter as function of x value in LLT vacancy structure (La 2/3-x Li 3x TiO 3 ) 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 26

38 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 t 2. 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 cm 2 /s. This result has good agreement with the experimental data [5, 24, 42, 45]. In addition, according to Arrhenius equation D = D 0 exp ( Ea RT ) (3.2) where E a is diffusion energy barrier, T is temperature and R is gas constant. If we take the logarithm respectively, log D = log D 0 Ea 1 R T (3.3) where D 0 is pre-exponential factor. The slope of logd over 1/T gives activation energy 27

39 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 R 2 values higher than 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 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 ( C) for composition Li 0.34 La 0.5l TiO 2.94 [5-8, 29]. The simulated diffusion energy barrier fell well in the range of the two values. 28

40 Mean square displacement (A 2 ) Mean square displacement (A 2 ) K 686K 600K 480K 400K Time (ps) (a) K 686K 600K 480K 400K Time (ps) (b) Figure 3.3: (a) Mean square displacement of lithium lanthanum titanate (LLT) (b) MSD in logarithm. 29

41 Activation Energy(eV) Diffusion coefficient(10-5 cm 2 /s) X value Figure 3.4: Lithium ion diffusion coefficients as a function of x value in La 2/3-x Li 3x TiO X Value Figure 3.5: Lithium ion diffusion energy barrier as a function of x value in La 2/3-x Li 3x TiO 3. 30

42 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 31

43 Lattice Energy (ev) 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 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) 32

44 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]. 33

45 Figure 3.7: Trajectories of lithium ions for composition Li 0.2 La 0.6 TiO 3. 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.) 34

46 Step A Li ion tries to migrate through bottleneck. Step B Li ion is close to bottleneck. Step C Li ion is within the bottleneck. Step D 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) 35

47 3.8 Total Ionic Conductivity Calculation The total ionic conductivity of a solid can be expressed as σ = i n i Z i μ i (3.4) where n i is the charge carrier concentration, Z i 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 La 2/3-x Li 3x TiO 3 (Figure 3.10) with the maximum at x= The available lithium ion concentration in La 2/3-x Li 3x TiO 3 can be expressed as N Li : 3x/V s, in which V s is the unit cell volume, and lithium ion vacancy concentration expressed a (0.33-2x)/V s. Assuming the total A site concentration N=N v +N Li to be identical in terms of symmetry and energies, the total charge carrier concentration can be expressed as [51]. n Li = N LiN V N = (x 6x2 ) (0.33+x)V s (3.5) 36

48 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 = z2 e 2 nd 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 The maximum ionic conductivity was obtained to be S/cm at x= Experimental conductivity measurements also showed a similar shape although the exact maximum position was slightly different: x equals 0.11 vs 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 (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 (Figure 3.5). The coincidence of the highest lithium ion diffusion coefficient and lowest activation energy barrier at a composition with x= (around 40% lithium ion vacancy) from simulations is in good agreement with the maximum behavior of 37

49 Ionic Conductivity (S/cm) Charge Carrier Concentration (10 20 cm -3 ) experimental conductivity data X Value Figure 3.10: Charge carrier concentration as a function of x value in La 2/3-x Li 3x TiO X value Figure 3.11: Ionic conductivity as a function of x value in La 2/3-x Li 3x TiO 3 38

50 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 39

51 from Ti 4+ to Ti 3+, 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 40

52 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 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 Li 2 O 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 41

53 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/v 2 O 5 crystal interface which is similar to our system [60]. They created different surface terminations (vanadium and oxygen) of (001) and (010) surfaces of V 2 O 5 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, 42

54 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] Initiation of Glass System Wide composition range in the Li 2 O-SiO 2 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 Li 2 O-SiO 2 glasses [64]. In our simulations, however, we only consider homogeneous glasses. The glass compositions simulated are xli 2 O-(1-x)SiO 2 with x=0.1, 0.2, 0.3, 0.33, 0.4, and 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 43

55 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 Atom number Li 2 O SiO 2 (g/cc)* O Si Li LS LS LS LS LS LS * Density data from ref. [65] 44

56 Figure 4.1: The structure of the lithium silicate oxide Li 2 O-2SiO 2. Golden pyramids: silicon oxygen tetrahedrons, red ball: oxygen, blue ball: lithium ions 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 La 2/3-x Li 3x TiO 3 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 45

57 interface is Å. 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. 46

58 (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. 47

59 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 Li 2 O 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 Li 2 O 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. 48

60 g (r) S N (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] Li 2 O 10 mol% Li 2 O 20 mol% Li 2 O 30 mol% Li 2 O 40 mol% Li 2 O 46 mol% r (Angstrom) Figure 4.4: Li-O pair distribution function as a function of Li 2 O concentration in lithium silicate glasses. Arrow points to increase of Li 2 O concentration. 49

61 The average coordination number of lithium ions indeed increases from 3.4 to 3.8, as Li 2 O 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 Li 2 O 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 Li 2 O-70SiO 2 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 50

62 Li Coord. Number 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] Li Cood. # BO# NBO# FO# Li 2 O (mol %) Figure 4.5: Change of lithium ion coordination number as a function of Li 2 O concentration. 51

63 Percentage Percentage Li 2 O-70SiO 2 Average Li Coord Li Coord. Number Figure 4.6: Distribution of lithium ion coordination number of 30 Li 2 O-70SiO 2 compositions Q1 Q2 Q3 Q4 Q2 exp Q3 exp Q4 exp Li 2 O(mol%) Figure 4.7: Q n distribution as a function of Li 2 O concentration (Experimental data from Ref [68]). 52

64 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 Å and Å for a and c, respectively, in the tetragonal unit cells of LLT. This compares well with experimental cell parameters Å and 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 53

65 interface. 100 Atom number density (number/nm 3) 80 Si Li La O Ti 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 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 54

66 calculations in the literature varied greatly: Pedone et al. used 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 Li 2 O concentrations, for example the R 2 values for linear fitting are for LS40 and for LS10 for the high temperature range, while the R 2 values are for LS40 and 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 The energy barrier of the lithium silicate glasses decreases with increasing Li 2 O 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 55

67 MSD(nm 2 ) MSD(nm 2 ) 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] time(ps) (a) time(ps) (b) Figure 4.9: (a) Linear (a) and logarithm (b) mean square displacement of lithium ions in lithium disilicate (LS33) glass. 56

68 Activation Energy(eV) Figure 4.10: Diffusion coefficients of lithium ions in glass crystal interface and different composition of glasses for different temperatures (The unit D is cm 2 /s) Glass Glass Interface Crystal Li O(mol%) 2 Figure 4.11: Diffusion energy barrier for lithium disilicate glass, LLT crystal, and the glass-crystal interfacial structure. 57

69 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 La 3+ -rich and La 3+ -deficit layer. The lithium ions can only migrate two-dimensionally within the La 3+ -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 58

70 mentioned some results above, the lanthanum layer (La 3+ -rich) and lithium layer (La 3+ -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, 59

71 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. 60

72 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 61

73 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=5x10 7 V/m, but shows non-linear behavior above E=5x10 7 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 :LiClO 4 with adjustment electrical field was also studied by Wang et al.. They have found that the conductivity of the poly(ethylene oxide) 10 :LiClO 4 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 62

74 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 63

75 Diffusion Coeficient (10-5 cm 2 /s) field. The vertical direction is z-direction, and horizontal direction is x-direction Z-axis X-axis Electrical Field (MV/m) Figure 5.1: Lithium ion diffusion coefficient under different external electrical field along X-axis and Z-axis for composition Li 0.2 La 0.6 TiO 3 (MD simulations at 600 K). Figure 5.2: Trajectories of lithium ions with 43MV/m external electrical field for composition Li 0.2 La 0.6 TiO 3. MD simulations at 600 K for 160 ps with electrical field applied along z-axis. 64

76 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 65

77 [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. 66

78 Diffusion Coefficient (Log D) LLT LGT LYT /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 67

79 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 Å 3 for LLT, Å 3 for LGT, and Å 3 for 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. 68

80 Free Volume (Angstrom 3 ) Energy barrier (ev) (a) Free Volume Energy barrier Ionic Radius (Angstrom) 0 (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]. 69