STUDY ON Na2x[NixTi1-x]O2 AS BI-FUNCTIONAL ELECTRODE MATERIAL FOR SODIUM-ION BATTERIES By Qian Chen A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Chemical Engineering—Doctor of Philosophy 2021 ABSTRACT STUDY ON Na2x[NixTi1-x]O2 AS BI-FUNCTIONAL ELECTRODE MATERIAL FOR SODIUM-ION BATTERIES By Qian Chen The rapid development of renewable energy resources in response to growing concerns about fossil fuels dependence and greenhouse gas emissions has raised challenges to developing large-scale energy storage systems for the smooth integration of intermittent energy resources into the grid. Increasing research efforts have been devoted to the study of the sodium-ion battery technology as a promising solution to these challenges, especially in the search for better electrode materials. In this thesis, we focused on the investigation of Na2x[NixTi1-x]O2, a promising bi- functional electrode material that can be used as either positive or negative electrode material in sodium-ion batteries. This thesis aims to gain fundamental understandings of this material using a combination of experimental and computational techniques and to provide insights into the exploration and design of electrode materials for sodium-ion batteries. Firstly, the average and local structural properties and energetics of atomic distribution in P2-Na2/3[Ni1/3Ti2/3]O2 were investigated using Rietveld refinement on neutron diffraction datasets and atomistic simulations based on the Buckingham and Morse interatomic potential models. Both computational and experimental results showed similar nuclear density maps and higher occupancies of Na at edge-sharing sites than face-share sites. The simulations based on both potential models suggested that it is energetically favorable to have an equal amount of Na and transition metal in each of the two layers. The atomic and electronic structure changes during cycling were studied based on density functional theory (DFT) calculations. DFT-based molecular dynamics (MD) simulations showed an expansion of ab plane and contraction of c axis upon Na insertion and a small change of lattice parameters upon Na extraction. DFT calculations revealed that Ni and Ti play a dominant role in the redox reactions upon Na extraction and insertion respectively, along with the participation of O. A higher in-plane electronic conductivity was observed compared to the through-plane one, with both increasing when Na ions were inserted or extracted. The quasi-elastic neutron scattering (QENS) experiments showed that Na ion diffusion can be well described by the Singwi-Sjölander jump diffusion model, where the obtained mean jump length matched the distances between the neighboring edge-share and face-share Na sites. The MD simulations based on DFT calculations showed a better consistency with experimental results than MD based on interatomic potential (IP) models in terms of diffusivity and activation energy. Faster diffusion was observed for compositions with less sodium, i.e., more vacancies. Both Na-deficient phase Na5/9[Ni1/3Ti2/3]O2 and Na-rich phase Na7/9[Ni1/3Ti2/3]O2 showed higher ionic conductivity compared to the pristine phase. Finally, the diffusion and ionic conduction for P2 and O3 Na2x[NixTi1-x]O2 are calculated with machine learning based interatomic potential models. Higher diffusivity and ionic conductivity were observed for the P2 structure. ACKNOWLEDGEMENTS First of all, I would like to express my sincere gratitude to my adviser, Dr. Wei Lai, for his support and guidance during my Ph.D. study. He has set the best role model for me for a wonderful mentor and researcher: always kind and patient, incredibly knowledgeable yet still enthusiastic about learning new things, open to new ideas while maintaining a critical perspective, passionate and devoted to research work. I am grateful for the opportunity to work with him and I could not have imagined having a better PhD advisor. I’d like to extend my thanks to other members of my thesis committee, Dr. Scott Calabrese Barton, Dr. Jose Mendoza, and Dr. Pengpeng Zhang for their guidance and advice on this thesis. I’d like to thank my former and current fellow labmates that I have worked with: Dr. Matt Klenk, Dr. Junchao Li, Jin Dai, Yue Jiang, and Yining He. It was a great pleasure working with them. Being the only chemical engineer in the group, every time I turned to them for insights from material scientists' perspectives, they were always willing to share helpful and valuable ideas in our discussions. I have learned so much from each of them in many ways. Especially Jin, we began our Ph.D. studies and joined the group at the same time and have spent most of our time together in the office. Your friendship and support are priceless. I’d like to thank my parents for their unconditional love throughout these years and for always supporting every decision I’ve made. Thank you to my friends who made my life in a foreign country far away from my family fun and colorful. Last but not least, I’d like to thank my incredible husband Xiao for always being there for me and having faith in me through the hard times. I couldn't have made it through the difficulties of my PhD journey without his encouragement and support. iv TABLE OF CONTENTS LIST OF TABLES ........................................................................................................................ vii LIST OF FIGURES ..................................................................................................................... viii Introduction ........................................................................................................... 1 1.1 Background .......................................................................................................................... 1 1.2 Sodium-ion batteries vs. lithium-ion batteries ..................................................................... 2 1.2.1 Sodium vs. lithium .................................................................................................... 2 1.2.2 The working principle of sodium-ion batteries ......................................................... 5 1.3 Electrode materials for sodium-ion batteries ....................................................................... 6 1.3.1 Positive electrode materials ...................................................................................... 8 1.3.2 Negative electrode materials ................................................................................... 14 1.3.3 Bi-functional electrode materials ............................................................................ 16 1.4 The Na2x[NixTi1-x]O2 group ............................................................................................... 16 1.5 Thesis overview ................................................................................................................. 18 Methods ............................................................................................................... 20 2.1 Computational methods ..................................................................................................... 20 2.1.1 Density functional theory ........................................................................................ 20 2.1.2 Interatomic potential model .................................................................................... 22 2.1.3 Molecular dynamics simulation .............................................................................. 27 2.2 Experimental methods ....................................................................................................... 28 2.2.1 Sample preparation ................................................................................................. 28 2.2.2 Neutron diffraction and Rietveld refinement .......................................................... 29 2.2.3 Quasi-elastic neutron scattering .............................................................................. 30 Structural properties of P2-Na2/3[Ni1/3Ti2/3]O2 .................................................... 33 3.1 Introduction ........................................................................................................................ 33 3.2 Average structural information from Rietveld refinement ................................................ 33 3.3 Energy minimization based on two interatomic potential models ..................................... 40 3.4 Energetics of Atomic Distribution and Ordering ............................................................... 43 3.5 Simulated Average Structure ............................................................................................. 45 3.6 Conclusions ........................................................................................................................ 47 Structural and electronic properties change during cycling ................................ 49 4.1 Introduction ........................................................................................................................ 49 4.2 DFT calculations ................................................................................................................ 49 4.3 Structural changes upon Na extraction/insertion ............................................................... 50 4.4 Electron density distribution and the atomic charge .......................................................... 54 4.5 Density of states ................................................................................................................. 56 4.6 Electronic conductivity ...................................................................................................... 58 4.6.1 Electronic conductivity from the Kubo-Greenwood approach ............................... 58 4.6.2 Electronic conductivity from the Boltzmann transport equations .......................... 59 4.7 Conclusions ........................................................................................................................ 62 v Dynamic properties of P2-Na2/3[Ni1/3Ti2/3]O2 ..................................................... 64 5.1 Introduction ........................................................................................................................ 64 5.2 Self-diffusion of Na ions from QENS ............................................................................... 64 5.3 Ionic conductivity and Na self-diffusion from IP-based MD ............................................ 68 5.3.1 Construction of the polarizable interatomic potential model .................................. 68 5.3.2 IP-based MD simulations ........................................................................................ 70 5.3.3 Extraction of diffusivity and ionic conductivity from IP-based MD ...................... 73 5.4 First principles MD simulations......................................................................................... 79 5.5 Conclusions ........................................................................................................................ 85 Study on P2/O3 Na2x[NixTi1-x]O2 with machine learning based interatomic potential models ............................................................................................................................ 87 6.1 Introduction ........................................................................................................................ 87 6.2 Construction of the neural network potential..................................................................... 88 6.3 Testing the NN potential .................................................................................................... 91 6.4 MD simulations with the NN potential .............................................................................. 93 6.4.1 Calculation of lattice parameters ............................................................................ 94 6.4.2 Diffusion and ionic conduction ............................................................................... 96 6.5 Conclusions ........................................................................................................................ 98 Conclusions and future work ............................................................................ 100 7.1 Conclusions ...................................................................................................................... 100 7.2 Future work ...................................................................................................................... 103 BIBLIOGRAPHY ....................................................................................................................... 106 vi LIST OF TABLES Table 1: The Comparison between sodium and lithium ................................................................. 3 Table 2: Structural information of Na2/3[Ni1/3Ti2/3]O2 using Rietveld refinement of neutron diffraction datasets. ....................................................................................................................... 35 Table 3: Interatomic potential parameters of the Buckingham model and Morse model. ............ 42 Table 4: Potential transferability across selected materials containing Na, Ni, Ti, and O. (The mean absolute errors of the Buckingham and Morse potentials are 1.4 and 2.8%, respectively). 43 Table 5: Polarizable interatomic potential parameters.................................................................. 69 Table 6: Structures used in the training and validation set ........................................................... 88 Table 7: Parameters of symmetry functions for all atoms ............................................................ 89 vii LIST OF FIGURES Figure 1: The schematic of a sodium-ion battery ........................................................................... 6 Figure 2: Schematics of (a)P2 and (b)O3 structure for NaxTMO2.................................................. 9 Figure 3: Schematics of structures of (a) olivine-type NaFePO4, (b) Na3V2(PO4)3 and (c) Na2FePO4F .................................................................................................................................... 11 Figure 4: Schematics of the NaFeF3 structure .............................................................................. 13 Figure 5: Structure of NaxM1[M2(CN)6]y .................................................................................... 14 Figure 6: The distribution of Na ions in edge-sharing and face-sharing sites in P2- Na2x[NixTi1- x]O2 viewed (a) along the ab plane (b) along the c axis................................................................ 17 Figure 7: The schematics of non-polarizable fixed charge model and polarization models including fluctuating charge model, core-shell model and induced dipole model142 ................................... 24 Figure 8: The schematic of a neural network architecture to construct interatomic potentials. Gi are the atomic descriptors. Eat is the atomic energy. 𝑦𝑖𝑗 is the output value at node i in layer j. 𝑎𝑘, 𝑖𝑗 − 1, 𝑗 is the weights connecting node k in layer j-1 to the node i in the layer j. 𝑏𝑖𝑗 is the bias for node i in layer j. ....................................................................................................................... 27 Figure 9: Rietveld refinement of Na2/3[Ni1/3Ti2/3]O2 powders characterized at 15K using detector bank 1 (a) and bank 4 (b), at 300K using bank 1 (c) and bank 4 (d). ........................................... 34 Figure 10: 3D nuclear density maps and 2D slices on the {001} plane of Na. 3D isosurface level is 0.5 Å-3 while 2D isosurface level is between 0 and 1 Å-3. Results from the Rietveld refinement at 15 K, Buckingham potential, and Morse potential are shown for each column. ...................... 37 Figure 11: 3D nuclear density maps and 2D slices on the {001} plane of Ni. 3D isosurface level is 0.5 Å-3 while 2D isosurface level is between 0 and 1 Å-3. Results from the Rietveld refinement at 15 K, Buckingham potential, and Morse potential are shown for each column. ...................... 38 Figure 12: 3D nuclear density maps and 2D slices on the {001} plane of Ti. 3D isosurface level is 0.5 Å-3 while 2D isosurface level is between 0 and 1 Å-3. Results from the Rietveld refinement at 15 K, Buckingham potential, and Morse potential are shown for each column. ...................... 39 Figure 13: 3D nuclear density maps and 2D slices on the {001} plane of O. 3D isosurface level is 0.5 Å-3 while 2D isosurface level is between 0 and 1 Å-3. Results from the Rietveld refinement at 15 K, Buckingham potential, and Morse potential are shown for each column. ...................... 40 Figure 14: Energy distribution curve from the Buckingham and Morse potential for 3x2x1 supercells with four different scenarios of atom distribution: Na equal and Ni equal, Na random and Ni equal, Na equal and Ni random, and Na random and Ni random. .................................... 45 viii Figure 15: Example structures of (a) P2 and (b, c) O2 stacking for the supercell used in the atomistic simulation. ..................................................................................................................... 47 Figure 16: Calculated lattice parameters of P2-Nax[Ni1/3Ti2/3]O2 vs. sodium content x (black squares), compared to an XRD study by Fielden et al.182............................................................. 51 Figure 17: (a) The distribution of Ni-O bond lengths (b) an example of a JT-inactive Ni ion (c) an example of a JT-active Ni ion. ...................................................................................................... 53 Figure 18: Electron density difference (positive in yellow and negative in cyan) upon (a) Na insertion, (b) Na extraction. The isosurface level is 0.005 Å-3. The Na, Ni, Ti and O atoms are represented by yellow, gray, blue and red spheres, respectively. ................................................. 55 Figure 19: The atomic charge distribution for each atom at different sodium content x in Nax[Ni1/3Ti2/3]O2 based on (a) DDEC model; (b) Bader model.................................................... 56 Figure 20: The density of states of Nax[Ni1/3Ti2/3]O2 with (a)x=0.44, (b) x=0.67, (c) x=0.89. The Fermi energy is set to be zero. ...................................................................................................... 57 Figure 21: The electronic conductivity of Nax[Ni1/3Ti2/3]O2 at different x from the Kubo- Greenwood approach .................................................................................................................... 59 Figure 22: The electrical conductivity of Nax[Ni1/3Ti2/3]O2 at different x. The left, middle, and right panel represents results for Na5/9[Ni1/3Ti2/3]O2, Na2/3[Ni1/3Ti2/3]O2, Na7/9[Ni1/3Ti2/3]O2, respectively. Yellow, green, and purple points represent the calculated electrical conductivity along x, y, and z direction, respectively. .................................................................................................................. 61 Figure 23: Experimental QENS spectra S(Q,E) for various temperatures at Q=0.3 Å-1. ............. 65 Figure 24: (a) Fit of the QENS spectra at 700 K and Q=0.3 Å-1. (b) The HWHM of the Lorentzian function at 700 K as a function of Q2, fitted to the Fickian model at small Q (red line) and the SS model (black line). ........................................................................................................................ 66 Figure 25: Diffusional properties from QENS at 450 to 700K: (a) The Fickian self-diffusivity of Na. (b) The residence time and (c) jump length of Na diffusion with the SS model. .................. 68 Figure 26: The comparison of IP forces (blue circles) and DFT forces (black line) on each type of atom............................................................................................................................................... 70 Figure 27: Lattice parameters of pristine P2-Na2/3[Ni1/3Ti2/3]O2 at different temperatures obtained from neutron diffraction by Shanmugam et al178 (red stars), DFT-based MD (blue circles), and IP- based MD (green squares). ........................................................................................................... 72 Figure 28: The nuclear density map of Na during 500 ps simulation at 1100 K. (a) 3D view with an isosurface level of 0.2 Å-3, (b) 2D slice on the (001) lattice plane with an isosurface level of 0~0.5 Å-3. “E” and “F” represent edge-share sites and face-share sites ...................................... 73 ix Figure 29: (a) The I(Q,t) at 1100 K (b) Γ vs. Q2 (black circles), with the CE fit (black line) and linear fit for the three smallest Q (red line). ................................................................................. 75 Figure 30: Diffusional properties from MD simulations at 900 to 1200 K: (a) The Fickian self- diffusivity of Na. (b) The residence time and (c) jump length of Na diffusion with the Chudley- Elliott model.................................................................................................................................. 76 Figure 31: The real part of coherent charge-current correlation at 1100 K in (a) full frequency span. (b) the low-frequency region. The blue and red curves represent the in-plane and through-plan ionic conduction, respectively. ..................................................................................................... 78 Figure 32: The calculated in-plane ionic conductivity of Na2/3[Ni1/3Ti2/3]O2 compared with experimental values from Shanmugam et al.104, Shin et al. 103, Smirnova et al.200 ...................... 79 Figure 33: The nuclear density map of Na from 20 ps MD simulation trajectory at 1100 K. (a) 3D view with an isosurface level of 0.2 Å-3, (b) 2D slice on the (001) lattice plane with an isosurface level of 0~0.5 Å-3. ......................................................................................................................... 80 Figure 34: Calculated Fickian diffusivity from FPMD simulation (red) compared to results from IPMD(green) and QENS(blue). .................................................................................................... 81 Figure 35: Calculated Fickian diffusivity from FPMD simulation at 900 K for different sodium content. .......................................................................................................................................... 82 Figure 36: Calculated in-plane ionic conductivity of Na2/3Ni1/3Ti2/3]O2 compared with computational values from Chen et al.202 and experimental data from Shanmugam et al.104, Shin et al.103, and Smirnova et al.200.......................................................................................................... 84 Figure 37: The calculated in-plane ionic conductivity from FPMD simulation at 900 K for different sodium content. ............................................................................................................................. 85 Figure 38: The comparison of NN-IP energies (red circles for P2 structures and blue circles for O3 structures) and DFT energies (black line) for the training set. ..................................................... 91 Figure 39: The comparison of NN-IP forces (blue circles) and DFT energies (black line) for the test set for each type of atom in (a) P2 structures and (b) O3 structures. ..................................... 92 Figure 40: Lattice parameters of P2-Na2/3[Ni1/3Ti2/3]O2 at different temperatures obtained from NN-IP based MD simulations (yellow circles) compared to results from neutron diffraction by Shanmugam et al178 (red stars), DFT-based MD (blue diamonds), and IP-based MD (green squares). ....................................................................................................................................................... 94 Figure 41: Lattice parameters of O3 Na0.83[Ni0.42Ti0.58]O2 at different temperatures obtained from NN-IP based MD simulations (yellow circles) compared to results extrapolated from XRD by Fielden et al.105 (red stars) ............................................................................................................. 95 Figure 42: Fickian diffusivity of Na in P2-Na2/3[Ni1/3Ti2/3]O2 and O3-Na0.83[Ni0.42Ti0.58]O2 calculated from IP-NN based MD simulations (yellow squares for P2-Na2/3[Ni1/3Ti2/3]O2 and green x squares for O3-Na0.83[Ni0.42Ti0.58]O2) compared to results for P2-Na2/3[Ni1/3Ti2/3]O2 from FPMD(green squares) and QENS(blue circles). ........................................................................... 96 Figure 43: Calculated in-plane ionic conductivity of P2-Na2/3[Ni1/3Ti2/3]O2 (blue squares) and O3- Na0.83[Ni0.42Ti0.58]O2 (blue circles) from IP-NN based MD simulations compared with results from DFT-MD (black) and experimental data for P2-Na2/3[Ni1/3Ti2/3]O2 and O3-Na0.8[Ni0.4Ti0.6]O2 from Shanmugam et al.104 (red), Shin et al.103 (green), and Smirnova et al.200 (yellow). Results for P2 and O3 structures are represented as squares and circles respectively. ........................................ 98 xi Introduction 1.1 Background With increasing concerns about fossil fuels dependence and greenhouse gas emissions, renewable energy technologies have developed rapidly in recent decades, like solar and wind energy. The rapid development of these intermittent renewable sources brought in new challenges to developing large-scale electrical energy storage systems with low cost, as they are essential for the smooth grid integration and peak load management. Among the energy storage technologies suitable for grid-scale applications, electrochemical storage systems, i.e., batteries, provide a promising option to store the extra energy. Compared to other energy storage technologies like pumped hydroelectric storage and compressed air storage, advantages of batteries include flexibility, low environmental impacts, high round-trip efficiency, and low maintenance cost1,2. Among battery technologies, lithium-ion batteries (LIB) have been dominating the electrochemical energy storage market since the first carbon/LiCoO2 cell was introduced into the market by Sony in 19913, due to their overwhelming advantages of high power density, high energy density, and long lifetime. Lithium-ion batteries have been widely used in portable electric devices like cell phones, tablets, and laptops as well as electric vehicles. However, applications of lithium-ion batteries in large-scale stationary energy storage are likely to be limited by the high cost and limited availability of lithium resources4. Studies based on the analysis of the geological resources of lithium have predicted that the estimated lithium reserve may be insufficient to meet the massive demand for lithium-ion batteries5,6. Furthermore, the low recovery rate of lithium may result in the problem that the recoverable lithium is unable to satisfy the growing lithium demand7. Even if lithium resources are abundant, lithium-ion batteries are not an economically feasible option for the grid-scale energy storage. Considering the insufficient abundance and geographical 1 concentration of lithium reserves, as well as the relatively high cost, the search for an earth- abundant and affordable alternative material to lithium is essential for grid-scale electrochemical energy storage. Sodium is also in the series of alkali metal elements in the periodic table as lithium, thus sharing some similar chemical properties with lithium. Consequently, as one of the most earth- abundant elements, sodium has attracted researchers’ attention as a potential alternative to lithium for battery applications. Sodium-ion batteries, where sodium ions serve as charge carriers, have been studied as early as the late 1970s8-10. However, the research development of sodium-ion batteries decelerated as the research focus has been shifted to lithium-ion batteries since a high energy density was reported for lithium-ion batteries11. Until in recent years, with growing interests in grid-scale electrochemical storage systems, sodium-ion batteries became attractive as a cost- effective alternative to lithium-ion batteries for large-scale stationary energy storage applications, because of the low cost and huge natural abundance of sodium resources12,13. Thus, increasing research efforts have been devoted to the study of sodium-ion batteries14-17. 1.2 Sodium-ion batteries vs. lithium-ion batteries 1.2.1 Sodium vs. lithium Table 1 shows the comparison of some properties related to the battery performance between sodium and lithium. As shown in the table, sodium is slightly heavier than lithium with an atomic weight of 23 g/mol compared to the 6.9g/mol atomic weight of lithium. The small atomic weight of lithium has been a significant advantage of lithium batteries, leading to a high gravimetric capacity. Similarly, the larger ion size results in a smaller volumetric capacity of sodium-ion batteries. Although sodium is the lightest and smallest element sharing similar properties with lithium (compared to magnesium and potassium), these slight differences make it 2 hard for sodium-ion batteries to compete with lithium-ion batteries regarding capacity and energy density. The low redox potential of lithium (E0(Li+/Li)= -3.04 V vs. SHE) enables the high voltage and high energy density of lithium-ion batteries. Sodium has a close standard electrode potential of -2.71 V, which is also suitable for battery applications. The theoretical capacity of lithium metal is around three times of sodium metal. With the same redox couple, the difference between theoretical capacities becomes smaller, as shown in the example of NaCoO2 and LiCoO2 in Table 1. Higher atomic weight, lower reducing potential, and larger ion size contribute to the major constraints for the sodium-ion batteries to achieve as high gravimetric and volumetric energy density as lithium-ion batteries. However, as shown in Table 1, sodium-ion batteries showed tremendous advantages over lithium-ion batteries at cost and material abundance. The price for sodium carbonates is around a hundred times lower than the price of lithium carbonates. While sodium resources are almost unlimited everywhere, lithium resources are constrained by insufficient abundance and uneven distribution18. Table 1: The Comparison between sodium and lithium Na Li Atomic Weight 23 g/mol 6.9 g/mol E0 vs. SHE -2.71 V -3.04V Ionic Radius 0.98Å 0.69Å Capacity (metal) 1165 mAh/g 3829 mAh/g Capacity (ACoO2) 235 mAh/g 274 mAh/g Melting Point 97.7 °C 190.5°C Coordination Preference octahedral and prismatic octahedral and tetrahedral 3 Table 1 (cont’d) Price (Carbonates)19 $150/ton $13000/ton Resources19 unlimited 80 million tons The difference between the physical properties of sodium and lithium leads to different chemistries in lithium-ion batteries and sodium-ion batteries. Besides abundance and cost, studies have found that sodium-ion batteries exhibit some other advantages over lithium-ion batteries. With a larger ion size than lithium ions, sodium ions have a preference for octahedral and tetrahedral sites, leading to O- type and P-type structures of the sodium transition metal oxides (these two structures will be discussed in the following section). First principles calculations showed that the migration barrier of sodium ions is lower than that of lithium ions in layered structures20. While the similar sizes of lithium ions and transition metal ions would result in thermodynamic instability of lithium-ion transition metal oxides21, the larger ion size of sodium leads to the larger ionic size difference between sodium and transition metal ions, providing more options of sodium compounds for researchers. For grid-scale battery systems, the shortcomings in capacity and energy density can be made up by increasing the number of cells, where abundance and cost are more critical than the energy density of a single stack. Despite the high competitiveness in energy density of lithium-ion batteries, sodium-ion batteries still show great potential in substituting lithium-ion batteries for large-scale energy storage applications where the energy density is not very crucial, considering the unlimited abundance and low cost of sodium. 4 1.2.2 The working principle of sodium-ion batteries The working mechanism of sodium-ion batteries is almost identical to that of lithium-ion batteries. Both sodium-ion batteries and lithium-ion batteries are among the rechargeable “rocking-chair” type batteries, as the charge carriers, either sodium ions or lithium ions, are shuttling back and forth between the electrodes during charging and discharging processes. The schematic of a sodium-ion battery is shown in Figure 1. A typical sodium-ion battery consists of two electrodes, positive and negative, both made with sodium insertion materials, separated by the electrolyte that allows sodium ions but not electrons to travel through. A voltage is produced from the chemical potential difference between the positive and negative electrode of the cell. Such potential difference drives sodium ions to be extracted from the negative electrode, pass through the electrolyte and be inserted into the positive electrode. In the meantime, electrons are lost at the negative electrode and accepted by the positive electrode, traveling through the external circuit, producing an electronic current. In such a process which we call discharge, the chemical energy is converted to the electric energy. The reverse process, when an external voltage is applied to drive sodiums ions and electrons to move in an opposite direction from the positive electrode to the negative electrode is called charge, where the electric energy is converted into chemical energy and stored in the batteries. 5 Figure 1: The schematic of a sodium-ion battery 1.3 Electrode materials for sodium-ion batteries The selection of electrode material is critical for the development of sodium-ion batteries as the performance of the cell is largely dependent on the properties of electrode materials. The basic requirement for the operation of the battery is that the sodium ions can be inserted or extracted from the electrodes while the stable structures can be still maintained. The insertion or the extraction of the Na ions occurs with the reduction or oxidation reaction of some other elements in the electrode material, which in most cases are multivalent transition metals. 6 Energy density is important for batteries as it determines how much energy can be stored with a certain volume or mass. Since the energy density is the product of the operating voltage and the specific capacity, higher energy density can be accomplished by improving these two factors. The operating voltage of the cell is theoretically proportional to the chemical potential difference between the positive and negative electrode materials according to V = −∆μ/zF , where Δμ is the chemical potential difference between the positive and negative electrodes, z the number of elementary charges carried by ions (1 for Na), F the Faraday constant. To achieve higher operating voltages and higher energy densities, a positive electrode material with a higher potential and a negative electrode material with a lower potential are desired. The theoretical specific capacity of the electrode is determined by how many charge carriers are contained in the unit mass material. As a result, the more Na ions can be inserted into or extracted from the unit mass electrode, the higher the theoretical specific capacity can be. However, the actual capacity of the cell that can be achieved is usually much less than the theoretical one as it is limited by the amount of sodium that can be extracted and inserted reversibly without irreversible phase transformations. Thus, the good structural stability of the electrode material at different sodium content is essential for high reversible capacity as well as long cycle life and good capacity retention. High rate capability is also required for a battery to deliver high power and to charge fast. The rate capability is also closely correlated to the properties of the electrode material as it can be limited by sodium diffusion coefficients, sodium diffusion pathways, ionic conductivity, and electronic conductivity of the electrodes. Better rate performance can be achieved by improving the ion and electron transport in the electrode material. In addition, safety, cost, and environmental-friendliness are also important factors to consider when choosing electrode materials for sodium-ion batteries. 7 For the development of sodium-ion batteries with high performance, the exploration and design of electrode materials that can fulfill these requirements are essential. Tremendous research efforts have been devoted to the search for electrode materials for better performance. A wide range of materials have been proposed as potential positive and negative electrode materials for sodium-ion batteries. 1.3.1 Positive electrode materials Lithium transition metal oxides with layered structure(LixTMO2, TM stands for the transition metals like Mn, Co, Ni, V) have been widely investigated and used as positive electrode materials for lithium-ion batteries3 because of their high energy densities. Thus, their sodium analogs NaxTMO2 have attracted research interests. It has been found that they are promising candidates for electrode materials as they exhibited high ionic conductivity and large reversible capacities22-24. Moreover, sodium transition metal oxides showed advantages of safety over their lithium counterparts as the capacity loss caused by spinel formation is less likely to happen in these materials, mainly due to the larger ionic radius of sodium25,26. The structures of sodium transition metal oxides mainly fall into two categories: P-type and O-type according to Delmas27, with sodium atoms occupying prismatic and octahedral sites, respectively. A number of studies have been conducted on P2-type (2 TM layers with ABBA stacking and Na occupying prismatic sites) and O3-type (3 TM layers with ABCABC stacking and Na occupying octahedral sites) NaxTMO2. Structures of P2 and O3 types are shown in Figure 2. Both P2 and O3 type layered NaxTMO2 compounds have been investigated extensively as potential electrode materials for sodium-ion batteries. LiCoO3, the commercialized cathode material in lithium-ion batteries, is one of the representative materials of the O3 structure. However, the O3 Na materials exhibit different properties from the O3 Li materials as there is an O3-O’3-P3 8 transformation during Na extraction28,29. The P2-type materials tend to present higher capacity and longer cycle life than O3 type materials, as the sodium ions are more stable in larger prismatic sites than in octahedral sites, despite the fact that the O3 type Nax[TM]O2 can be synthesized at a lower temperature compared to the P2 analogs and possess more sodium based on the 30,31 stoichiometry . Furthermore, the sodium ions showed higher diffusion and lower diffusion barriers in P2-type NaxTMO2, leading to a better rate performance 32,33. (a) (b) Figure 2: Schematics of (a)P2 and (b)O3 structure for NaxTMO2 Early studies on NaxTMO2 started dating back to the 1980s when the electrochemical properties of NaxCoO2 were studied28. With resurgent research interests in sodium-ion batteries, NaxCoO2 has been revisited by researchers as electrode materials for sodium-ion batteries. Comparing to their analogs LixCoO2, which are widely used as cathode materials in Li-ion batteries, NaxCoO2 presents more complex phase transitions during the insertion and extraction of sodium ions, with multiple voltage drops and plateaus in the electrochemical curve34. The phase diagram of P2-type NaxCoO2 was studied by Berthelot et al. with a combination of in-situ x-ray diffraction and electrochemical methods35. A higher capacity but worse cycling stability was observed for the 9 sodium manganese oxides P2-NaxMnO2. The O3-type NaMnO2 studied by Ma et al.36 showed a capacity of 185 mAh/g for the first cycle but a large capacity loss after several cycles. A capacity of 140 mAh/g in P2-type Na00.6MnO2 was reported by Caballero et al.37. However, the capacity faded after a few cycles. The poor structure stability of NaxMnO2 might be related to the Jahn- Teller distortion of Mn3+. Improved capacity retention was observed with different synthesis routes and electrolytes, while the cycling stability can be further improved by doping Mg into P2- Na0.67MnO2. 38. While most layered sodium transition metal oxides with a single type of transition metal may suffer from low capacity and cycling stability, attempts have been made to increase the cycling and structural stability by introducing additional transition metal elements into the material. The mixing of transition metals can improve the structural stability and affect the sodium diffusion barriers in NaxTMO2 39,40. A variety of binary systems were proposed and studied as promising candidates for positive electrodes such as the combination of Fe/Mn41-44, Mg/Mn45,46, Ni/Mn47-49. Some ternary systems were also investigated such as NaNi1/3Mn1/3Co1/3O250, Na2/3Ni1/3Mn2/3- 51 xTixO2 , Na0.8Ni0.3Co0.2Ti0.5O252, NaFe0.4Ni0.3Ti0.3O253 and Na0.67Mn0.65Fe0.20Ni0.15O254. Besides the layered sodium transition metal oxide, other materials were also studied as potential positive electrodes for sodium-ion batteries, for example, the phosphates materials, which showed better stability than the oxide materials. Moreover, phosphates materials present higher voltage than that of the metal oxides, mainly due to the inductive effect of PO43- polyanion55,56. Among the phosphates materials, olivine-type NaFePO4 (Figure 3(a)) has attracted the most attention for the high theoretical specific capacity it could achieve as well as the success of LiFePO4 in commercialized lithium-ion batteries. Studies have found that olivine-type NaFePO4 can be synthesized through the ion exchange from LiFePO457. Ali et al. 58 developed a high- 10 performance cathode for sodium-ion batteries using olivine-type NaFePO4, obtaining an excellent retention capacity of 94% after 100 cycles and a reversible capacity of 142 mAh/g. The NASICON-type phosphate materials are also promising options for positive electrodes with a flat voltage plateau and an open framework to enhance sodium transport. A storage capacity of 107 mAh/g and 93% capacity retention after 80 cycles were achieved using Na3V2(PO4)3 (Figure 3(b)) with as cathode and carbon nanocomposites as anodes for sodium-ion batteries in a recent study by Jian et al59. The electrochemical performance of Na3V2(PO4)3 can be further improved by combining it with carbon materials60,61. Ellis et al62. investigated the possibility of using fluorophosphate Na2FePO4F ((Figure 3(c))) as cathodes in Na-ion batteries. With two-dimensional sodium transport pathways, it delivered a capacity of 135 mAh/g, which is 85% of the theoretical value. (a) (b) Figure 3: Schematics of structures of (a) olivine-type NaFePO4, (b) Na3V2(PO4)3 and (c) Na2FePO4F 11 Figure 3 (cont’d) (c) Fluoride compounds like NaMF3 (M stands for the transition metal like Fe, Mn, Co, Ni, V) or MF3 have also been studied as positive electrode materials with high voltage resulting from the highly ionic metal-ligand bonds. The initial study on NaFeF3 (Figure 4) showed an initial discharge capacity of 130 mAh g−1 while both NaMnF3 and NaNiF3 both showed poor electrochemical performance in a Na/Na+ cell63. Kitajou et al.64 synthesized highly crystalline NaFeF3, with the obtained NaFeF3 achieved an initial discharge capacity of 197 mAh/g in the voltage range of 1.5 V to 4.5 V, with good Coulombic efficiency. The performance of perovskite-type metal trifluorides for both Li and Na batteries, including FeF3, TiF3, VF3, MnF3, and CoF3 was studied by Nishijima et al.65. The FeF3 system, with a high theoretical capacity of 712 mAh/g, showed a reversible capacity of 100 mAh/g with a discharge voltage of 2.2 V. FeF3·xH2O/graphene showed improved electrochemical performance with the addition of graphene 66. 12 Figure 4: Schematics of the NaFeF3 structure Other materials being studied as positive electrode materials for sodium-ion batteries include Prussian blue analogs with a chemical formula of NaxM1[M2(CN)6]y·nH2O. The open framework of Prussian blue analogs promises efficient sodium diffusion during cycling. A few systems were investigated such as Na0.61Fe[Fe(CN)6]0.9467, Na2Mn[Mn(CN)6]68, Na2Zn3[Fe(CN)6]2·nH2O69, Na2Mn[Fe(CN)6]·nH2O70. It is worth mentioning that some organic materials are also attractive candidates for positive electrodes with high gravimetric energy density and flexibility. A reversible capacity of 484 mAh/g was achieved with disodium rhodizonate (Na2C6O6)71. The sodium 4,4′-stilbene-dicarboxylate electrode delivered a reversible capacity of 105 mAh/g at a current density of 2 A/g72. 13 Figure 5: Structure of NaxM1[M2(CN)6]y 1.3.2 Negative electrode materials Carbon-based materials are common materials used as anodes in batteries with high electrical conductivity and low cost. Although graphite has been widely used as anode materials in lithium-ion batteries, it is not suitable for sodium-ion batteries as sodium atoms could hardly be inserted into graphitic materials73. The potential of other carbon-based materials being used as anodes in sodium-ion batteries has been investigated. Hard carbon has been identified as a promising candidate for anode materials as similar reversible insertion/extraction mechanisms have been observed for sodium and lithium in hard carbon73. A high reversible capacity of 300 mAh/g was obtained by using hard carbon as the anode material for sodium-ion batteries in a study by Stevens et al74. A study by Komaba et al.75 attained a high overall capacity of more than 200 mAh/g and excellent reversibility using hard carbon as the anode and layered NaNi2Mn0.5O2 as cathodes for the sodium-ion battery. With a modified synthesis method, a large reversible capacity of 478 mAh/g could be achieved with hard carbon76. A study by Ding et al.77 showed that the carbon nanosheets with larger intergraphene spacing could achieve a stable capacity of 298mAh/g and excellent cycle retention. Zhu et al.78 developed a novel material using 30 nm Sb nanoparticles 14 encapsulated in 400 nm carbon fibers, leading to excellent cycle stability with only 0.06% decay per cycle and a high overall capacity with an initial capacity of 422 mAh/g. Some transition metal oxides are also promising negative electrode materials for sodium- ion batteries with high capacity. Jiang et al.79 investigated the performance of sodium-ion batteries using a series of transition metal oxides as anodes, including Fe2O3, Mn3O4, Co3O4, and NiO. The results showed that the cell with Fe2O3 anode exhibited a large capacity of 385 mAh/g after 200 cycles. Anatase titanium dioxide TiO2 showed ∼150 mAh/g over 100 cycles as the negative electrode in Na cells80. Some other titanium-based compounds exhibit low potential, thus suitable for negative electrode materials. For example, Na2Ti3O7 with the layered structure was identified as a promising negative electrode material for sodium-ion batteries with an average potential of 0.3 V vs Na+/Na081. A charge capacity of 188 mAh/g of the cell was obtained using synthesized Na2Ti3O7/carbon black composite as the negative electrode and sodium metal as the counter electrode82. O3-type NaTiO2 delivered a reversible capacity of 152 mAh/g, corresponding to ~0.5 Na reversible intercalation83. The titanium-containing sodium transition metal oxides P2-type Na0.66[Li0.22Ti0.78]O2 and O3‐type Na0.66Mg0.34Ti0.66O2 as well as NASICON-type NaTi2(PO4)3 have also been demonstrated to be promising candidates as negative electrode materials for sodium-ion batteries 84-86. Alloy-based negative electrode materials attract research interests as they can achieve a much higher theoretical capacity compared to intercalation‐based materials. Sn, Si, P, Sb, and Ge- based materials have been studied for sodium-ion batteries87-91. However, the alloy-based negative electrode materials usually suffer from large volume changes during cycling. 15 1.3.3 Bi-functional electrode materials In most cases, materials with either high or low potential vs Na electrodes are considered to be suitable for positive or negative electrode materials of sodium-ion batteries, respectively. Some intercalation-based electrode materials are found to be able to function as either positive or negative electrodes, usually due to the coexistence of both high and low redox couples. Such bi- functional electrode materials enable the construction of symmetric cells where the positive and negative electrodes are made of the same material, bring additional advantages of simpler design and lower manufacturing cost. A few bi-functional electrode materials for sodium-ion batteries have been proposed and studied, including layered sodium transition metal type materials with both P2 and O3 structures such as Na0.6[Cr0.6Ti0.4]O292, Na0.66[Ni0.17Co0.17Ti0.66]O293, Na0.5[Ni0.25Mn0.75]O294, NASICON-type materials Na3V2(PO4)395,96, and organic materials Na4C8H2O697. 1.4 The Na2x[NixTi1-x]O2 group Among the layered sodium transition metal oxide materials, the Na2x[NixTi1-x]O2 series have been identified as promising bi-functional electrode materials for sodium-ion batteries with coexistence of high redox potential (E = 3.7 V vs. Na/Na+) for Ni and low redox potential (E = 0.7 V vs. Na/Na+) for Ti. The P2-Na2/3[Ni1/3Ti2/3]O2 showed good electrochemical performance, delivering a reversible capacity around 75 mAh/g as either cathodes or anodes98. The symmetric cell built with O3-type Na0.8[Ni0.4Ti0.6]O2 as both positive and negative electrode delivered a reversible discharge capacity of 85 mAh/g with good rate capability99. The structures of P2-type and O3-type Na2x[NixTi1-x]O2 are shown in Figure 2. As discussed above, in P2-type Na2x[NixTi1-x]O2, Ni/Ti atoms are distributed in the 2 transition metal layers in the unit cell with Na ions occupying the Prismatic sites. In P2 Na2x[NixTi1-x]O2, the Na 16 ions are distributed in two sites, the edge and face-sharing sites (Na_e and Na_f in Figure 6)), depending on whether Na prisms are sharing edges or faces with the TMO6 octahedra. In O3 type Na2x[NixTi1-x]O2, Ni/Ti atoms are distributing in the 3 transition metal layers in the unit cell with Na ions occupying the Octahedral sites. In both P2 and O3 Na2x[NixTi1-x]O2, no evidence of the ordering of Ni/Ti or Na/vacancy was found from powder X-ray diffraction experiments100,101. The disordered distribution of Ni/Ti and Na/vacancy (especially for P2-type with Na occupying two different sites) lead to complex local structural features of Na2x[NixTi1-x]O2 and also raise questions on how these structural features affect the dynamic and electronic properties of the material. (a) (b) Figure 6: The distribution of Na ions in edge-sharing and face-sharing sites in P2- Na2x[NixTi1-x]O2 viewed (a) along the ab plane (b) along the c axis. A few studies have been performed to study the properties of Na2x[NixTi1-x]O2. Electrochemical experiments have been conducted for both P2 and O3 Na2x[NixTi1-x]O2 to assess the cycling performance98,102 as well as the transport properties103,104. The structural stability of Na2x[NixTi1-x]O2 during cycling was investigated using in situ x-ray diffractions105,106. However, the fundamental understandings of this material series and the underlying mechanism during the electrochemical processes are still limited. 17 1.5 Thesis overview With the general background information provided above, including an introduction of the sodium-ion batteries, a discussion of its significance to address the environmental challenges and a review of the research progress on some electrode materials for sodium-ion batteries, this thesis aims to seek the answers to the fundamental research questions related to Na2x[NixTi1-x]O2 as bi- functional electrode materials for sodium-ion batteries. For example, how would the complex local structure caused by disordered Na/vacancy and Ni/Ti impact the dynamics of Na in the material? How would the atomic and electronic structure change during the electrochemical processes when the Na ions are being inserted and extracted from the material? What redox reactions would occur when the material is used as either a positive or negative electrode? How do Na ions diffuse in the material and how to enhance diffusion? How does the diffusion mechanism differ in P2 and O3 structures? How are the atomic-scale features related to the macroscopic electrochemical properties and further the battery performance? These questions will be addressed in the following chapters. The answers to them not only help us to gain a better atomic-level understanding of this material but also provide insights on further improving the performance of similar materials as well as the discovery and design of the new electrode materials. CHAPTER 2 provides a brief introduction of the theoretical backgrounds of the computational and experimental methods and how they are applied in this thesis. CHAPTER 3 investigates the average and local structure of P2-Na2/3[Ni1/3Ti2/3]O2 using a combination of Rietveld refinement on neutron diffraction datasets and atomistic simulation. Two potential sets are applied for the atomistic simulation and the energetics of atomic distribution and ordering are discussed. In CHAPTER 4, the structural and electronic properties of P2-Nax[Ni1/3Ti2/3]O2 are investigated using density functional theory. The structural and electronic change of P2- 18 Nax[Ni1/3Ti2/3]O2 are examined as Na ions inserted and extracted from the material during charging/discharging. The density of states, electron density distribution, and electronic conductivities are calculated and discussed in this chapter. In CHAPTER 5 the diffusion mechanism as well as ionic conductivity P2-Na2/3[Ni1/3Ti2/3]O2 are investigated with a combination of experimental and computational techniques. The quasi-elastic neutron scattering (QENS) experiments and molecular dynamics (MD) simulations based on interatomic potential (IP) and density functional theory (DFT) are performed to identify the diffusion mechanism. CHAPTER 6 extends the analysis to the O3 system with the use of machine learning based interatomic potential models. CHAPTER 7 summarizes the work in this thesis and discusses the future directions based on this work. 19 Methods 2.1 Computational methods 2.1.1 Density functional theory Density functional theory (DFT)107 is a computational method to investigate the ground state electronic structures of materials and has been a useful tool to study properties of electrode materials including the phase stability108, defect energetics109, order/disorder phenomenon110, 20 conduction pathways, and migration activation energies . In Kohn-Sham111 DFT calculations, the interacting many-body system is simplified to non-interacting single-electron systems with an effective potential Veff, resulting in a set of single-electron Schrödinger-like equations as in Eq. 1. ħ2 2 Eq. 1 [− ∇ + 𝑉𝑒𝑓𝑓 (𝑟)] 𝜓𝑖 (𝑟) = 𝜀𝑖 𝜓𝑖 (𝑟) 2𝑚 Where 𝜓_𝑖 (𝒓) are the Kohn-Sham orbitals, which relate to the spatially dependent electron density 𝑛(𝐫) according to 𝑛(𝐫) = ∑𝑁 2 𝑖=1 |𝜑𝑖 (𝐫)| . The total electron energy E of many-body systems is determined as functionals of 𝑛(𝐫): 𝐸[𝑛(𝐫)] = 𝑇[𝑛(𝐫)] + 𝑉𝑒 𝑛[𝑛(𝐫)] + 𝐸𝐻 [𝑛(𝐫)] + 𝐸𝑥c [𝑛(𝐫)] Eq. 2 Where T is the non-interacting kinetic energy, Ven the electron-nuclei interaction energy, EH the Hartree term for the electron-electron Coulomb interaction, and Exc the exchange- correlation energy. The first three terms in Eq. 2 can be determined according to the following equations: 𝑁 ħ2 𝑇[𝑛(𝐫)] = − ∑ ∫ 𝜓𝑖∗ (𝑟)∇2 𝜓𝑖 (𝑟)𝑑𝑟 Eq. 3 2𝑚 𝑖=1 𝑍𝑒 2 𝑉𝑒𝑛 [𝑛(𝐫)] = − ∑ ∫ 𝑛(𝐫)d𝐫 Eq. 4 |𝐫𝑗 − 𝐑𝑗 | 𝑗 20 𝑒2 ′ 𝜌(𝐫)𝜌(𝐫 ′ ) 𝐸H [𝜌] = ∫ 𝑑𝐫∫ 𝑑𝐫 Eq. 5 2 |𝐫 − 𝐫 ′ | The exchange-correlation energy 𝐸𝑥c [𝑛(𝐫)] consists of exchange energy 𝐸𝑥 [𝑛(𝐫)] and correlation energy 𝐸c [𝑛(𝐫)], accounting for the exchange and correlation interactions between electrons, respectively. 𝐸𝑥c [𝑛(𝐫)] is the only unknown term in Eq. 2 and can be approximated by functionals in DFT calculations, for example, local-density approximation (LDA) and generalized gradient approximation (GGA). In LDA, 𝐸𝑥c [𝑛(𝐫)] depends on the local densities only as in Eq. 6: 𝐿𝐷𝐴 Eq. 6 𝐸xc [𝑛(𝐫)] = ∫ 𝑑𝐫𝑛(𝐫)𝜖xc (𝑛(𝐫)) where 𝜖xc is the exchange-correlation energy of homogeneous electron gas of density n(r). In GGA, 𝐸𝑥c [𝑛(𝐫)] depends on not only the local densities but also the first derivative of densities as in Eq. 7: GGA 𝐸XC [𝑛(𝐫)] = ∫ 𝑓 GGA [𝑛(𝐫), ∇𝑛(𝐫)]𝑑𝐫 Eq. 7 where 𝑓 GGA is a function of electron densities 𝑛(𝐫) and their gradients ∇𝑛(𝐫) . GGA exchange-correlation functionals that are commonly used include PBE112, PW91113, BLYP114,115 116 and PBEsol . We used PBE functionals for all the DFT calculations of the Na2x[NixTi1-x]O2 series in this work. While LDA and GGA approximations have been successfully applied to predict properties for most materials, they can produce large errors when dealing with the electron correlation effects in the 3d localized orbitals of transition metal oxides117. A GGA+U approach118 was employed for Na2x[NixTi1-x]O2 in this work to account for the electron localization of Ni ions. It is also worth mentioning that the standard Kohn-Sham DFT calculations do not include the long-range correlations to treat weak van der Waals (vdW) interactions in layered materials. This can be remedied by adding an atom-pairwise dispersion term Edisp to Eq. 2. The dispersion-corrected PBE 21 functional with Becke-Jonson damping (PBE-D3BJ) functional119,120 was used for layered Na2x[NixTi1-x]O2 in this thesis. With the self-consistent solutions calculated in DFT, the ground state electron density for minimized structure is obtained. The topological analysis of electron density can be utilized to investigate materials properties, for example, atomic charges, bond orders, atomic energies, and atomic volumes. The common models to assign electron densities to each atom include Density Derived Electrostatic and Chemical (DDEC) charges121, Bader charges122, Hirshfeld charges123, and Mulliken charges124. Considering that the Hirshfeld charge model tends to underestimate the charge125,126 and the Mulliken charge model is sensitive to the basis sets127,128, DDEC6 and Bader models are applied to calculate the atomic charges in this thesis. The DDEC charge model assigns electron densities to atoms by reproducing the chemical states while the Bader charge model assigns the charges based on the zero flux surfaces. 2.1.2 Interatomic potential model While the DFT calculations provide relatively accurate evaluations of energies and forces, the high computational cost has limited the time and length scale accessible to DFT. Compared to DFT calculations, the interatomic potential (IP) model with much lower costs can be applied to investigate systems with hundreds of thousands of atoms and up to microseconds. The interatomic potential model describes the interatomic interactions as a function of the atomic positions thus the potential energy of a system U is expressed as Eq. 8: 𝑈 = ∑ 𝑈1,𝑖 (𝒓𝑖 ) + ∑ ∑ 𝑈2,(𝑖,𝑗) (𝒓𝑖 , 𝒓𝑗 ) 𝑖 𝑖 𝑖<𝑗 Eq. 8 + ∑ ∑ ∑ 𝑈3,(𝑖,𝑗,𝑘) (𝒓𝑖 , 𝒓𝑗 , 𝒓𝑘 ) + ⋯ 𝑖 𝑖<𝑗 𝑗<𝑘 22 where U1, U2, U3 are the one-body, two-body and three-body terms, r the positions of the atoms. 2.1.2.1 Empirical interatomic potential model The empirical IP model is based on analytical functions that are derived from physical principles. Parameters for these analytical functions in the empirical IP model can be obtained either from the first-principles results129 or based on the empirical fit to experimental data130,131. A widely-used empirical interatomic potential model is the pair potential Lenard-Jones (LJ)132 potential, which defines the potential U as a function of the distance between two particles r with two parameters 𝜀 and 𝜎: 𝜎 12 𝜎 6 U Lennard−Jones (𝑟) = 4𝜀 [( ) − ( ) ] Eq. 9 𝑟 𝑟 The LJ potential consists of a short-range repulsive term (r-12) and a long-range attractive term (r-6). A wide range of different IP models have been developed to address different systems 133 and different problems, for example, the embedded atom method (EAM) and Finnis-Sinclair potentials134 for metallic materials, Tersoff135 and Stillinger–Weber136 for covalent materials, ReaxFF137 and Charge Optimized Many Body potential (COMB)138 as reactive potentials for more complex systems. For ionic materials like oxides materials Na2x[NixTi1-x]O2 studied in this thesis, pairwise potentials are commonly used, typically consisting of the long-range electrostatic interactions defined by the Coulombic law and the short-range interactions for the repulsion between atoms. For the electrostatic interactions, the atoms can be represented as either fixed point charges at atomic centers or polarization models, with the former neglecting the effect of polarization. Classical polarization models include the core-shell (or Drude oscillator) model139, induced-dipole model140, and fluctuation charge model141. The schematics of the non-polarizable model and these 23 polarization models are shown in Figure 7142 In the core-shell model, each atom is split into two charged sites, core and shell, with a fixed total charge. The shell is connected to the core atom with a harmonic spring. In the induced dipole model, the polarizable atoms are represented as induced dipoles with an induced dipole moment μ proportional to the external electric field. In the fluctuation charge model or the chemical potential equilibration model, the atomic charges can be changed during the simulation to are redistributed to make the electronegativity equal to the chemical potential. Figure 7: The schematics of non-polarizable fixed charge model and polarization models including fluctuating charge model, core-shell model and induced dipole model142 The core-shell model and induced dipole model are applied in this thesis. The common models to describe the short-range pairwise interactions include Buckingham143 and Morse144 potential model used in this work, with the potentials expressed as the following equations: 𝑈𝐵𝑢𝑐𝑘𝑖𝑛𝑔ℎ𝑎𝑚 (𝑟) = 𝐴𝑒 (−𝑟/𝜌) − 𝐶𝑟 (−6) Eq. 10 2 𝑈 𝑀𝑜𝑟𝑠𝑒 (𝑟) = 𝐷 [(1 − 𝑒 (−𝑎(𝑟−𝑟0)) ) − 1] Eq. 11 where r is the distance between two interacting atoms, A, C, 𝜌, D, r0 are the parameters in the models. 24 2.1.2.2 Machine-learning based interatomic potential model Unlike empirical interatomic potential models, machine learning (ML) based potentials do not use physics-derived functional forms to describe the atomic interactions but mathematical functions to represent the potential energy surface. The ML-based IP model is constructed using machine learning techniques with a reference date set containing accurate energies and forces for different configurations. The reference data set is typically generated with quantum mechanical calculations such as DFT. With the reference data, descriptors representing the local environment of each atom in a mathematical way are used as inputs for the machine learning process. Commonly used atomic descriptors include atom-centered symmetry functions (ACSF)145, smooth overlap of atomic positions (SOAP)146, Coulomb matrix147, bispectrum of the atomic neighbor density148. A few approaches to construct ML-based potentials have been developed with different descriptors and machine learning algorisms, such as neural network (NN) potentials149, Gaussian approximation potentials (GAP)148, spectral neighbor analysis potentials (SNAP)150 and moment tensor potentials151. NN potential models with atom-centered symmetry functions (ACSF) were developed for Na2x[NixTi1-x]O2 in this thesis. Both radial symmetry functions 𝐺𝑖2 (Eq. 12) and angular symmetry functions 𝐺𝑖4 (Eq. 13) are used as descriptors. 2 𝐺𝑖2 = ∑ 𝑒 −𝜂(𝑅𝑖𝑗−𝑅𝑠 ) ⋅ 𝑓c (𝑅𝑖𝑗 ) Eq. 12 𝑗 all 𝜁 2 2 2 𝐺𝑖4 = 21−𝜁 ∑ (1 + 𝜆cos 𝜃𝑖𝑗𝑘 ) ⋅ 𝑒 −𝜂(𝑅𝑖𝑗+𝑅𝑖𝑘+𝑅𝑗𝑘) 𝑗,𝑘≠𝑖 Eq. 13 ⋅ 𝑓c (𝑅𝑖𝑗 ) ⋅ 𝑓c (𝑅𝑖𝑘 ) ⋅ 𝑓c (𝑅𝑗𝑘 ) 25 𝑅𝑖𝑗 is the distance between atoms i and j. 𝜃𝑖𝑗𝑘 is the angle formed by 𝑅𝑖𝑗 and 𝑅𝑖𝑘 , centered at atom i. 𝜂, 𝑅𝑠 , 𝜆 and 𝜁 are parameters in the symmetry functions. 𝑓c (𝑅𝑖𝑗 ) is the cutoff function defined as Eq. 14 with the cutoff radius 𝑅c . 𝜋𝑅𝑖𝑗 0.5 ⋅ [cos ( ) + 1] for 𝑅𝑖𝑗 ≤ 𝑅c 𝑓𝑐 (𝑅𝑖𝑗 ) = { 𝑅c Eq. 14 0 for 𝑅𝑖𝑗 > 𝑅c Neural networks consisting of an input layer, an output layer, and one or more hidden layers in between are used to construct NN potentials. For each atom, a set of atomic descriptors {Gi} is supplied in the input layer, then passed to the hidden layers, to calculate the energy contribution of this atom in the output layer. The schematic of such neural network architecture with two hidden 𝑗 layers is shown in Figure 8. At each node i in the layer j, the value 𝑥𝑖 is calculated from a linear 𝑗−1 combination of the output values 𝑦𝑘 at each connected node k in the preceding layer j-1 as in 𝑗 𝑗 Eq. 15. A nonlinear activation function 𝑓𝑖 is applied to yield 𝑦𝑖 at node i in layer j through 𝑗 𝑗 𝑗 𝑦𝑖 = 𝑓𝑖 (𝑥𝑖 ). 𝑁𝑗−1 𝑗 𝑗 𝑗−1,𝑗 𝑗−1 𝑥𝑖 = 𝑏𝑖 + ∑ 𝑎𝑘,𝑖 ⋅ 𝑦𝑘 Eq. 15 𝑘=1 Weight parameters {a} and the bias weights {b} are optimized during the training process to minimize the root mean squared error (RMSE) of the energies and forces of the train set. 26 Figure 8: The schematic of a neural network architecture to construct interatomic 𝑗 potentials. Gi are the atomic descriptors. Eat is the atomic energy. 𝑦𝑖 is the output value at 𝑗−1,𝑗 node i in layer j. 𝑎𝑘,𝑖 is the weights connecting node k in layer j-1 to the node i in the layer j. 𝑗 𝑏𝑖 is the bias for node i in layer j. 2.1.3 Molecular dynamics simulation The molecular dynamics (MD) simulation records the time evolution of the atoms and molecules in the system based on Newton’s equation. Depending on how the potential energy is determined, MD simulations can be carried out with different physical models including the first principles calculations like DFT and parametric models like IP models. The forces are calculated as the derivatives of the potential energy 𝑭 = −∇𝑈(𝒓) . The motions of the particle follow Newton’s law as in classical mechanics: d2 𝑟𝑖 (𝑡) 𝑭𝒊 = mi Eq. 16 d𝑡 2 Where 𝑭𝒊 is the force acting on particle i at position r with mass mi at time t. With initial positions and velocities assigned for each particle in the system, the next positions of particles can 27 be predicted based on Eq. 16 after a short time step Δt. Numerical integration methods are combined with Eq. 16 to evolve the motions and predict the position ri(t+Δt) at time t+Δt based on known position ri(t) at time t. The Verlet integration is commonly used as in Eq. 17 which provides good numerical stability and time-reversibility. 1 d𝑈(𝒓(𝑡)) 2 𝒓(𝑡 + Δ𝑡) = 2𝒓(𝑡) − 𝒓(𝑡 − Δ𝑡) − Δ𝑡 Eq. 17 𝑚 d𝒓 To study the system under different thermodynamic states, MD simulations are performed with different ensembles such as microcanonical ensemble (NVE), canonical ensemble (NVT), and isobaric-isothermal ensemble (NPT). The NVE ensemble corresponds to an isolated system with the number of atoms (N), the volume of the system (V), and the total energy (E) conserved. In the NVT ensemble, N and V along with the temperature of the system (T) are conserved. The system is coupled to a thermostat, which regulates the temperature through the modifications of the velocities of the particles. Popular thermostats used in MD simulations include the Berendsen thermostat152, Anderson thermostat153, and Nosé-Hoover thermostat154,155. The number of atoms (N), the pressure (P), and the temperature of the system (T) are conserved in an NPT ensemble. A barostat is applied along with a thermostat to control the pressure and temperature, such as Berendsen thermostat152 and Parrinello-Rahman barostat156. 2.2 Experimental methods 2.2.1 Sample preparation In this thesis, Na2/3[Ni1/3Ti2/3]O2 powders were all synthesized by solid-state reactions. The solid-state synthesis method has been commonly used to synthesize ceramic materials and has been used to prepare Na2x[NixTi1-x]O2 successfully in previous studies98,157. Stoichiometric amounts of Na2CO3 (≥99.5%,), NiO (99%), and TiO2 (≥99%) precursor powders, all from Sigma- 28 Aldrich were mixed, dry-milled, and fired at 900 oC for 12 hours. 10% excess of Na2CO3 powders were added to compensate for volatility losses during the high temperature process. 2.2.2 Neutron diffraction and Rietveld refinement Neutron diffraction, based on the elastic scattering of neutrons, is a powerful materials characterization technique to extract the structural information of the sample material. As neutrons are scattered by the sample, there is an exchange of momentum ℏ𝑸 = ℏ(𝒌𝒔 − 𝒌𝒊 ), where ℏ is reduced Planck’s constant, 𝒌𝒊 and 𝒌𝒔 the incident and scattered wave vectors respectively and 𝑸 the scattering vector. In the elastic neutron scattering process, the energy of incident neutrons equals the energy of the scattered neutrons. Compared to X-ray diffraction, neutron diffraction provides a more reliable and accurate estimation of the atomic displacement parameters (ADP) as neutrons are scattered by nuclei. Moreover, with similar numbers of electrons but different neutron scattering power, Ni and Ti can be more easily differentiated in neutron scattering experiments than in X-ray diffraction. From the diffraction experiments, the structural information of the materials such as the lattice parameters and atomic positions can be obtained based on a quantitative analysis of the diffraction pattern such as the peak positions and peak intensities. The Bragg’s peak positions are determined based on the relationship between the distance between lattice planes d, the scattering angle θ and the wavelength λ described as Bragg’s Law λ = 2d sin θ. For a diffraction peak from (hkl) planes in perfect crystal materials, the peak intensity 𝐼_ℎ𝑘𝑙is related to the structure factor 𝐹_ℎ𝑘𝑙 by 2 𝐼ℎ𝑘𝑙 ∝ |𝐹ℎ𝑘𝑙 |2 = |∑ 𝑏𝑗 ⋅ 𝑒 2𝜋𝑖(ℎ⋅𝑥𝑗 +𝑘⋅𝑦𝑗+𝑙⋅𝑧𝑗) | Eq. 18 𝑗 where bj is the neutron scattering length of the jth atom and depends on the isotope. 29 Rietveld refinement158 is commonly used as a quantitative analysis of the diffraction data to extract detailed information of the structure, based on a least-square fitting of the calculated pattern to the observed pattern. With an initial guess of the structure, the diffraction pattern can be calculated with known peak positions and intensities. The structural parameters including lattice parameters, atomic positions, occupancies, atomic displacement parameters, as well as the profile parameters including background, peak shape and width are refined to minimize the sum of squared residuals: 2 ∑ 𝑤𝑖 (𝑦𝑖𝑜𝑏𝑠 − 𝑦𝑖𝑐𝑎𝑙𝑐 ) Eq. 19 𝑖 where 𝑦𝑖𝑜𝑏𝑠 and 𝑦𝑖𝑐𝑎𝑙𝑐 are the observed and calculated intensity of the ith data point, respectively, wi the weighting factor. Different parameters can be calculated to assess the quality of the fitting, for example, the commonly used parameter weighted profile R-factor (Rwp): 1/2 2 2 𝑅𝑤𝑝 = {∑ 𝑤𝑖 (𝑦𝑖𝑜𝑏𝑠 − 𝑦𝑖𝑐𝑎𝑙𝑐 ) / ∑ 𝑤𝑖 {(𝑦}𝑜𝑏𝑠 𝑖 ) } Eq. 20 𝑖 𝑖 2.2.3 Quasi-elastic neutron scattering In interactions between neutrons and the sample, the elastic scattering of neutrons with zero energy transfer can be used to determine the structure information as in neutron diffraction experiments. The rotational and translational motions of atoms and molecules can result in a distribution of energy exchanges in a small range (typically within 1 meV). Such a scattering process is called quasi-elastic neutron scattering (QENS) and can be used to investigate the diffusion in the sample159. QENS experiments provide both temporal and microscopic spatial information on atomic dynamics and can provide experimental verification to the computational 30 results on diffusion behaviors. QENS experiments are commonly used accompanied by MD simulations as they probe the diffusion behaviors at similar length and time scales160,161. With both energy transfer and momentum transfer occurring in QENS, the intensity is measured as a function of momentum/wave vector Q and energy/ frequency ω in the QENS experiment, related to the dynamical structure factor S(Q, ω). The incoherent and coherent structure factors 𝑆inc (𝑸, 𝜔) and 𝑆coh (𝑸, 𝜔) are the Fourier transform of self and collective intermediate scattering functions 𝐼self (𝑸, 𝑡) and 𝐼coll (𝑸, 𝑡), respectively: 1 +∞ 𝑆inc (𝑸, 𝜔) = ∫ 𝐼 (𝑸, 𝑡)exp (−𝑖𝑤𝑡)𝑑𝑡 2𝜋 −∞ self Eq. 21 1 +∞ 𝑆coh (𝑸, 𝜔) = ∫ 𝐼 (𝑸, 𝑡)exp (−𝑖𝑤𝑡)𝑑𝑡 Eq. 22 2𝜋 −∞ coll 1 𝐼self (𝑸, 𝑡) = ⟨∑ exp(−𝑖𝑸 ⋅ [𝑹𝑖 (𝑡) − 𝑹𝑖 (0)])⟩ Eq. 23 𝑁 𝑖 1 𝐼coll (𝑸, 𝑡) = ∑ ∑ ⟨𝑒𝑥𝑝(−𝑖𝑸 ⋅ [𝑹𝑗 (𝑡) − 𝑹𝑖 (0)])⟩ Eq. 24 𝑁 𝑗 𝑖 For continuous long-range translational diffusion following Fick’s Law, the self intermediate scattering function is related to the self-diffusivity D according to 𝐼self (𝑸, 𝑡) = 𝑒𝑥𝑝 (−Γ𝑡) = exp (−𝐷𝑸𝟐 𝑡) Eq. 25 In energy space, the incoherent scattering function is a Lorentzian peak: 1 Γ 1 𝐷𝑄 2 𝑆𝑖𝑛𝑐 (𝑄, 𝜔) = = Eq. 26 𝜋 𝜔 2 + (Γ)2 𝜋 𝜔 2 + (𝐷𝑄 2 )2 which can be observed as a broadening of the elastic peak with a half-width-half-maximum (HWHM) or Γ of DQ2. At smaller distances or large Q, the local motions of the particles may not occur following Fick’s law but in jumps, where the Q-dependence of Γ would deviate from Γ= DQ2. In this case, 31 the Q-dependence of Γ can be described by jump diffusion models such as Chudley-Elliott (CE) jump model162 and the Singwi-Sjölander (SS) jump model163 we applied in this work: 1 𝑠𝑖𝑛(𝑄𝑑) CE model: 𝛤 = 𝜏 (1 − ) Eq. 27 𝑄𝑑 1 𝑄 2 <𝑟 2 >/6 SS model: 𝛤 = 𝜏 (1+𝑄2<𝑟 2>/6) Eq. 28 where the jumps are characterized by the residence time τ and the constant jump length d in the CE model or the mean jump distance r in the SS model. 32 Structural properties of P2-Na2/3[Ni1/3Ti2/3]O2 3.1 Introduction While sodium host materials rely on the rapid ion and electron conduction processes to achieve good sodium intercalation properties, it is important to understand the average and local structural features of P2-Na2/3[Ni1/3Ti2/3]O2, which are closely related to these conduction processes. In P2-Na2/3[Ni1/3Ti2/3]O2, the partial occupancy of Na atoms over two different sites and disordered distribution of Ni and Ti atoms lead to complex local structural features of P2- Na2/3[Ni1/3Ti2/3]O2, which have not been fully understood yet. The average structure of P2-Na2/3[Ni1/3Ti2/3]O2 can be obtained from time-of-flight neutron diffraction based on the Rietveld refinement. Besides the structural information from the previous study based on x-ray diffraction100, the atomic displacement parameters (ADP) at different temperatures, which are also correlated with partial occupancies, can be more accurately obtained by neutron diffraction. The refined structure from neutron diffraction can provide an initial configuration for atomistic modeling. Classical simulations based on force fields are particularly useful to study disordered materials as they allow the exploration of the broad energy landscape by evaluating a large number of initial configurations164-169. In this chapter, the average and local structural properties and energetics of atomic distribution in P2-Na2/3[Ni1/3Ti2/3]O2 are investigated using neutron diffraction and force-field based atomistic simulations. 3.2 Average structural information from Rietveld refinement Time-of-flight (TOF) neutron diffraction experiments were performed at 15 K and 300 K with the POWGEN diffractometer at Spallation Neutron Source (SNS) of Oak Ridge National Laboratory (ORNL) to study the average structure of P2-Na2/3[Ni1/3Ti2/3]O2. The software package JANA2006170 was used to perform the Rietveld refinement to analyze the neutron diffraction data 33 and plot the nuclear density maps. Although anisotropic thermal displacement is physically more realistic in these complex oxides, we obtained negative atomic displacement parameters (ADP) values for transition metal atoms during the refinement procedure. Hence, an isotropic model was applied for nickel and titanium atoms while anisotropy was maintained for the remaining atoms. The structural model based on the P63/mmc space group simulated the observed Bragg peak positions accurately as shown in Figure 9. The bump at the d spacing of ~4.2 Å could not be indexed by the main and impurity phase reflections, which might be due to the complex local ordering of the sodium and/or transition metal atoms. (a) (b) 1200 2500 Iobs 103 103 1000 15K,Bank 1 15K,Bank4 Iobs Ical 2000 Ical 800 Iobs-Ical Iobs-Ical Bragg positions 1500 Bragg positions 600 210 Intensity Intensity 216 400 313 203 1000 106 006 101 107 104 100 200 500 002 004 0 0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 d-spacing (Å) d-spacing (Å) (c) (d) 1200 2500 300K,Bank 1 Iobs 300K,Bank 4 Iobs 1000 Ical 2000 Ical 800 Iobs-Ical Iobs-Ical Bragg positions 1500 Bragg positions 600 Intensity Intensity 1000 400 200 500 0 0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 d-spacing (Å) d-spacing (Å) Figure 9: Rietveld refinement of Na2/3[Ni1/3Ti2/3]O2 powders characterized at 15K using detector bank 1 (a) and bank 4 (b), at 300K using bank 1 (c) and bank 4 (d). 34 Table 2 provides obtained structural information from Rietveld refinement of neutron diffraction datasets at different temperatures. Estimated lattice parameters are comparable with the previous XRD report on the same material100. The powders show a tendency for higher occupancy of ‘2c’ sodium sites (edge-sharing or Nae) compared to ‘2b’ sites (face-sharing or Naf), i.e. 0.505 vs 0.162, which may be resulting from the energy penalty in occupying the ‘2b’ sodium sites (0, 0, 1/4) due to the repulsive interactions as they are positioned directly in-line with the transition metal ‘2a’ sites (0, 0, 0). At ambient temperatures (300 K), the higher thermal energy allows for slightly increased occupancy of the energetically unfavorable ‘2b’, i.e., 0.169 vs 0.162 at 15 K. The occupancy of face-sharing Na was 0.198 in the previous XRD study100. Calculated thermal expansion coefficients between 15 and 300 K are: αa=5.91×10-6 K-1, αc=1.57×10-5 K-1. These coefficients are fairly positive and a large anisotropy is observed, typical of isostructural layered oxides such as P2-NaxMnO2171. Table 2: Structural information of Na2/3[Ni1/3Ti2/3]O2 using Rietveld refinement of neutron diffraction datasets. a. Temperature=15 K Element x y z Occupancy Uiso(Å2) U11(Å2) U33(Å2) (Site) Naf (2b) 0 0 0.25 0.162(7) - 0.009(6) 0.001(9) Nae (2c) 0.3333 0.6667 0.25 0.505(7) - 0.018(2) 0.009(3) Ni (2a) 0 0 0 0.3333 0.006(3) - - Ti (2a) 0 0 0 0.6667 0.015(5) - - O (4f) 0.6667 0.3333 0.09583(13) 1 - 0.0067(4) 0.0062(6) Lattice parameters: a=b=2.9593Å, c=11.1014Å; goodness of fit: Rwp=9.30%. 35 Table 2 (cont’d) b. Temperature=300 K Element x y z Occupancy Uiso(Å2) U11(Å2) U33(Å2) (Site) Naf (2b) 0 0 0.25 0.169(8) - 0.022(8) 0.004(10) Nae (2c) 0.3333 0.6667 0.25 0.498(8) - 0.031(3) 0.012(4) Ni (2a) 0 0 0.0 0.3333 0.009(3) - - Ti (2a) 0 0 0.0 0.6667 0.019(6) - - O (4f) 0.6667 0.3333 0.09523(15) 1.0 - 0.0084(6) 0.0088(8) Lattice parameters: a=b=2.9643Å, c=11.1510Å; goodness of fit: Rwp=8.86%. Table 2 can be visualized as nuclear density maps where the density of each species is shown as spheres (isotropic) or ellipsoids (anisotropic) in the left column in Figure 10 to Figure 13. For example, Figure 10(a) depicts the nuclear density map of Na in the refined structure at 15 K. For the shown isosurface level of 1 Å-3, Nae density spots appear larger than Naf due to the higher fractional occupancy (Table 2a). In addition, sodium thermal ellipsoids also appear considerably oblate with the major axis along the ‘ab’ plane. The degree of oblateness is quantified using U33 displacement ratio (r = √ ⁄U − 1) which compares ‘in-plane’ (U11 ) with the ‘out-of-plane’ 11 ( U33 ) deviation. At 15 K, Nae and Naf atoms have negative ‘r’ values of -0.29 and -0.67, respectively, which are consistent with the oblate sodium ellipsoids. The lower ‘r’ value for Naf is due to the repulsion from Ni and Ti atoms. These negative ‘r’ values are lower compared to other materials with layered structure like O3-NaCrO2 (r=-0.18) 172. The ‘in-plane’ repulsive interactions are lower in P2 than O3 phases due to the lower sodium content and hence the P2 phase displays more pronounced ‘in-plane’ deviations and oblateness. The transition metal atoms were taken to be isotropic for the reason discussed previously and hence appear as perfect spheres in Figure 11(a, 36 d) for Ni and in Figure 12(a, d) for Ti. We also allowed Ni and Ti to take different ADP parameters in order to yield positive ADP values. Oxygen atoms were refined anisotropically but they appear spherically in Figure 13(a, d) since U11 ≈ U33. Rietveld refinement Buckingham Morse (a) Na-3D (b) Na-3D (c) Na-3D (d) Na-2D (e) Na-2D (f) Na-2D Nae Naf Naf Nae Nae Naf Figure 10: 3D nuclear density maps and 2D slices on the {001} plane of Na. 3D isosurface level is 0.5 Å-3 while 2D isosurface level is between 0 and 1 Å-3. Results from the Rietveld refinement at 15 K, Buckingham potential, and Morse potential are shown for each column. 37 Rietveld refinement Buckingham Morse (a) Ni-3D (b) Ni-3D (c) Ni-3D (d) Ni-2D (e) Ni-2D (f) Ni-2D Figure 11: 3D nuclear density maps and 2D slices on the {001} plane of Ni. 3D isosurface level is 0.5 Å-3 while 2D isosurface level is between 0 and 1 Å-3. Results from the Rietveld refinement at 15 K, Buckingham potential, and Morse potential are shown for each column. 38 Rietveld refinement Buckingham Morse (a) Ti-3D (b) Ti-3D (c) Ti-3D (d) Ti-2D (e) Ti-2D (f) Ti-2D Figure 12: 3D nuclear density maps and 2D slices on the {001} plane of Ti. 3D isosurface level is 0.5 Å-3 while 2D isosurface level is between 0 and 1 Å-3. Results from the Rietveld refinement at 15 K, Buckingham potential, and Morse potential are shown for each column. 39 Rietveld refinement Buckingham Morse (a) O-3D (b) O-3D (c) O-3D (d) O-2D (e) O-2D (f) O-2D Figure 13: 3D nuclear density maps and 2D slices on the {001} plane of O. 3D isosurface level is 0.5 Å-3 while 2D isosurface level is between 0 and 1 Å-3. Results from the Rietveld refinement at 15 K, Buckingham potential, and Morse potential are shown for each column. 3.3 Energy minimization based on two interatomic potential models The disordered distribution of Na_e/Na_f in the Na layers and Ni/Ti in the TM layers in Na2/3[Ni1/3Ti2/3]O2, which may affect Na dynamics, is critical for us to understand the local structure. We explored the energy landscape and investigated the overall energetics with the randomized supercells containing 24 formula units of Na2/3[Ni1/3Ti2/3]O2, with two NaO6 trigonal prism layers and two NiO6/TiO6 octahedral layers. The equivalent orthorhombic supercell in space group Cmcm (#63) was constructed by converting the unit cell in the original space group P63/mmc (#194) through the following transformation matrix without any origin shift: 40 3 2 0 T = (0 4 0 ) 0 0 1 The 16 sodium atoms in each supercell can be allocated equally or disproportionately to the two individual sodium layers, while still maintaining the overall composition. The presence of plentiful vacant sites in the sodium layer leads to a large number of possible configurations even for structures with equal sodium count between the layers. Likewise, the 24 transition metal atoms can be distributed equally or disproportionately while keeping the overall Ti:Ni ratio at 2:1. Using these randomized configurations, we explore the energy landscape and investigate the overall energetics using atomistic simulation. Since there are a large number of possible configurations for Na/vacancy and Ni/Ti distribution, 5000 randomized structures were sampled to yield statistically relevant results for four different scenarios based on Na/vacancy and Ni/Ti distribution in the two layers: Na equal and Ni equal, Na random and Ni equal, Na equal and Ni random, and Na random and Ni random. The energy minimization for each configuration was performed in General Utility Lattice Program (GULP)173 with the lattice parameters fixed. The long-range interactions were defined by Coulombic law, while two different types of potential models, Buckingham and Morse, as specified in Table 3, were used for the short-range interactions. For the Buckingham potential model, titanium and oxygen atoms were treated as polarizable and represented with the Dick- Overhauser core-shell approach while remaining atoms were treated as non-polarizable174. The model parameters for Ti-O interactions were taken from a defect study of anatase130, while the parameters for other interactions were derived in GULP to fit the experimentally obtained structure for P2-Na2/3[Ni1/3Ti2/3]O2 at 15 K. For the Morse potential model, the Morse function with a Lennard-Jones repulsive term added was used. The empirical Morse potential parameters were obtained from a self-consistent model for evaluating oxides and silicates131. Electronic polarization 41 effects were neglected in Morse potential. The short-range potential cutoff radius was set to be 10 Å for the Buckingham model and 12 Å for the Morse model. Table 3: Interatomic potential parameters of the Buckingham model and Morse model. a. Buckingham model 𝑈𝐵𝑢𝑐𝑘𝑖𝑛𝑔ℎ𝑎𝑚 = 𝐴𝑒 −𝑟/𝜌 − 𝐶𝑟 −6 Interactions A (eV) ρ (Å) C (eV Å6) Na-O 1172.75 0.281 0.0 Ni-O 868.03 0.317 0.0 Ti-O 5111.70 0.2625 0 O-O 12420.50 0.2215 29.07 Core-Shell Potential Parameters Species k (eV Å -2) Y (e) Ti 314 -0.1 O 31 -2.8 b. Morse model 𝑈 𝑀𝑜𝑟𝑠𝑒 = 𝐷[(1 − 𝑒 −𝑎(𝑟−𝑟0) )2 − 1] + 𝐴𝑟 −12 Morse Parameters Interactions Dij (eV) aij (Å -2) r0 (Å) A (eV Å12) Na-O 0.0234 1.76 3.01 5.0 Ni-O 0.0294 2.68 2.50 3.0 Ti-O 0.0242 2.25 2.71 1.0 O-O 0.0424 1.38 3.62 22.0 We evaluated the validity of the atomistic potential models by examining their ability to reproduce experimental lattice parameters of a selected group of materials containing Na, Ni, Ti, and O, including NiO (cubic), TiO2 (tetragonal), NiTiO3 (rhombohedral), Na2Ti3O7 (monoclinic) and Na8Ti16O32 (monoclinic). The comparison of computed lattice parameters from both Buckingham and Morse potential sets and experimental values are shown in Table 4. The mean absolute errors employing Buckingham and Morse potential sets are 1.4 and 2.8%, respectively, 42 demonstrating the general validity of the studied pairwise atomic interactions and transferability across various structure polytypes, except for Na2Ti3O7 and Na2Ti6O13 in the Morse model. Good reproducibility of lattice parameters for Na2/3[Ni1/3Ti2/3]O2 at 15 K were confirmed by small deviations for both Buckingham (ε=0.08%) and Morse (ε=1.2%) model. Table 4: Potential transferability across selected materials containing Na, Ni, Ti, and O. (The mean absolute errors of the Buckingham and Morse potentials are 1.4 and 2.8%, respectively). Buckingham Lattice NiO TiO2 NiTiO3 Na2Ti3O7 Na2Ti6O13 Cubic Tetragonal Hexagonal Monoclinic Monoclinic Expt. Calc. Expt. Calc. Expt. Calc. Expt. Calc. Expt. Calc. a (Å) 4.178 4.173 4.594 4.428 5.032 5.090 9.133 9.186 15.131 15.296 b (Å) - - - - - - 3.806 3.717 3.745 3.698 c (Å) - - 2.959 3.068 13.792 13.766 8.566 8.612 9.159 9.098 Morse Lattice NiO TiO2 NiTiO3 Na2Ti3O7 Na2Ti6O13 Cubic Tetragonal Hexagonal Monoclinic Monoclinic Expt. Calc. Expt. Calc. Expt. Calc. Expt. Calc. Expt. Calc. a (Å) 4.178 4.188 4.594 4.571 5.032 5.074 9.133 9.613 15.131 15.550 b (Å) - - - - - - 3.806 3.605 3.745 3.577 c (Å) - - 2.959 3.004 13.792 13.978 8.566 8.872 9.159 9.625 3.4 Energetics of Atomic Distribution and Ordering Energy distribution histograms of optimized structures using two potential sets are shown in Figure 14. It was found that the distribution of sodium atoms and distribution of nickel and titanium atoms, i.e., four scenarios, have different effects on the energy distribution. The structures with random Ni/Ti distribution, Figure 14(c, d, g, h), exhibit two energy peaks with higher energy values while those with Ni/Ti equal distribution, Figure 14(a, b, e, f), exhibit just one energy peak with lower energy values. When taking a closer look at configurations corresponding to the two 43 energy peaks presented in the energy distribution histogram for equal sodium and random Ni/Ti distribution, Figure 14(c, g), it was found that the higher energy peak is associated to configurations with large deviation in the number of same type atoms per layer. These configurations tend to have 6 Ni atoms and 6 Ti atoms in one layer, and 2 Ni atoms and 10 Ti atoms in the other layer. The structures that contribute towards the low energy peak tend to have equal number of the same type atoms distributed in each layer (4 Ni atoms and 8 Ti atoms in each layer), i.e., similar to Figure 14(a, e). If we compare (a) and (b) or (e) and (f), we can see that random distribution of Na in two layers does not add extra energy peaks but leads to higher energy of the peak position. This trend suggests that the distribution of transition metal atoms plays a larger role in the overall energetics than sodium atoms and equal distribution of both sodium atoms and transition metal atoms provides the lowest energy values and hence the most favorable configurations. We further examined the in-plane distribution of Ni and Ti in these configurations and no Ni/Ti ordering was observed. 44 Buckingham Morse Na equal, Ni equal (a) Na equal, Ni equal (e) Energy distribution (arb. unit) Energy distribution (arb. unit) Na random, Ni equal (b) Na random, Ni equal (f) Na equal, Ni random (c) Na equal, Ni random (g) Na random, Ni random (d) Na random, Ni random (h) -2570 -2565 -2560 -2555 -2550 -970 -968 -966 -964 -962 -960 -958 -956 -954 -952 Energy (eV) Energy(eV) Figure 14: Energy distribution curve from the Buckingham and Morse potential for 3x2x1 supercells with four different scenarios of atom distribution: Na equal and Ni equal, Na random and Ni equal, Na equal and Ni random, and Na random and Ni random. 3.5 Simulated Average Structure Atomic coordinates of optimized structures in Figure 14(a, e) were used to generate nuclear density maps for each individual species. After energy minimization, atomic coordinates were shifted to have one Ni or Ti atom at the origin to facilitate the comparison of different structures. The nuclear probability density maps were generated by calculating the probability of atomic distribution in individual pixels (resolution ~0.1 Å). 3D nuclear density maps and 2D slices were visualized using VESTA175. This provides a graphical representation of the simulated average 45 structures. Because atomic densities in the two layers are similar, we only presented results for one of the layers. As shown in Figure 10(b, c), sodium density spots obtained from atomistic simulation are compressed into oblate spheroids (major axis in the ‘ab’ plane) for both Buckingham and Morse potential models, which is consistent with the nuclear density map and negative sodium displacement ratio from the Rietveld refinement. The spheroidal elongation suggests that sodium gets thermally displaced in-plane and this might lead to the low diffusional activation energy. 2D isosurface sections are sliced along the {001} family to distinguish density spots between two sodium sites. For the Buckingham potential, most sodium clusters around edge-sharing sites, with the rest of sodium atoms distributed in the channels between edge- and face-sharing sites. On the other hand, sodium atoms from the Morse potential are localized at edge and face-sharing sites with higher concentration at edge sites, showing a close resemblance to the Rietveld analysis. The transition metal (Ni/Ti) and O densities from atomistic simulation in Figure 11 to Figure 13 remain fairly spherical for both Buckingham and Morse potentials, although the distribution of atoms based on the Morse model is more concentrated. This suggests that the isotropic thermal displacement assumption in Rietveld analysis is reasonable. However, the 2D view along the {001} plane of Morse density maps also shows extra density spots that were not observed in experimental results, i.e. (f) of Figure 11 to Figure 13, suggesting possible gliding of Ni/TiO6 layers. In these offset structures, one of the transition metal oxide layers glides along the ab plane, leading to the transition from P2 stacking to a O2 stacking, where the sodium atoms occupy octahedral sites. An example of this O2 structure is shown in Figure 15(b, c) from two different perspectives, while an example of the P2 structure is shown in Figure 15(a). Such P2-O2 phase transformations have been reported for other sodium transition metal oxides including 46 Na2/3[Ni1/3Mn2/3]O2176 and Na0.7Fe0.4Mn0.4Co0.2O2177 during electrochemical insertion/extraction. On the other hand, O2 stacking was not observed in the simulated density maps based on the Buckingham potential. Energetics analysis of O2 structures from the Morse potential indicates that they also have a Gaussian distribution (e.g., Figure 14) with energy values comparable to normal structures (P2). As the O2 stacking does not add extra peaks to the diffraction pattern, the existence of O2 stacking is not able to be verified by the neutron diffraction results. Further studies are needed to better understand the possible static and dynamic features of the O2 stacking of this material. (a) P2 (b) O2 (c) O2 Na Ti Ni Figure 15: Example structures of (a) P2 and (b, c) O2 stacking for the supercell used in the atomistic simulation. 3.6 Conclusions The average and local structural properties and energetics of atomic distribution in P2- Na2/3[Ni1/3Ti2/3]O2 were investigated using neutron diffraction and interatomic potential based atomistic simulations. The average crystal structure can be adequately described by a unit cell based on the P63/mmc space group, in which face-sharing sites have lower occupancy compared with edge-sharing sites. In addition, refinement of anisotropic atomic displacement has revealed considerable oblateness in sodium motion along the ab plane, possibly caused by the repulsion from the transition metal oxide layer. Atomistic simulation based on the Buckingham and Morse 47 potential models suggest that it is energy favorable to have equal distributions of Na and transition metal in each of the two layers in the unit cell. While nuclear density maps from the Morse potential overall agree well with those from neutron diffraction, they also show signs of transition metal layer gliding that leads to O2 stacking. 48 Structural and electronic properties change during cycling 4.1 Introduction In this chapter, we focused on the structural and electronic properties of the layered P2- type Nax[Ni1/3Ti2/3]O2 using density functional theory. As Na ions are electrochemically inserted and extracted during the charge/discharge processes, the structural and electronic change of Nax[Ni1/3Ti2/3]O2 upon Na insertion and extraction were examined. The electronic conductivity, which is closely correlated to the battery performance, was also investigated in order to better our understanding of how to improve material performance in batteries. 4.2 DFT calculations Considering the disordered distribution of Na_e/Na_f in the Na layers and Ni/Ti in the TM layers, the input configurations were determined based on the potential energy landscape analysis for a large number of randomized configurations explicated in CHAPTER 3. As it was found to be most energetically favorable to have equal Na ions in each Na layer and equal Ni ions in each TM layer in the unit cell178, such equal Na/equal Ni distribution was applied in all simulation cells in this study. The most probable configurations from the energy distribution were selected as the representative structures for the DFT calculation. 3×3×1 Nax[Ni1/3Ti2/3]O2 supercells containing 18 formula units with varying sodium contents (x=0.33, 0.44, 0.56, 0.67, 0.78, 0.89 and 1) were used as the input configuration. All DFT calculations were performed using the Vienna Ab initio Simulation Package (VASP)179 within the projector augmented-wave (PAW) approach180. The dispersion-corrected Perdew-Burke-Ernzerhof with Becke-Jonson damping (PBE-D3BJ) functional119,120 was used. To account for the electron localization of Ni ions, the Hubbard U correction was applied with UNi=6.2 eV181. 49 4.3 Structural changes upon Na extraction/insertion To study the structural stability of Nax[Ni1/3Ti2/3]O2 during cycling, the structural evolution of Nax[Ni1/3Ti2/3]O2 upon sodium insertion and extraction was investigated by DFT-based MD simulations. The MD simulations were carried out at 300 K with the NPT ensemble for 3 ps with a time step of 1 fs. A plane-wave energy cut-off of 450 eV and single Γ point were used for the k- mesh. The lattice parameters of Nax[Ni1/3Ti2/3]O2 at 300 K obtained from DFT-based MD calculations were plotted with varying sodium content x as in Figure 16. The error bars indicate the fluctuation (standard deviation) of lattice parameters during MD simulation, which is primarily determined by the cell size. An expansion along a axis and a contraction of c were observed upon sodium insertion from the pristine state, exhibiting the same trend with previous in-situ XRD study182. The expansion of ab plane is probably due to the increase of the total number of Na ions within the layer leading to repulsion between the Na ions, while the decrease of c can be attributed to the increasing Coulomb attraction between the layers resulting from increasing sodium ions. The structural changes introduced by Na extraction from the pristine state are relatively small. For both a and c direction, lattice constants stay at a stable value with small fluctuations, which might be related to the balance of Coulomb forces due to the decrease of Na ions and increase of TM charges. While a possible P2 to O2 phase transition was observed in our previous ex situ XRD study98 when x in Nax[Ni1/3Ti2/3]O2 drops below 1/3, such transition was not found in the x range of 1/3~1 either in that experimental study or in this work. With no phase transition and a relatively small lattice constant change leading to a volume change less than 2% observed in the studied Na content range, we expect a small possibility of structure collapse and an overall good structural stability of the material in the Na content range of 1/3 to 1, as confirmed by experiments98. 50 Na extraction Na insertion 3.00 DFT XRD: Fielden 2.97 a (Å) 2.94 2.91 11.3 0.33 0.44 0.56 0.67 0.78 0.89 1.00 11.2 11.1 c (Å) 11.0 10.9 10.8 10.7 0.33 0.44 0.56 0.67 0.78 0.89 1.00 x in NaxNi1/3Ti2/3O2 Figure 16: Calculated lattice parameters of P2-Nax[Ni1/3Ti2/3]O2 vs. sodium content x (black squares), compared to an XRD study by Fielden et al.182 For further analysis of the structural change with respect to x, especially for the sodium extraction region, the TM-O bond lengths of the average structure for each composition were calculated. The local Jahn-Teller (JT) distortion of TM-centered octahedra can be identified from the unequal distribution of six TM-O bond lengths (two longer bonds and four shorter bonds), which provides information on the oxidation state of the TM ions. It was found that none of the Ti ions are JT active in any of the compositions we studied, while in some compositions, some of the Ni ions are JT active with two longer Ni-O bonds around 2.4 Å and four shorter bonds around 1.9 Å. Figure 17(a) presents the distribution of the four shortest Ni-O bonds (red circles) and the two longest Ni-O bonds (blue diamonds) connected to each Ni. For the pristine and sodium-rich material (x>=0.67), bond lengths for the four shortest Ni-O bonds and two longest Ni-O bonds of 51 each Ni atom are well separated, indicating all Ni ions are JT active with one example shown in Figure 17(c). With Na ions extracted from the pristine state, some of the Ni ions become JT inactive with an example shown in Figure 17(b), with the bond lengths of two longest Ni-O overlapping with four shortest bonds in Figure 17(a). The decrease of JT distortion upon Na extraction can be an indication of the redox change of the Ni ions, as also observed in LiNiO2 during lithium deintercalation by X-ray absorption fine structure (XAFS)183. Such JT transition adds the complexity of overall lattice change when Na ions being extracted, providing a possible reason for the plateau of lattice parameters upon Na extraction. When furtherly examining the local environment of the Ni atoms that undergo such JT transition, we did not find a direct correlation between the positions of these Ni ions and either the positions of removed Na or the local Ni/Ti ordering. Considering the dynamics of the system, the JT transition of Ni was mostly likely to be affected by the global structural fluctuations, rather than the local environment. 52 (a) 2.8 Na extracion Na insertion 2 longest bonds connected to each Ni 2.6 Ni-O bond lengths 2.4 2.2 2.0 1.8 4 shortest bonds connected to each Ni 0.33 0.44 0.56 0.67 0.78 0.89 1.00 x in NaxNi1/3Ti2/3O2 (b) (c) Figure 17: (a) The distribution of Ni-O bond lengths (b) an example of a JT-inactive Ni ion (c) an example of a JT-active Ni ion. 53 4.4 Electron density distribution and the atomic charge To study the local electron distribution change and redox reactions that occurred during cycling, the electron density distribution and atomic charge for different sodium content in P2- Nax[Ni1/3Ti2/3]O2 were studied using DFT. The configuration for DFT calculation is the average structure obtained from MD. A plane-wave energy cut-off of 450 eV and a Gamma-centered k- mesh (6×6×6) were used for electronic structure calculations. To visualize the electron transfer involved in the redox reactions during Na insertion and extraction, the electron density difference between the pristine cell and the cell upon Na insertion and extraction are calculated by DFT and plotted using VESTA175 as in Figure 18 with an isosurface level of 0.005 Å-3. While the significant electron density change around the inserted/extracted Na ion is directly correlated to the insertion or extraction of Na, the electron change around the surrounding TM ions provides information on the redox reaction occurring upon Na insertion/extraction. The electron density around Ni decreased as one Na ion was extracted (Figure 18(a)), indicating the oxidation of Ni. No noticeable electron density change was found around the Ti ions, implying that the Ti ions are not active in the redox reactions during Na extraction. Similarly, the electron gain around Ti ions (Figure 18(b)) indicates the reduction of Ti upon Na insertion, where the Ni ions are electrochemically inactive. It is worth mentioning that the surrounding O ions bonded to the Na ion also lose and gain electrons upon Na extraction and insertion, respectively, suggesting the participation of O in the redox reactions, which have also been observed in similar materials P2-Nax[Ni1/3Mn2/3]O2184 and P2- Na2/3[Mn1/3Co2/3]O2185. 54 (a) (b) Figure 18: Electron density difference (positive in yellow and negative in cyan) upon (a) Na insertion, (b) Na extraction. The isosurface level is 0.005 Å-3. The Na, Ni, Ti and O atoms are represented by yellow, gray, blue and red spheres, respectively. The redox processes during Na extraction/insertion could be associated with the change of atomic charges. The common charge models to assign electron densities to each atom include Density Derived Electrostatic and Chemical (DDEC) charges121, Bader charges122, Hirshfeld charges123, and Mulliken charges124. Considering that the Hirshfeld charge model tends to underestimate the charge125,126 and the Mulliken charge model is sensitive to the basis sets127,128, we only applied the DDEC6 and Bader models to calculate the atomic charge in this study. Figure 19(a) shows the distribution of calculated atomic charges of each atom for different compositions as well as the average atomic charges for each type of atom. The DDEC and Bader results gave the same trends for the atomic charge. During Na insertion and extraction, the charge of Na ions stays at a stable value around 1. For transition metal ions, Ni and Ti act differently upon Na insertion and extraction. With Na ions inserted, the charge of Ti ions decreased while the charge of Ni maintained at the same level. On the contrary, the charge of Ti ions stayed flat and the charge of some Ni ions increased, corresponding to the Jahn-Teller transition of Ni discussed before. The trend of the atomic charge confirms that the Ni is redox active during Na extraction and Ti is redox 55 active during Na insertion, consistent with what we observed from electron density distribution. The increasing O charge with increasing x provides another evidence that the oxygen ions are also involved in the redox reactions, also consistent with the observed electron density change around O. (a) (b) 2.25 2.25 Ti Ti 2.00 2.00 1.25 1.25 DDEC charge (e) Bader charge (e) 1.00 Na 1.00 Ni 0.75 0.75 Na 0.50 Ni 0.50 -0.75 -0.75 O O -1.00 -1.00 -1.25 -1.25 -1.50 -1.50 0.33 0.44 0.56 0.67 0.78 0.89 1.00 0.33 0.44 0.56 0.67 0.78 0.89 1.00 x in NaxNi1/3Ti2/3O2 x in NaxNi1/3Ti2/3O2 Figure 19: The atomic charge distribution for each atom at different sodium content x in Nax[Ni1/3Ti2/3]O2 based on (a) DDEC model; (b) Bader model. 4.5 Density of states To gain a further understanding of the change of electronic structures during cycling, the density of states (DOS) of Nax[Ni1/3Ti2/3]O2 with different sodium contents (x=0.44, 0.67, 0.89) was calculated. The total DOS, as well as the partial DOS for Na, d-orbitals of Ni and Ti, and p- orbital for O, are presented in Figure 20. 56 Total Ni-d Ti-d O-p Na 200 x=0.44 (a) DOS 100 0 200 x=0.67 (Pristine) (b) DOS 100 0 200 x=0.89 (c) DOS 100 0 -5 0 5 Energy (eV) Figure 20: The density of states of Nax[Ni1/3Ti2/3]O2 with (a)x=0.44, (b) x=0.67, (c) x=0.89. The Fermi energy is set to be zero. When Na is extracted from the pristine state, e.g., from x=0.67 (Figure 20(b)) to x=0.44 (Figure 20(a)), the mixed Ni-d and O-p band at the top of the valence band split into two bands with one fully occupied and the other unoccupied. The unoccupied Ni/O band suggests partial electron loss, i.e., oxidation of Ni and O upon Na extraction. An insignificant change of Ti-d band was found from x=0.67 to x=0.44, suggesting that the Na extraction had less influence on Ti oxidation state. In the meanwhile, the split band leads to a smaller bandgap compared to the pristine material. When Na is inserted from the pristine state, e.g., from x=0.67 (Figure 20(b)) to x=0.89 (Figure 20(c)), the Fermi level moved toward the higher energy end, leading to a partially occupied Ti-d band. The partial occupancy of the Ti-d band indicates the electron gaining of Ti, consistent 57 with the reduction of Ti during Na insertion. The main Ni bands stayed fully filled before and after Na insertion, suggesting that the Ni ions are inactive in the redox reactions during Na insertion. 4.6 Electronic conductivity As high electronic conductivity is desired for electrode materials, we calculated the electronic conductivity for different compositions using two approaches based on the electronic structure calculated from DFT. 4.6.1 Electronic conductivity from the Kubo-Greenwood approach DFT calculations were carried out with a plane-wave energy cut-off of 450 eV and a Gamma- centered k-mesh (6×6×6). The electronic conductivities were derived from the frequency- dependent complex dielectric function based on the Kubo-Greenwood approach186,187. Based on the Kubo-Greenwood formula, the transverse conductivity 𝜎⊥ (𝒒, 𝜔)is related to the transverse component of the dielectric function 𝜖⊥𝑟 (𝒒, 𝜔) by: i 𝜖⊥𝑟 (𝒒, 𝜔) = 1 + 𝜎 (𝒒, 𝜔) Eq. 29 𝜖0 𝜔 ⊥ where q and ω are the wave vector and frequency, respectively. Thus, with q→0 the electronic conductivity σe is related to the imaginary part of dielectric function ε2 as σe (ω)=ωε2(ω). The DC electronic conductivity was obtained by the extrapolation of σe as ω→0. The obtained electronic conductivities of P2-Nax[Ni1/3Ti2/3]O2 at different x were plotted in Figure 21. At all compositions, the electronic conductivities for the Cartesian xx and yy direction are similar, both higher than along the zz direction, suggesting electron conduction in P2- Nax[Ni1/3Ti2/3]O2 is mainly two-dimensional along the ab plane. The electronic conductivities in all directions increased when either inserting and removing Na atoms, which may be due to the increasing electron delocalization during atom oxidation and reduction processes. 58 xx Na extraction Na insertion yy Electronic conductivity (S/cm) zz 100 10 0.33 0.44 0.56 0.67 0.78 0.89 1.00 x in NaxNi1/3Ti2/3O2 Figure 21: The electronic conductivity of Nax[Ni1/3Ti2/3]O2 at different x from the Kubo- Greenwood approach 4.6.2 Electronic conductivity from the Boltzmann transport equations To account for the dynamics in the material, 50 input structures were randomly selected from the MD trajectory for each composition to calculate the electronic conductivity. The DFT electronic structure calculations were conducted for each configuration with a cut-off energy of 450 eV and a Gamma-centered k-mesh of 4×4×3. The electrical conductivities were calculated using maximally localized Wannier functions as basis functions with the BoltzWann module12 in the Wannier90 package2 based on the semiclassical Boltzmann transport equations. The current density J and the electronic heat current 𝑱𝑄 is related to the conductivity 𝝈 as Eq. 30 and Eq. 31 respectively: 𝑱 = 𝝈(𝑬 − 𝑺𝜵𝑇) Eq. 30 59 𝑱𝑄 = 𝑇𝝈𝑺𝑬 − 𝑲𝜵𝑇, Eq. 31 where E is the electric field; S is the Seebeck coefficient; 𝜵𝑇 is the temperature gradient. 𝑲 is related to the thermal conductivity according to 𝜿 = 𝑲 − 𝑇𝝈𝑺2 . The conductivity depending on chemical potential 𝜇 and temperature T can be derived as: +∞ 𝜕𝑓(𝐸, 𝜇, 𝑇) Eq. 32 [𝝈]𝑖𝑗 (𝜇, 𝑇) = 𝑒 2 ∫ 𝑑𝐸(− )𝛴𝑖𝑗 (𝐸) −∞ 𝜕𝐸 where 𝑓(𝜀, 𝜇, 𝑇) is the Fermi-Dirac distribution function as Eq. 33 and 𝛴𝑖𝑗 (𝐸) expressed as Eq. 34: 1 Eq. 33 𝑓(𝜀, 𝜇, 𝑇) = 𝑒 (𝜀−𝜇)/𝐾𝐵 𝑇 +1 1 Eq. 34 𝛴𝑖𝑗 (𝐸) = ∑ 𝑣𝑖 (𝑛, 𝑘)𝑣𝑗 (𝑛, 𝑘)𝜏𝑛𝑘 𝛿(𝐸 − 𝐸𝑛,𝑘 ) 𝑉 𝑛,𝑘 where 𝜏𝑛𝑘 is the relaxation time for the electron on band n at wave vector, v and E the velocity and energy respectively, V the volume. The relaxation time was approximated as a constant value of 10 fs for all electrons. Figure 22 shows the calculated electronic conductivity in three directions along the Cartesian x, y, and z for three different compositions. Each data point represents the value calculated from one individual structure, while the box gives the standard deviation of the data. The numbers in the plot label the average calculated electronic conductivities for all configurations of a composition. Despite the limitation of the DFT+U method in choosing the semi-empirical U values, the electronic structure calculations still provide valuable insights in the Na concentration dependence of the electronic conductivity of this material. 60 x y z x y Electronic conductivity (S/cm) z 103 3.5102 4.8102 100 2.8102 2.4102 x y z 1.6101 2.2101 10-3 10-6 7.410-3 5.810-3 -9 10 1.210-4 Na5/9[Ni1/3Ti2/3]O2 Na2/3[Ni1/3Ti2/3]O2 Na7/9[Ni1/3Ti2/3]O2 10-12 Figure 22: The electrical conductivity of Nax[Ni1/3Ti2/3]O2 at different x. The left, middle, and right panel represents results for Na5/9[Ni1/3Ti2/3]O2, Na2/3[Ni1/3Ti2/3]O2, Na7/9[Ni1/3Ti2/3]O2, respectively. Yellow, green, and purple points represent the calculated electrical conductivity along x, y, and z direction, respectively. It is clear that electronic conductivity is sensitive to atomic structure as its value varies significantly, by orders of magnitude, for different configurations. For all compositions, electronic conductivities observed along the Cartesian x and y directions are about one order’s magnitude higher than that along the z (i.e. crystallographic c) direction, suggesting the mainly two- dimensional electron conduction. However, the electronic conductivity along the z direction (e.g. ~10 S/cm) of Na-rich and deficient phases is not negligible, unlike the case of ionic conductivity. There is a significant increase (about 5 order’s magnitude) in the electronic conductivity when sodium content is decreased or increased from the pristine phase by 16.7% (i.e. from x=2/3 to x=5/9 and from x=2/3 to x=7/9). This trend is consistent with the above results using a single configuration calculation based on the Kubo-Greenwood approach. This significant increase of electronic conductivity is not surprising as the pristine phase of Na2/3[Ni1/3Ti2/3]O2 contains 61 nominally Ni2+ and Ti4+, while decrease or increase of Na content will activate Ni2+/Ni3+ or Ti4+/Ti3+ redox couple, respectively. 4.7 Conclusions In this chapter, we studied the structural and electronic properties of P2-type Nax[Ni1/3Ti2/3]O2 during electrochemical oxidation/reduction using DFT calculations. For structural evolutions during electrochemical processes, it was found that the insertion of sodium leads to expansion of the ab plane and contraction of c axis. On the other hand, the extraction of sodium brings in a small structural change, which might be related to the JT active to JT inactive transition for Ni ions during Na extraction. Electronic properties including the electron density distribution, the atomic charge, and density of states were analyzed, all leading to the same conclusion that Ni and Ti are the major redox-active ion during Na extraction and insertion, respectively, with O also participating in the redox reaction all the way. To assess the electrochemical performance of P2-type Nax[Ni1/3Ti2/3]O2, we examined the electronic conductivity of the material using two approaches based on the electronic structure calculated from DFT, Kubo-Greenwood approach and Boltzmann transport equations. Both approaches showed similar results that the dominant electronic conduction occurs within the 2D layers along ab plane and an increase of electronic conductivity can be resulted from both inserting and extracting Na ions. From the results based on Boltzmann transport equations, the pristine Na2/3[Ni1/3Ti2/3]O2 showed an electronic conductivity of ~10-3 S/cm, of which a significant increase can be achieved by either removing or inserting Na ions. Removing or inserting 16.7% of the Na can lead to an increase in the electronic conductivity of 5 orders. Such an improved understanding 62 of structural and electronic properties could be utilized to optimize materials and devices for better battery performance. 63 Dynamic properties of P2-Na2/3[Ni1/3Ti2/3]O2 5.1 Introduction To assess the rate performance of P2-type layered material Na2/3[Ni1/3Ti2/3]O2, we investigated the diffusion mechanism as well as ionic conductivity with a combination of experimental and computational techniques. The quasi-elastic neutron scattering (QENS) experiments and molecular dynamics (MD) simulations based on interatomic potential (IP) and density functional theory (DFT) were performed to identify the diffusion mechanism. 5.2 Self-diffusion of Na ions from QENS QENS experiments were conducted using the time-of-flight backscattering spectrometer (BASIS)188 at the Oak Ridge National Laboratory (ORNL). The sample Na2/3[Ni1/3Ti2/3]O2 powders were synthesized by solid-state reactions, heated to 200 oC for 10 hours, and then stored in an argon atmosphere to limit the exposure to moisture. The measurements were performed from 450 to 700 K to study the Na ion dynamics and at 30 K for the instrument resolution function. The data were normalized by the Vanadium measurement and reduced in the Mantid package189, with the energy transfer (E) range of ±100 μeV (0.4 μeV binning) and momentum transfer (Q) range of 0.3 to 1.9 Å-1 (0.2 Å-1 binning). The QENS spectra at each Q, with a convolution of the resolution function, were fitted with a combination of one delta function (for the elastic peak), a Lorentzian peak (for the quasi-elastic features), and a flat background using the DAVE package190. In QENS experiments, the quasi-elastic broadening of spectra was the direct evidence of diffusive motions, as shown in the measured QENS spectra S(Q, E) for Q=0.3 Å-1, i.e., Figure 23. The inset plot in Figure 23 shows the increased QENS broadening with increasing temperature, suggesting enhanced diffusion at higher temperatures. 64 1.2 30K (Resolution) 450K 500K 1.0 600K 650K 700K 0.8 0.02 S(Q,E) Increasing T 0.6 0.4 0.00 -15 -10 -5 0 5 10 0.2 Q=0.3Å-1 0.0 -100 -80 -60 -40 -20 0 20 40 60 80 100 E (eV) Figure 23: Experimental QENS spectra S(Q,E) for various temperatures at Q=0.3 Å-1. The quantitative analysis of spectra was conducted to further understand the Na diffusion behaviors in the sample. The measured QENS data at each temperature and each Q, was fitted with a combination of a flat background, one delta function for the elastic peak and a Lorentzian peak for the quasi-elastic features, convoluted with the resolution function. An example for 700 K and Q=0.3 Å-1 shows the fit and residuals in Figure 24(a). The Q-dependence of half-width-half- maximum (HWHM or Γ) of the Lorentzian peak characterizes the diffusive motions of Na. Figure 24(b) shows an example of Γ against Q at 700 K. 65 (a) (b) 1.0 Data 210 0.05 Delta Background 700K Lorentzian 0.04 180 0.04 S(Q,E) 0.02 150 S(Q,E) HWHM (eV) 0.5 120 0.03  (ps-1) 0.00 1 Q 2  r 2  /6 = ( ) -10 0 E (eV) 10  1 + Q 2  r 2  /6 90 T=700K 0.02 Q=0.3Å-1 0.0 60 0.01 Residual 0.01 30 0.00  = DQ2 -0.01 0 0.00 -100 -80 -60 -40 -20 0 20 40 60 80 100 0 1 2 3 4 E (eV) Q2 (Å-2) Figure 24: (a) Fit of the QENS spectra at 700 K and Q=0.3 Å-1. (b) The HWHM of the Lorentzian function at 700 K as a function of Q2, fitted to the Fickian model at small Q (red line) and the SS model (black line). In the low-Q region, a linear relationship between Γ and Q2 indicates continuous translational diffusion of Na ions following Fick’s law at large distances. The Fick’s model describes the long-range dynamics. The Fickian diffusivity D was calculated according to  (Q ) = DQ 2 in Q≤0.5 Å-1 range, as the slope of the red line in Figure 24(b). The Fickian diffusivity values extracted from the QENS spectra at different temperatures are plotted in Figure 25 (blue filled circles). The Na self-diffusivity in the temperature range of 450 to 700 K is on the order of 10-6 cm2/s, which falls in the same range with the similar layered compound Na0.8CoO2 from a QENS study191. Based on the Arrhenius fit for the temperature dependence of Na self-diffusivity, we obtained an activation energy of 0.15 ± 0.004 eV and a diffusivity of ~10-7 cm2/s at room temperature. The low activation energy barrier of this material is comparable to some P2-type compounds including Na2/3[Ni1/3Mn1/3Ti1/3]O2 (0.17 eV from FPMD192), Na0.8CoO2 (0.17eV at ~350K from QENS191), and Nax[Ni2/3Mn2/3]O2 with 1/3/6 𝛤(𝑄) = ( ) Eq. 35 𝜏 1 + 𝑄 2 < 𝑟 2 >/6 The SS model only applies to the short-range behaviors of Na dynamics at large Q. The mean jump length and residence time obtained from the SS model with the temperature dependence are plotted in Figure 25(b). The jump lengths are between 1 and 2 Å which fit into the dimension of the honeycomb sublattice in the Na layer, as the distance from an edge-share site to a nearest-neighbor face-share site and vice versa is around 1.7 Å. This confirms that Na primarily migrates between edge-share and face-share sites within the 2D diffusion pathway. The average jump length of Na increases with temperature, while the residence time decreases with temperature as more frequent and longer jumps can be activated by higher thermal energy. 67 T (K) (b) (a) 700 600 500 50 40 30  (ps) 20 10 1.4 1.6 1.8 2.0 2.2 1.9 (c) 1.8 1.7 1.6 d (Å) 1.5 1.4 1.3 1.2 1.1 1.4 1.6 1.8 2.0 2.2 1000/T (1/K) Figure 25: Diffusional properties from QENS at 450 to 700K: (a) The Fickian self- diffusivity of Na. (b) The residence time and (c) jump length of Na diffusion with the SS model. 5.3 Ionic conductivity and Na self-diffusion from IP-based MD IP-based MD simulations were performed to study dynamic properties including diffusion behaviors and ionic conductivity at the larger time and length scale. 5.3.1 Construction of the polarizable interatomic potential model Parameters for the polarizable interatomic potential model194 were obtained from DFT results using the force-matching method. The charge for each atom was taken from the DFT calculation based on DDEC3 approach195, as it provided better results in structure reproduction compared to DDEC6 charge121. The short-range pairwise interactions are described by a combination of Buckingham143 and Morse144 potential model as the following equation: 𝑈 𝑖𝑗 = 𝐴𝑒 −𝑟/𝜌 − 𝐶𝑟𝑖𝑗 −6 + 𝐷[(1 − 𝑒 −𝑎(𝑟𝑖𝑗−𝑟0) )2 − 1] Eq. 36 The O atoms were treated as induced dipoles while other atoms were non-polarizable. The obtained polarizability for O is 1.99 Å3. The Tang-Toennies damping functions196 were applied 68 for correction of short-range charge-dipole interactions between atom i and atom j as the following equation: k =0 (bij rij )k g (rij ) = 1 − c exp(−b rij )  ij ij ij Eq. 37 4 k! The obtained parameters are listed in Table 5. Table 5: Polarizable interatomic potential parameters. − a ( rij − r0 ) 2 a. short-range pairwise interactions U ij = Ae − r /  − Crij −6 + D[(1 − e ) − 1] Interactions A(eV) ρ(Å) C(eVÅ6) D(eV) a(Å-1) r0(Å) Na-O 1300.1 0.28 - - - - Ni-O 1003.8 0.17 - 0.002 3.95 2.46 Ti-O 1907.4 0.27 14.8 - - - O-O 1250.0 0.30 39.8 - - - k =0 (bij rij )k b. Tang-Toennies damping functions g (rij ) = 1 − c exp(−b rij )  ij ij ij 4 k! Interactions Na-O Ni-O Ti-O bij 2.86 3.46 2.96 cij 1.72 1.41 0.91 As the parameters for the IP model were obtained from DFT through force-matching, the goodness of fitting was assessed by the root mean squared error (RMSE) of calculated forces on each type of atom. The RMSE are 0.182, 0.586, 0.737, and 0.451 eV/Å for Na, Ni, Ti, and O, respectively. The comparison of forces calculated from the obtained IP model and DFT is shown in Figure 26. 69 (a) (b) 4 Na 3 2 IP force (eV/Å) 1 0 -1 -2 -3 -2 -1 0 1 2 3 DFT force (eV/Å) (c) (d) Figure 26: The comparison of IP forces (blue circles) and DFT forces (black line) on each type of atom. 5.3.2 IP-based MD simulations The IP-based MD(IPMD) simulation was performed with the FIST module implemented in cp2k197. A 9×9×3 Na2/3[Ni1/3Ti2/3]O2 supercell containing 1782 atoms was used. An NPT ensemble was performed at different temperatures from 300 to 1200 K with a simulation time of 50 ps to obtain the lattice parameters. Due to the computational cost concerns, the NVE runs with 50 ps 70 equilibrium time were conducted at only high temperatures from 900 to 1200 K to calculate diffusion and ionic conduction, using incoherent density correlation functions and coherent charge-current correlation functions198 respectively, as explicated below. The time step was 1 fs. Both DFT and IP-based MD simulations with the NPT ensemble were used to predict the structure of P2-Na2/3[Ni1/3Ti2/3]O2 at different temperatures for the validation of the two models. The calculated unit cell parameters were compared with experimental results obtained from the previous neutron diffraction study (ND) by Shanmugam et al. for the same material178 as shown in 3.2. For the lattice parameters at 300 K, calculated results from DFT and IP differ from the experiment by 0.41% and 1.42% along a, 1.09% and 0.11% along c, respectively. The reliabilities of the two models were confirmed by a small deviation between calculated and experimental results concerning both lattice constants and the thermal expansion along a and c direction. 71 ND: Shanmugam DFT FF 3.00 a (Å) 2.95 2.90 0 200 400 600 800 1000 1200 11.5 11.4 11.3 c (Å) 11.2 11.1 11.0 0 200 400 600 800 1000 1200 Temperature (K) Figure 27: Lattice parameters of pristine P2-Na2/3[Ni1/3Ti2/3]O2 at different temperatures obtained from neutron diffraction by Shanmugam et al178 (red stars), DFT-based MD (blue circles), and IP-based MD (green squares). For the visualization of Na diffusion pathways in P2-Na2/3[Ni1/3Ti2/3]O2, the nuclear density map of Na was generated by plotting the Na atom distribution along the MD simulation trajectory. For clarity, the Na densities in the simulation cell at 1100 K were projected to a smaller cell (3×3×1) using VESTA175 as in Figure 28. The Na trajectory reveals that Na ions are highly mobile along the ab plane at high temperatures, forming a honey-comb like 2D diffusion network connected by bridges between edge-share (E) and face-share (F) sites, similar to other layered P2 sodium insertion materials33,39. The regions enclosed by the isosurface are larger for the edge-share sites 72 than for the face-share sites, suggesting higher occupancy at edge-share sites, consistent with the results from XRD as discussed in the previous section 3.2. (a) (b) Figure 28: The nuclear density map of Na during 500 ps simulation at 1100 K. (a) 3D view with an isosurface level of 0.2 Å-3, (b) 2D slice on the (001) lattice plane with an isosurface level of 0~0.5 Å-3. “E” and “F” represent edge-share sites and face-share sites 5.3.3 Extraction of diffusivity and ionic conductivity from IP-based MD The diffusion and ionic conduction were calculated using incoherent density correlation functions and coherent charge-current correlation functions198. For the calculation of Na self- diffusion, the production time for NVE is ranged from 1 ns (for 1200 K) to 8.5 ns (for 900 K) to ensure an adequate time of Na jumps were sampled. The position data were utilized every 1 ps. The incoherent density correlation function, also known as the incoherent intermediate scattering function, was calculated based on the following equation: 73 1 𝑁𝛼 𝛼 𝛼 𝐼𝛼 (𝑄, 𝑡) = ⟨𝑁 ∑𝑛=1 𝑒 −𝑗𝑄⋅𝑟𝑛 (0) 𝑒 𝑗𝑄⋅𝑟𝑛 (𝑡) ⟩ 𝛼 Where Nα stands for the total number of atom α and r stands for the position vectors. Q is the wave vector transfer, given by Qnx ny nz = nx a + n y b + nz c , where a*, b*, c* are reciprocal unit * * * vectors and nx, ny, and nz are all integers. The I(Q,t) for different Q values were calculated from MD trajectories with values at 1100 K shown in Figure 29(a), for both through-plane Q (red curves) and in-plane Q (blue curves). The through-plane I(Q,t) have plateaus of around 1, suggesting no Na diffusion across the layer. This is consistent with the Na density map in Figure 28. The relaxation of the correlation function I(Q,t) can be described by the stretched exponential decay, also known as the Kohlrausch-Williams-Watts (KWW) function199: 𝐼(𝑄, 𝑡) = 𝑒𝑥𝑝[ − (𝛤 𝐾𝑊𝑊 𝑡)𝛽 ] Eq. 38 where ΓKWW represents the relaxation rate and β is the stretching factor ranged from 0 to 1. Both ΓKWW and β are Q-dependent and can be obtained by least-squared fitting. The mean 𝛤KWW 1 relaxation rate Γ is given by 𝛤 = 𝐺(𝛽), where G is the gamma function. The obtained mean 𝛽 relaxation rate Γ with Q dependency is shown in Figure 29(b). 74 (a) (b) 0.10 Equation Plot y=a+ \G(G) Weight No Weigh 1.0 1100K Intercept Slope 0 ± -- 0.11102 ± 0 Residual Sum of Squares 2.80555 Pearson's r 0.9980 increasing Q along c axis 0.08 R-Square(COD) Adj. R-Square 0.9961 0.9942 0.8 streched exponetial fit 0.6 0.06  (ps-1) I(Q,t)=exp[-( KWW*t)] I(Q,t)  = 1 (1− sin(Qd )) 0.4 0.04 Qd 0.2 0.02 0.0  = DQ2 increasing Q along ab plane 0.00 0 50 100 150 200 250 300 0 1 2 3 4 t (ps) 2 -2 Q (Å ) Figure 29: (a) The I(Q,t) at 1100 K (b) Γ vs. Q2 (black circles), with the CE fit (black line) and linear fit for the three smallest Q (red line). Within the small Q regime of Figure 29(b), the Fickian diffusivity D was obtained by a linear fitting of Γ vs. Q2 for the three smallest in-plane Q values according to 𝛤 = 𝐷𝑄 2 . The Fickian self- diffusivity from 900 to 1200 K are plotted in Figure 30(a). The activation energy for Na diffusion obtained by the Arrhenius fit is 0.53 ± 0.02 eV. The extrapolated Na diffusion coefficient at 300 K is around 3×10-12 cm2/s, one order higher than the measured value from a previous study (10- 14 ~10-13cm2/s)104. For the entire Q regime of Figure 29(a), the Chudley-Elliott jump model162 was used to describe the Na diffusion mechanism. According to the CE model, Γ was fitted to the 1 sin(Qd ) equation  = (1 − ) as a function of Q values to obtain the residence time τ and jump  Qd length d. The temperature dependence of τ and d are plotted in Figure 30(b) and (c). The residence time τ decreases with increasing temperature (activation energy of 0.62 ± 0.04 eV), suggesting more frequent jumps with higher temperature. The decreasing jump length d with increasing temperature may be due to the growing delocalization of Na ions as temperature rises. Considering 75 that the distance between closest edge-share and face-share site is around 1.7 Å, the mean jump length of Na from 900 to 1200 K is roughly 2-3 times of that distance. (a) (b) T (K) 1200 1100 1000 900 100  (ps) 0.62eV 10 0.8 1.0 1.2 (c) 4.2 4.0 3.8 d (Å) 3.6 3.4 3.2 3.0 0.8 1.0 1.2 1000/T (K-1) Figure 30: Diffusional properties from MD simulations at 900 to 1200 K: (a) The Fickian self-diffusivity of Na. (b) The residence time and (c) jump length of Na diffusion with the Chudley-Elliott model. The results of Na diffusion from IP-based MD were compared to the results from QENS experiment in Figure 30(a). The IP-based MD showed high in-plane Na self Fickian diffusivity in a range of 10-6~10-5 cm2/s from 900 to 1200 K, with an activation energy of 0.53 ± 0.02 eV. The Na Fickian diffusivity measured from QENS experiment is around 10-6 cm2/s in the temperature range of 450 to 700 K, with an activation energy of 0.15 ± 0.004 eV. Two different jump models are found to well describe the Na jump behaviors for IP-based MD and QENS results, the Chudley- Elliott and the Singwi-Sjölander jump model, respectively. For both MD and QENS results, more frequent jumps were found at higher temperatures. However, the two models showed different trends of jump distance with respect to temperatures. The analysis based on the MD trajectory reveals that the Na ions migrate between the adjacent edge-sharing and face-sharing sites, which 76 is associated with the jump distance of around 1.7Å observed from QENS measurements. The inconsistency between the MD simulation results and the QENS experimental results including the large deviation between activation energy, the different jump mechanism and the different jump distance trend regarding temperature might be due to the limitations of the interatomic potential model as it relies on parametric forces and tends to neglect the underlying electronic origin of the interactions. Although we found this potential model worked well in predicting the structural properties like the lattice parameters and thermal expansion for this material, the performance of it for dynamic properties predictions may not be as good as for static properties. The ionic conductivity was extracted from the IP-based MD trajectories using the coherent charge-current correlation function in the transverse polarization198, i.e., with two directions perpendicular to each Q probed. The transverse coherent charge-current correlation function was calculated based on the NVE trajectory ranged from 200 ps (for 1200 K) to 600 ps (for 900 K) sampling every 0.02 ps, with the following equation: 𝑒2 1 𝐶 𝑇 (𝑄, 𝑡) = ⟨(𝑄 × 𝐽(𝑄, 0)) 𝑉𝑘𝐵 𝑇 2𝑄 2 Eq. 39 ⋅ (𝑄 × 𝐽(−𝑄, 𝑡))⟩ where V is the volume of the system, kB the Boltzmann constant, and T the temperature. The charge current J(Q,t) is given by 𝐽(𝑄, 𝑡) = ∑𝑁 𝑛=1 𝑞𝑛 𝑣𝑛 (𝑡) 𝑒 −𝑗𝑄⋅𝑟𝑛 (𝑡) , where qn is the charge carried by the nth atom, and vn and rn are velocity and position vectors. A Fourier transform in time of CT(Q,t) can lead to ST(Q,ω) in the frequency domain as ∞ 𝑆 𝑇 (𝑄, 𝜔) = ∫ 𝐶 𝑇 (𝑄, 𝑡) 𝑒𝑥𝑝( 𝑖𝜔𝑡) 𝑑𝑡 Eq. 40 0 The DC ionic conductivity was approximated using ST(Q,ω) at smallest Q (Q001, Q010 and Q100) and smallest ω (0.12 THz with 50 ps correlation time). 77 Figure 31 shows the real part of its result in the frequency domain at 1100 K for the small Q values (Q100=Q010=0.27 Å-1 and Q001=0.18 Å-1), presenting four in-plane and two through-plane ionic conduction. The DC ionic conductivity was calculated by averaging the data points at the smallest frequency (around 0.12 THz as shown in Figure 31(b)). The in-plane ionic conductivity is several magnitudes higher than that of through-plane, as the ionic conduction in Na2/3[Ni1/3Ti2/3]O2 is mainly within the as the 2D ionic conduction layer formed by the NaO6 prisms. (a) (b) 80 1.8 1100K 70 1.6 60 1.4 In-plane S(Q,) (S/cm) S(Q,) (S/cm) In-plane Through-plane 1.2 50 40 1.0 0.8 30 0.6 20 0.4 10 0.2 Through-plane 0 0.0 0 20 40 60 80 100 120 140 0.0 0.2 0.4 0.6  (Thz)  (Thz) Figure 31: The real part of coherent charge-current correlation at 1100 K in (a) full frequency span. (b) the low-frequency region. The blue and red curves represent the in-plane and through-plan ionic conduction, respectively. The calculated ionic conductivity at 900, 1000, 1100 and 1200 K are shown in Figure 32 along with experimental results from Shanmugam et al.104, Shin et al. 103 , and Smirnova et al.200 The calculated in-plane ionic conductivities from the IP-based MD matched well with the experimental results. The calculated activation energy was 0.37 ± 0.03 eV, slightly higher than the experimental values. 78 T (K) 1200 1000 800 600 400 1 0.3 Ionic conductivity (S/cm) 7e V, IP, in- pla 0.2 ne 5eV ,E xp: 0.1 Sm irno 0.2 va 0.23 6eV eV, ,E Exp xp: : Sh Sh in anm 0.01 uga m 0.001 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 1000/T (1/K) Figure 32: The calculated in-plane ionic conductivity of Na2/3[Ni1/3Ti2/3]O2 compared with experimental values from Shanmugam et al.104, Shin et al. 103, Smirnova et al.200 5.4 First principles MD simulations The first-principles molecular dynamics(FPMD) simulations based on DFT with the NVT ensemble were performed at 900, 1000, and 1100 K with an equilibrium time of 3 ps and production run time ranging from 20 ps to 100 ps. The parameters for the DFT calculations were the same as in section 4.2. The non-spin-polarized calculations were performed with a plane-wave energy cut-off of 450 eV and a single Γ point. The starting structure for pristine Na2/3[Ni1/3Ti2/3]O2 was a 3 × 3 × 1 supercell containing 66 atoms. The configurations for Na7/9[Ni1/3Ti2/3]O2 and Na5/9[Ni1/3Ti2/3]O2 as the Na-rich and Na-deficient phases were generated by randomly adding and removing Na atoms from the pristine phase while maintaining the same number of Na atoms in each layer. For all compositions, the lattice parameters were determined based on an NPT ensemble run of 3ps. Convergence tests for the trajectory time and correlation time were performed 79 before extracting the diffusivity to make sure they are long enough. The time step was 1 fs for all temperatures. The FPMD trajectory recording the positions of all Na ions provided the direct visualization of Na motion. We plotted the Na density maps from the MD trajectories as shown in Figure 33. Na density maps showed traces along the 2D honeycomb diffusion pathway in the Na layer, consistent with the jump length obtained from the QENS measurement. (a) (b) Figure 33: The nuclear density map of Na from 20 ps MD simulation trajectory at 1100 K. (a) 3D view with an isosurface level of 0.2 Å-3, (b) 2D slice on the (001) lattice plane with an isosurface level of 0~0.5 Å-3. From the simulation, the Fickian self-diffusion coefficient along direction X was calculated by the Green-Kubo integration of velocity autocorrelation function: 𝑁𝑎 ∞ 1 𝑋 𝐷α = ∑ ∫ ⟨𝑣𝑖𝑋 (𝑡) ⋅ 𝑣𝑖𝑋 (0)⟩𝑑 𝑡 Eq. 41 𝑁α 0 𝑖=1 where Nα stands for the total number of atom α and 𝑣𝑖𝑋 the X-direction component of velocity for the ith atom. The Green-Kubo integral reached a plateau within a short correlation time. For 𝐷α𝑋 along the c axis, the integral plateaued around 0, suggesting no cross-plane diffusion, 80 consistent with what we observed from the Na density maps. The diffusion coefficients along a and b axis were obtained by averaging over a correlation time period in the plateau region (we selected 2-3 ps in this work). The Fickian self-diffusivity along a and b axis as well as the averaged value over two directions in the temperature range 900 to 1100 K are plotted in Figure 34. The diffusivity extracted from the MD simulation is about two times higher than those from the QENS experiment when extrapolating to the same temperature. The activation energy obtained from the Arrhenius equation for in-plane Na diffusion from 900 to 1100 K is 0.20 ± 0.02 eV which is slightly higher than QENS results (0.15 eV). Figure 34: Calculated Fickian diffusivity from FPMD simulation (red) compared to results from IPMD(green) and QENS(blue). While the FPMD simulation provides comparable results with the QENS experiment for the pristine material of sodium content x=2/3, to assess the Na diffusivity change during cycling, 81 we further calculated the diffusivity with the same approach for the other two compositions of x=5/9 and 7/9. Figure 35 shows the calculated Na in-plane Fickian diffusivity for the three compositions at 900 K. From x=2/3 to x=5/9, Na5/9[Ni1/3Ti2/3]O2 shows a comparable diffusivity to Na2/3[Ni1/3Ti2/3]O2 with only a 3% increase. This trend is consistent with the diffusivity measured in this x range for Nax[Ni1/3Ti2/3]O2104 as well as in other P2-type compounds184,192,201 using the Potentiostatic Intermittent Titration Technique (PITT) method. From x=2/3 to x=7/9, the insertion of Na ions into the pristine material leads to a significant drop in diffusivity, which can be attributed to the limited number of vacant sites. 1.8x10-5 a b Mean 1.6x10-5 D (cm2/s) 1.4x10-5 1.2x10-5 Na extraction Na insertion 1.0x10-5 0.5 0.6 0.7 0.8 x in NaxNi1/3Ti2/3O2 Figure 35: Calculated Fickian diffusivity from FPMD simulation at 900 K for different sodium content. As the sodium diffusivity predicted by our first-principles MD simulations showed good consistency with the experiments, the ionic conductivity was also assessed using the MD trajectories. The ionic conductivity was calculated based on the Green-Kubo relations: 82 ∞ 1 σ= ∫ ⟨𝐽 𝑋 (𝑡)𝐽 𝑋 (0)⟩ 𝑑𝑡 Eq. 42 𝑘𝐵 𝑇𝑉 0 where kB is the Boltzmann constant, V the volume of the system, and T the temperature. 𝑁 The JX represents the X-direction component of the charge current "𝑱 = ∑𝑖=1 𝑞𝑖 𝒗𝑖 (𝑡). The charge carried by the ith atom qi was calculated using Density Derived Electrostatic and Chemical (DDEC6) charge analysis in our previous study202. Similar to diffusion, the integral of charge current correlation along c axis converged to a plateau around zero within 1 ps, suggesting a 2D ionic conduction within the layers. The ionic conductivities along a and b axis were calculated from the mean integral of charge current correlation over a correlation time range from 2 to 3 ps. In-plane ionic conductivity at 900, 1000, and 1100 K were obtained by averaging the results along a and b axis as shown in Figure 36, along with computational results from our previous study based on the interatomic potential (IP) model202 and experimental results from Shanmugam et al.104, Shin et al. 103 , and Smirnova et al.200 Calculated in-plane ionic conductivities from the FPMD, if extrapolated to lower temperatures, are slightly higher than experimental results. The calculated activation energy was 0.27 ± 0.27 eV , comparable to the experimental values, while our previous work with the IP model had a larger activation energy of 0.37 eV. The large error in the activation energy in the fitting mainly comes from the limited number of data points. 83 T (K) 1200 1000 800 600 400 0.27e V, D 0.3 FT, i Ionic conductivity (S/cm) 1 n-pla 7e ne V, IP, in- pla ne 0.25 eV, 0.1 E xp: Sm 0.26 irno eV, va E xp: Sha nmu gam 0.01 0.23 eV, Exp : Sh in 0.001 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 1000/T (1/K) Figure 36: Calculated in-plane ionic conductivity of Na2/3Ni1/3Ti2/3]O2 compared with computational values from Chen et al.202 and experimental data from Shanmugam et al.104, Shin et al.103, and Smirnova et al.200 The Na concentration-dependence of the ionic conductivity was examined by calculating the conductivity values at 900 K for the other two compositions Na5/9[Ni1/3Ti2/3]O2 and Na7/9[Ni1/3Ti2/3]O2. As shown in Figure 37, both Na-deficient phase Na5/9[Ni1/3Ti2/3]O2 and Na- rich phase Na7/9[Ni1/3Ti2/3]O2 showed higher ionic conductivity compared to the pristine phase. From x=2/3 to x=5/9, when Na atoms are extracted from the pristine Na2/3[Ni1/3Ti2/3]O2, the increase of ionic conductivity is likely to be related to increased vacancies. From x=2/3 to x=7/9, when the Na atoms are inserted to Na2/3[Ni1/3Ti2/3]O2, higher ionic conductivity can be achieved with more charge carriers. 84 1.2 Na extraction Na insertion 1.1 Ionic conductivity (S/cm) 1.0 0.9 0.8 a 0.7 b Mean 0.6 0.5 0.6 0.7 0.8 x in NaxNi1/3Ti2/3O2 Figure 37: The calculated in-plane ionic conductivity from FPMD simulation at 900 K for different sodium content. 5.5 Conclusions In this chapter, the P2-type layered material Nax[Ni1/3Ti2/3]O2 was investigated with a combination of quasi-elastic neutron scattering and molecular dynamics to study the Na ion diffusion mechanism and ionic conduction. Firstly, based on the Q-dependence of the quasi-elastic broadening in QENS measurement, the localized Na ion diffusion behavior can be described by the Singwi-Sjölander jump model. The extracted average jump lengths are around 1.3~1.8 Å at 450~700 K, corresponding to the distance between the neighboring edge-share and face-share sites. This is also consistent with the diffusion mechanism revealed by the density maps from MD simulations: 2D local jumps occur between the edge-share and face-share Na sites. More frequent and longer jumps were observed at higher temperatures. From the IPMD simulations, the short- range behaviors can be described by the Chudley-Elliott jump model, with more frequent and shorter jumps at higher temperatures. For long-range diffusivity, Fick’s model was applied to 85 compare with the MD simulation results where Q→0. The QENS showed a Fickian diffusivity in the order of ~10-6 cm2/s from 450 to 700 K with Ea = 0.15 ± 0.004 eV. The IPMD simulations showed the Na self Fickian diffusivity in a range of 10-6~10-5 cm2/s from 900 to 1200 K, with an activation energy of 0.53 ± 0.02 eV. The FPMD simulation gave an in-plane diffusivity of ~10-5 cm2/s with Ea = 0.20 ± 0.02 eV in the temperature range of 900 to 1100 K. The diffusion coefficients were calculated for different compositions with the Na concentration x in the range of 5/9 to 7/9 using FPMD. Faster diffusion was observed for compositions with less sodium, i.e., more vacancies. Secondly, The IPMD simulations showed an in-plane ionic conductivity from 900 to 1200 K ranged in 0.4~1.4 S/cm, with an activation energy of 0.37 ± 0.03 eV. The FPMD simulations showed in-plane ionic conductivity values ~ 1 S/cm at 900 to 1100 K with an activation energy of 0.27 eV, which is in better consistency with experimental measurements in previous studies. Both Na-deficient phase Na5/9[Ni1/3Ti2/3]O2 and Na-rich phase Na7/9[Ni1/3Ti2/3]O2 showed higher ionic conductivity compared to the pristine phase. 86 Study on P2/O3 Na2x[NixTi1-x]O2 with machine learning based interatomic potential models 6.1 Introduction Besides P2-type Na2x[NixTi1-x]O2, O3-Na2x[NixTi1-x]O2 is also an important class in layered sodium transition metal compounds, sharing the same structure with the commercialized electrode material in lithium-ion batteries LiCoO3. In O3-Na2x[NixTi1-x]O2, Na ions occupy the octahedral sites with 3 TM layers in the unit cell. The O3 type Nax[TM]O2 can be synthesized at a lower temperature compared to the P2 analogs31 and possess more sodium based on stoichiometry. With the electrochemical performance of the O3 type Nax[NiyTi1-y]O2 as electrode material discussed in a few experimental studies102,182,203, further investigation is needed to understand the diffusion mechanism in this material, which will help us to better understand the relationship between structure and properties and to develop better electrode materials for sodium-ion batteries. In previous chapters, MD simulations based on empirical IP models and DFT calculations, as well as QENS experiments, have been employed to study the diffusion in P2-type layered material Na2/3[Ni1/3Ti2/3]O2. Empirical IP models showed an inconsistency between the MD results and QENS experimental results including the large deviation between activation energies, different jump mechanisms, and different jump distance trends regarding temperature, indicating an insufficient accuracy of the potential model. Higher accuracy can be obtained from first principles MD simulations based on DFT. However, the high computational cost of DFT calculations has limited DFT-based MD to small systems with short time frames, thus only conducted at high temperatures in most studies to ensure sufficient dynamic statistics, while most experiments are conducted at room temperature to measure diffusion and ionic conduction. To bridge the gap 87 between computational and experimental results, it is essential to develop a potential model with both low computational cost and high accuracy. Machine learning (ML) based interatomic potential (IP) models have emerged in recent years as a powerful tool to model the potential energy surface with comparable accuracy and higher efficiency than first principles calculations204,205. Among the ML potential models, neural network (NN) potentials149 are widely used and have been applied to study different materials206-208. In this chapter, we constructed NN potentials for both P2 and O3-type Nax[NiyTi1-y]O2. We first validated the reliability and accuracy of the NN potential. After that, we performed MD simulations based on the NN potential to investigate the diffusion and ionic conduction in P2-type Na2/3[Ni1/3Ti2/3]O2 and O3-type Na0.83[Ni0.42Ti0.58]O2. 6.2 Construction of the neural network potential The neural network (NN) potentials are trained using the SIMPLE-NN package209. The reference data set for the training is constructed with snapshots from first principles MD simulations based on DFT. All DFT calculations were performed using the Vienna Ab initio Simulation Package (VASP)179 using the same setup as in section 5.4. The MD simulations were conducted in a temperature span of 300 K to 2000 K for both P2 and O3 Na2x[NixTi1-x]O2 with different compositions. The structures included in the training and validation set are listed in Table 6. A total of 10243 structures are included in the training set and 1135 structures in the validation set. Gaussian density function (GDF) weighting210 was used to improve the sampling for the training process. Table 6: Structures used in the training and validation set Number P2/O3 Composition of atoms P2 Na0.67[Ni0.33Ti0.67]O2 66 88 Table 6 (cont’d) P2 Na0.44[Ni0.33Ti0.67]O2 62 P2 Na0.56[Ni0.33Ti0.67]O2 64 P2 Na0.61[Ni0.33Ti0.67]O2 65 P2 Na0.72[Ni0.33Ti0.67]O2 67 P2 Na0.78[Ni0.33Ti0.67]O2 68 P2 Na0.89[Ni0.33Ti0.67]O2 70 P2 Na0.67[Ni0.33Ti0.67]O2 264 O3 Na0.67[Ni0.33Ti0.67]O2 99 O3 Na0.78[Ni0.39Ti0.61]O2 204 O3 Na0.89[Ni0.44Ti0.56]O2 105 O3 Na0.83[Ni0.42Ti0.58]O2 184 Atom-centered symmetry functions (ACSF)145 were used as descriptors for the local atomic environment. 8 radial symmetry functions for each pair and 18 angular symmetry functions for each triplet are applied with the parameters same as a previous study on TiO2208. The parameters are shown in Table 7. Same symmetry functions with a cutoff radius of 6 Å are used for all the atoms. Each atomic NN has two hidden layers with 30 nodes in each layer, leading to a 212-30- 30-1 NN architecture. The hyperbolic tangent function is used as the activation function. The sum of mean square errors for energies and forces is minimized with an Adam optimizer211 during the training process. The root mean squared errors (RMSE) of energy and force for the training set are 15 meV and 0.7 eV/Å, respectively. Table 7: Parameters of symmetry functions for all atoms a. Parameters in radial symmetry functions 2 𝐺𝑖2 = ∑ 𝑒 −𝜂(𝑅𝑖𝑗−𝑅𝑠 ) ⋅ 𝑓c (𝑅𝑖𝑗 ) 𝑗 𝑅𝑠 = 0 No. η(Å-2) 1 0.003214 89 Table 7 (cont’d) 2 0.035711 3 0.071421 4 0.124987 5 0.214264 6 0.357106 7 0.714213 8 1.428426 b. Parameters in radial symmetry functions all 𝜁 2 2 2 𝐺𝑖4 =2 1−𝜁 ∑ (1 + 𝜆cos 𝜃𝑖𝑗𝑘 ) ⋅ 𝑒 −𝜂(𝑅𝑖𝑗+𝑅𝑖𝑘+𝑅𝑗𝑘) 𝑗,𝑘≠𝑖 ⋅ 𝑓c (𝑅𝑖𝑗 ) ⋅ 𝑓c (𝑅𝑖𝑘 ) ⋅ 𝑓c (𝑅𝑗𝑘 ) No. η(Å-2) ζ λ 1 0.000357 1 -1 2 0.028569 1 -1 3 0.089277 1 -1 4 0.000357 2 -1 5 0.028569 2 -1 6 0.089277 2 -1 7 0.000357 4 -1 8 0.028569 4 -1 9 0.089277 4 -1 10 0.000357 1 1 11 0.028569 1 1 12 0.089277 1 1 13 0.000357 2 1 14 0.028569 2 1 15 0.089277 2 1 16 0.000357 4 1 17 0.028569 4 1 18 0.089277 4 1 90 Figure 38: The comparison of NN-IP energies (red circles for P2 structures and blue circles for O3 structures) and DFT energies (black line) for the training set. 6.3 Testing the NN potential To assess the reliability of the NN potential, we calculated energies and forces for structures not included in either the training or validation set using this potential and compared them to the results from DFT. The overall RMSE of energies for the P2 and O3 test set is 16.9 and 9.47 meV per atom, respectively. Figure 39 shows the comparison of forces calculated from NN- IP potential and DFT for a P2 and an O3 test set for each type of atom. The force RMSE of Na, Ni, Ti, and O for the P2 test set is 0.12, 0.45, 0.34, and 0.48 eV/Å, respectively, with an overall RMSE of 0.36 eV/Å. The force RMSE of Na, Ni, Ti, and O for the O3 test set is 0.13, 0.47, 0.35, and 0.47 eV/Å, respectively, with an overall RMSE of 0.35 eV/Å. The test results demonstrated good transferability and ability of this potential in predicting energy and force for both P2 and O3 systems. 91 (a) Figure 39: The comparison of NN-IP forces (blue circles) and DFT energies (black line) for the test set for each type of atom in (a) P2 structures and (b) O3 structures. 92 Figure 39 (cont’d) (b) 6.4 MD simulations with the NN potential MD simulations were performed based on the NN potential using LAMMPS212 for both P2 type and O3 type Na2x[NixTi1-x]O2. MD simulations were conducted for a P2 Na2/3[Ni1/3Ti2/3]O2 supercell containing 1782 atoms from 300 to 1200 K and an O3 Na0.83[Ni0.42Ti0.58]O2 supercell containing 3312 atoms from 300 to 900 K. The time step was 1 fs for all MD simulations. An NPT ensemble was performed with a simulation time of 20 ps followed by an NVT ensemble with trajectory length ranging from 100 ps to 1 ns at different temperatures. 93 6.4.1 Calculation of lattice parameters The average lattice parameters at different temperatures were calculated for P2 Na2/3[Ni1/3Ti2/3]O2 and O3 Na0.83[Ni0.42Ti0.58]O2 using NN-IP based MD simulations with the NPT ensemble. The results for P2 Na2/3[Ni1/3Ti2/3]O2 from 300 to 1200 K are compared to the experimental results measured from neutron diffraction (ND)178 and simulation results calculated from MD simulations based on DFT and empirical polarizable IP models in Figure 40. Comparing to the ND results, the lattice parameters for P2-Na2/3[Ni1/3Ti2/3]O2 at 300K are 1.24% lower along both a and c, with the thermal expansion coefficient slightly larger along a and similar along c. Figure 40: Lattice parameters of P2-Na2/3[Ni1/3Ti2/3]O2 at different temperatures obtained from NN-IP based MD simulations (yellow circles) compared to results from neutron diffraction by Shanmugam et al178 (red stars), DFT-based MD (blue diamonds), and IP-based MD (green squares). 94 Lattice parameters for O3 Na0.83[Ni0.42Ti0.58]O2 obtained from NN-IP based MD are shown in Figure 41 compared to experimental data. The experimental data for Na0.83[Ni0.42Ti0.58]O2 are extrapolated based on data for Na0.8[Ni0.4Ti0.6]O2 and Na0.85[Ni0.48Ti0.52]O2 at room temperature in an XRD study by Fielden105 et al. The calculated lattice parameters from NN-IP at 300 K differ from experiment by 0.08% along a and 1.03% along c. The thermal expansion coefficient is 2.53×10-5 K-1 along both a and c, which is higher than the thermal expansion coefficient 2.53×10- 5 K-1 of a similar composition O3-type Na0.8[Ni0.4Ti0.6]O2 reported in a previous experimental study200. The reliability of the NN potential model is confirmed with an overall good consistency in lattice parameters for both P2 and O3 Na2x[NixTi1-x]O2 with other studies. Figure 41: Lattice parameters of O3 Na0.83[Ni0.42Ti0.58]O2 at different temperatures obtained from NN-IP based MD simulations (yellow circles) compared to results extrapolated from XRD by Fielden et al.105 (red stars) 95 6.4.2 Diffusion and ionic conduction The diffusion coefficient of Na ions in P2-Na2/3[Ni1/3Ti2/3]O2 and O3-Na0.83[Ni0.42Ti0.58]O2 are calculated based on the mean square displacement in the MD trajectories according to D = 1 ⟨[𝒓𝑖 (𝑡) − 𝒓𝑖 (0)]2 ⟩ , where 𝒓𝑖 (𝑡) is the position vector of particle I at time t. d is the 2𝑑𝑡 dimensionality of the diffusion, which is 2 in this case as the diffusion is only observed within the ab plane for both P2 and O3 systems. The diffusivity of Na from 500 to 1200 K calculated from IP-NN based MD simulations for P2 Na2/3[Ni1/3Ti2/3]O2 is shown in Figure 42 compared with results from FPMD simulations and QENS experiments. The diffusivity values for P2 Na2/3[Ni1/3Ti2/3]O2 are 3×10-7 to 2×10-5 cm2/s from 500 to 1200K, slightly lower than both FPMD and QENS results. The activation energy obtained from the Arrhenius equation is 0.32 ± 0.01 eV, which is higher than that from FPMD and QENS. Figure 42: Fickian diffusivity of Na in P2-Na2/3[Ni1/3Ti2/3]O2 and O3- Na0.83[Ni0.42Ti0.58]O2 calculated from IP-NN based MD simulations (yellow squares for P2- 96 Na2/3[Ni1/3Ti2/3]O2 and green squares for O3-Na0.83[Ni0.42Ti0.58]O2) compared to results for P2- Na2/3[Ni1/3Ti2/3]O2 from FPMD(green squares) and QENS(blue circles). Figure 42 shows a comparison of Na diffusion coefficients in P2-Na2/3[Ni1/3Ti2/3]O2 and O3-Na0.83[Ni0.42Ti0.58]O2. From 600 to 700 K, O3-Na0.83[Ni0.42Ti0.58]O2 showed a lower Na diffusivity and higher activation energy (0.35 eV) than the P2 system in the same temperature range, consistent with the observations of similar materials such as NaxCoO2 from first principles MD simulations33 and Nax[Ni1/3Mn2/3]O2 from GITT measurement184. Such difference is mainly due to the different Na diffusion mechanisms in different structures. As discussed in previous chapters, Na ions mainly diffuse between the neighboring edge-share and face-share prismatic sites, providing larger space for Na ions to move than in O3 type materials, where Na ions diffuse through narrow triangular faces between octahedrons. The ionic conductivity was calculated based on the Einstein-Helfand equation as Eq. 43. kB is the Boltzmann constant, T the temperature, and V the volume. 𝑞𝑖 is the charge of atom i at position 𝒓𝑖 , calculated using Density Derived Electrostatic and Chemical (DDEC6) charge analysis. 𝑁 𝑁 2 1 1 𝜎= lim ⟨(∑ 𝑞𝑖 𝒓𝑖 (𝑡) − ∑ 𝑞𝑖 𝒓𝑖 (0) ) ⟩ Eq. 43 𝑘𝐵 𝑇𝑉 𝑡→∞ 2𝑡 𝑖=1 𝑖=1 The ionic conductivity of P2-Na2/3[Ni1/3Ti2/3]O2 and O3-Na0.83[Ni0.42Ti0.58]O2 are calculated and compared to results from other studies in Figure 43. For P2-Na2/3[Ni1/3Ti2/3]O2, results from NN-IP showed the in-plane ionic conductivity in the range of 0.1 to 1 S/cm from 500 to 1200 K, with an activation energy of 0.22 ± 0.01 eV. Both conductivity values and activation energy for P2-Na2/3[Ni1/3Ti2/3]O2 showed a good match with the experimental results. O3- Na0.83[Ni0.42Ti0.58]O2 showed slightly lower ionic conductivities with P2-Na2/3[Ni1/3Ti2/3]O2 and 97 higher activation energy of 0.26 eV, following the same trend as in diffusion. Compared to experimental data from other studies, the NN-IP gives an overestimation of the conductivity value and an underestimation of the activation energy. Figure 43: Calculated in-plane ionic conductivity of P2-Na2/3[Ni1/3Ti2/3]O2 (blue squares) and O3-Na0.83[Ni0.42Ti0.58]O2 (blue circles) from IP-NN based MD simulations compared with results from DFT-MD (black) and experimental data for P2-Na2/3[Ni1/3Ti2/3]O2 and O3- Na0.8[Ni0.4Ti0.6]O2 from Shanmugam et al.104 (red), Shin et al.103 (green), and Smirnova et al.200 (yellow). Results for P2 and O3 structures are represented as squares and circles respectively. 6.5 Conclusions In this chapter, an NN potential is constructed with energies and forces from DFT calculations for P2 and O3-type Nax[NiyTi1-y]O2. The reliability of the NN potential is confirmed with low RMSE of energies and forces as well as a good match in lattice parameters and thermal expansion coefficient with other studies. MD simulations were performed based on the NN potential to investigate the diffusion and ionic conduction in P2-type Na2/3[Ni1/3Ti2/3]O2 and O3- 98 type Na0.83[Ni0.42Ti0.58]O2. The calculated Na diffusivity in P2-Na2/3[Ni1/3Ti2/3]O2 is in the range of 3×10-7 to 2×10-5 cm2/s from 500 to 1200K. A lower Na diffusivity and higher activation energy (0.35 eV) were observed in O3-Na0.83[Ni0.42Ti0.58]O2 than P2-type Na2/3[Ni1/3Ti2/3]O2 (0.32eV). P2- Na2/3[Ni1/3Ti2/3]O2 showed the in-plane ionic conductivity in the range of 0.1 to 1 S/cm from 500 to 1200 K, with an activation energy of 0.22 eV, consistent with other experimental studies. O3- Na0.83[Ni0.42Ti0.58]O2 showed slightly lower ionic conductivities and higher activation energy of 0.26 eV compared to P2-Na2/3[Ni1/3Ti2/3]O2, following the same trend as in diffusion. 99 Conclusions and future work 7.1 Conclusions In this thesis, we investigated the properties of Na2x[NixTi1-x]O2 as bi-functional electrode materials for sodium-ion batteries using a combination of computational and experimental techniques. Firstly, the average and local structural properties and energetics of atomic distribution in P2-Na2/3[Ni1/3Ti2/3]O2 were investigated using Rietveld refinement on neutron diffraction datasets and atomistic simulations based on the Buckingham and Morse IP models. The calculated average structure showed a good agreement with the experiment results, where the face-sharing sites have lower occupancy compared with edge-sharing sites. In addition, the nuclear density spots of Na showed an elongation within the ab plane, suggesting an anisotropic atomic displacement in sodium motion along the ab plane, possibly caused by the repulsion from the transition metal oxide layer. It was found that it is energy favorable to have an equal distribution of Na and transition metal in each of the two layers in the unit cell, which provides insights on the determination of the representative structures for the simulations. With in-depth understandings of the structure of P2-Na2/3[Ni1/3Ti2/3]O2, the atomic and electronic structure changes during cycling were studied based on DFT calculations. Structural evolutions during electrochemical processes were studied using DFT-based MD simulations. It was found that the insertion of sodium would lead to expansion of the ab plane and contraction of the c axis. On the other hand, the extraction of sodium brought in small structural changes. Electronic properties including the electron density distribution, the atomic charge, and density of states were analyzed using DFT, all leading to the same conclusion that Ni and Ti are the major redox-active ion during Na extraction and insertion, respectively, with O also participating in the 100 redox reaction all the way. A higher in-plane electronic conductivity was observed compared to the through-plane one, with both increasing when either inserting or extracting Na ions. The diffusion mechanism and ionic conduction in P2 Na2/3[Ni1/3Ti2/3]O2 using a combination of QENS experiments and MD simulations. Firstly, based on the Q-dependence of the quasi-elastic broadening in QENS measurement, the localized Na ion diffusion behavior can be described by the Singwi-Sjölander jump model. The extracted average jump lengths are around 1.3~1.8 Å at 450~700 K, corresponding to the distance between the neighboring edge-share and face-share sites. This is also consistent with the diffusion mechanism revealed by the density maps from MD simulations: 2D local jumps occur between the edge-share and face-share Na sites. More frequent and longer jumps were observed at higher temperatures. For long-range diffusivity, Fick’ s model was applied to compare with the MD simulation results where Q→0. The QENS showed a Fickian diffusivity in the order of ~10-6 cm2/s from 450 to 700 K with Ea = 0.15 eV. The MD simulations based on a polarizable interatomic potential model showed an in- plane Na self Fickian diffusivity in a range of 10-6~10-5 cm2/s from 900 to 1200 K in P2 Na2/3[Ni1/3Ti2/3]O2, with an activation energy of 0.53 ± 0.02 eV. The MD simulations based on DFT calculations gave an in-plane diffusivity of ~10-5 cm2/s with Ea = 0.20 ± 0.02 eV in the temperature range of 900 to 1100 K, which showed a better consistency with experimental results. The diffusion coefficients were calculated for different compositions with the Na concentration x in the range of 5/9 to 7/9 based on first-principles MD simulations. Faster diffusion was observed for compositions with less sodium, i.e., more vacancies. The in-plane ionic conductivity calculated using the polarizable IP-based MD for 900 to 1200 K, showed a range of 0.4~1.4 S/cm, with an activation energy of 0.37 ± 0.03 eV, which is slightly higher than experimental results. The first principles MD simulations showed in-plane ionic conductivity values ~ 1 S/cm at 900 to 1100 K 101 with an activation energy of 0.27 eV, which is in better consistency with experimental measurements in previous studies. Both Na-deficient phase Na5/9[Ni1/3Ti2/3]O2 and Na-rich phase Na7/9[Ni1/3Ti2/3]O2 showed higher ionic conductivity compared to the pristine phase. Machine learning interatomic potentials were constructed for P2 and O3 Na2x[NixTi1-x]O2. MD simulations based on neural network interatomic potentials were performed for P2-type Na2/3[Ni1/3Ti2/3]O2 and O3-type Na0.83[Ni0.42Ti0.58]O2. The calculated Na diffusivity in P2- Na2/3[Ni1/3Ti2/3]O2 is in the range of 3×10-7 to 2×10-5 cm2/s from 500 to 1200Km, with an activation energy of 0.32 eV. A lower Na diffusivity and higher activation energy of 0.35 eV were observed in O3-Na0.83[Ni0.42Ti0.58]O2. The in-plane ionic conductivity of P2-Na2/3[Ni1/3Ti2/3]O2 is 0.1 to 1 S/cm from 500 to 1200 K, with an activation energy of 0.22 eV, consistent with other experimental studies. O3-Na0.83[Ni0.42Ti0.58]O2 showed slightly lower ionic conductivities and higher activation energy of 0.26 eV compared to P2-Na2/3[Ni1/3Ti2/3]O2, following the same trend as in diffusion. The above findings not only provided us fundamental understandings of properties and underlying mechanisms in the Na2x[NixTi1-x]O2 series, but also can be extended to other systems and shed light on the design and improvement of electrode materials for sodium-ion batteries in the future. The analysis of the active redox pairs upon the insertion and extraction showed that the activation of different redox couples can affect the structural stability and electronic conductivities and essentially the stability and rate performance of the battery. These findings can provide insights on the selection of transition metal redox pairs when designing electrode materials, especially for bi-functional electrode materials when different redox reactions are involved in the charge and discharge process. The comparative study on P2 and O3 type Na2x[NixTi1-x]O2 showed higher Na diffusivity and ionic conductivity in the P2 system, suggesting P2 materials are better choices for electrode materials over O3 materials in terms of rate capabilities. However, the 102 diffusivity in P2 materials would decrease with more sodium ions in the composition. While higher capacities are typically promised by O3 systems or P2 systems with higher sodium content which possess more sodium ions in the structure, a tradeoff between rate performance and capacity needs to be considered when choosing the right electrode material for practical applications. 7.2 Future work The comprehensive investigation on Na2x[NixTi1-x]O2 as bi-functional electrode materials in this thesis has provided us a better understanding of this group of materials and provide atomic- level insights on our future study of similar layered compounds and further exploration and design of the sodium electrode with better performance. In this thesis, we studied the structural properties, electronic properties, and dynamic properties for Na2x[NixTi1-x]O2. However, there is still much future work we can do based on the current work to fully realize the potential of not only this materials series but also sodium-ion batteries. For instance, we were not able to identify the O2 phase for Na2/3[Ni1/3Ti2/3]O2 from the neutron diffraction experiments but observed it in our static energy calculations based on the Morse potential model. While we didn’t observe a P2-O2 phase transition in the sodium content range studies in this thesis, it may still occur at lower sodium contents as in other similar compounds like Na2/3[Ni1/3Mn2/3]O2176 and Na0.7Fe0.4Mn0.4Co0.2O2177. This can be verified by experimental techniques like diffraction studies, scanning transmission electron microscopy as well as computational techniques. The redox activities and electronic structures studied in this thesis can be supported by investigations through experiments such as X- ray absorption spectroscopic (XAS) techniques, X-ray photoelectron spectroscopy (XPS), and resonant inelastic X-ray scattering (RIXS). In this thesis, both classical and first principles MD simulations were performed at high temperatures to investigate the dynamic properties with the results compared to experiments conducted at low temperatures. The classical MD simulations 103 relying on the parametric potential energy surface are less accurate as it tends to neglect the underlying electronic origin of the interactions, while the high cost of first principles simulations have limited their applications for systems with small number of atoms and time scale thus may lead to insufficient diffusion statistics at low temperature. To probe the dynamics at low temperature and bridge the gap between computational and experimental results, besides developing potential models with high accuracy and low cost, efforts can also be made to accelerate the dynamics using methods like enhanced sampling, bias potentials, parallel replica methods, and kinetic Monte Carlo simulations. Besides the bulk properties of electrodes, the interfaces, such as the grain boundaries and electrode/electrolyte interfaces, especially the ion and electron transport at these interfaces are critical for the performance of the overall battery system. The diffusion pathways, the conduction mechanism, and activation energy can be completely different at the grain boundaries, which needs further examination. It is also important to understand how the electrode materials studied in this thesis interact with the common liquid electrolyte materials like NaClO4, NaPF6, NaFSI, NaBF4, and solid-state electrolytes like Na3PS4, Na3Zr2Si2PO12 in practical use. By modeling these solid/liquid or solid/solid interfaces we can gain a better understanding of the complex transport processes and reaction mechanisms at these interfaces and improve the rate performance. 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