In this thesis, computational studies of some of the promising classes of Li/Na ion conducting solids are presented. The first chapter of the thesis is devoted to reviewing ongoing research on Li/Na ion conducting inorganic solids.
Ag + Ion Conductor
First reported by Kuhset al.back in the argyrodites is a center of research interest for a long time. Structural analysis over the years has concluded that there are two phases, the low temperature phase which stabilizes in various ordered structures and the higher temperature cubic phase with F¯43mspace group.
Li + and Na + Ion Conductors
- LISICON
- Argyrodite
- Lithium Garnets
- Perovskite-type
- NASICON-type
- Na 2 M 2 TeO 6 (M = Ni, Co, Zn or Mg)
Tilementet al.[243] investigated the effect of mixed doping (FeII/FeIII) on ionic conductivity in NASICON. Efforts were also made to stabilize the highly conductive phase, i.e. rhombohedral phases of NASICON at room temperature.
Summary
Studies have also shown that the ordering of framework cations, such as the Si/P ordering in the NASICON framework, plays a crucial role in determining ion transport in solids. Many computational studies, such as that of Na2Ni2TeO6, suggest that an optimal correlation between ions thermodynamically favors higher disorder of the mobile sub-lattice of ions, resulting in enhanced ion transport [297].
Objective of the thesis
Convergence of the profile (plotted as a function of the reaction coordinate, λi) with the progression of the MTD procedure shown in Figure 4.11. Thus, a microscopic profile of the free energy of ions in the migration channel is constructed.
Molecular Dynamics
- Classical MD
- Ewald Summation
- Aspects of Different Ensembles
- Ab Initio Molecular Dynamics
The standard algorithm starts with an initial set of atomic coordinates of the system of interest (usually obtained from experimental data) and their velocities (usually generated from a Maxwellian distribution). In other words, one simulation field boundary under PBC always "feels" the opposite field boundary. The accuracy of the results obtained by classical MD depends largely on the shape of the interatomic potential and the parameters used for the simulation.
The potential atr = 0, which arises due to the self-interaction between the Gaussian and the corresponding point charge, must be subtracted. A numerical calculation of the force on the nuclei using ~FI= −~∇I〈Ψ0|Hbe|Ψ0〉 is very expensive and can be highly inaccurate for simulations.
Brief Overview of Density Functional Theory
The second-order time derivative in Eq. (2.39b) clearly indicates that, unlike BOMD, the electronic part in this case is a dynamic quantity[ref]. Using electron density instead of wave function reduces the degrees of freedom of the many-body problem from 3Neto 3. Vext(~r)n(~r) (2.42) The third term in Eq. (2.42) is the external potential, the first and second terms are electronic kinetic energy and electron-electron interaction energy, respectively.
Nocc represents the number of occupied orbitals and fi is the occupancy number for statesi and related to the number of electrons (Ne) according to the following relationship. The term electronic kinetic energy is thus approximated as: T. Effectively, the actual many-body problem is now transformed into a non-interacting single-body problem with the same electron density.
Metadynamics
Nudged Elastic Band method
The mathematical details of the technique and its implementation are discussed in more detail in the respective chapters. This causes the image to climb up the band at maximum energy to reach the saddle point (note that the image is allowed to climb down the energy, but only perpendicular to the band).
Basic Analysis
- Radial Distribution Function
- Diffusivity and Conductivity
The decrease of M-O bond lengths with the size of the M ion is reflected in the radial distribution functions (shown in Fig. 2). As mentioned earlier (Figure 5.7), this particular path is part of a long link along their axis. The theoretical background of the methods used for the studies is described in Chapter 2.
The free energy landscape is later reconstructed from the information of the stored external potentials. Convergence of free energy barriers with simulation progress is demonstrated.
Simulation Details
However, in view of the smaller Li radius, the Li – Li and Li – O parameters are adjusted to get the conductivity of LiZr2(PO4)3 in accordance with the previously experimental reports [14-16]. The Ai j for the O-O pair is also slightly modified, as required for the high temperature stability of the LiM2(PO4)3 system. The ionic radii of the framework cations, M = Zr, Hf, Sn and Ti, are chosen from previous estimates [17].
Simulations on LiZr2(PO4)3 using larger supercell of 5 × 5 × 2 unit cells (consisting of 5400 atoms) showed excellent convergence of structural and dynamic properties, confirming the adequacy of the system size. The time step for the integration of the equation of motion is taken as 2 fs for all the simulations.
Results and Discussions
The average M-O bond lengths (given in Table 3.2) show a systematic reduction with the size of the M ion. After confirming the structural stability of the simulated systems, the dynamic properties related to lithium ion transport were examined. The self-diffusion of Li ions in these systems is deduced using Einstein's relation.
Basically, the higher the diffusivity of Li ions, the more diffuse the nature of Li–Li rdfs. Instead, an energy barrier limiting the mobility of Li+ ions is observed at the Li2 sites (3.3 Å).
Conclusions
It should be noted that the free energy of a 'process' (apart from a negligible additive due to the reference state) as a function of CV (in the present case, the position of the labeled. The total free energy barrier is calculated from accumulated bias potential for the MTD simulation, as described in Chapter 4. During the 750 ps of MTD, the labeled Li+ is found to explore a large part of the simulation box (Figure 5.4c).
The underlying free energy barrier is also calculated from the information on the bias potential deposited on the tagged Li+ ion. The total free energy barrier calculated from the information on the applied bias potential is found to be 0.53 eV.
Simulation Details
The initial structure of NASICONs was taken from the X-ray study of Boilotet al. Free Energy Calculation: One of the most useful pieces of information from MTD simulation is the process mechanism and its underlying free energy landscape. In the present work, the free energy profile for Na a+ along the migration channel is calculated by plotting F( ¯rs ), as a function of site distance, ¯rs , relative to any two of its nearest Na1 sites , λ, under periodic boundary conditions.
Note that mapping F( ¯rs ) from space coordinates ¯rs to F(λ ) is in the spirit of introducing a 'reaction coordinate', which reduces the dimensionality of the information and helps visualize the free energy profile along the migration channel. Potential energy profile: the average potential energy of individual N a+ ions at a given location, ¯ri, due to all other ions in the system.
Results and Discussions
- NaZr 2 (PO 4 ) 3
- Na 4 Zr 2 (SiO 4 ) 3
TheN a+−N a+RDF (Figure 4.4), calculated from the MTD trajectory, provides insight into the ion transport in the system. The first peak of the dynamic RDF corresponds to the Na1–Na2 distance (3.4 Å), the second peak to the Na2–Na2 distance (4.9 Å), and the third and fourth peaks. Calculation of the energy profile, F, for Na4Zr2(SiO4)3 has been carried out from the accumulated external potential in the same way as in the previous case.
The barrier heights of the potential energy profiles are about 0.15 eV, which is much lower than the barrier obtained from the free energy profile (~0.34 eV) as well as the experimental activation energy. On the other hand, for the Na4Zr2(SiO4)3 system, there is a local maximum of the formed external potential. Figure 4.13: Left: Three-dimensional map of hills deposited on the labeled N a+)-ion in Na4Zr2(SiO4)3.
Conclusion
This confirms the integrity of the framework structure throughout the entire running time of the MTD simulation. As shown in Figure (5.4d), the diffusion of the tagged Li+ ultimately also caused movement of the other Li+ ions in the system, and thus also movement. Another path was observed consisting of the sections of the path along the axis described previously (Figure 5.9).
A different approach may be required to determine the energy barriers for these different pathways from bias potential information. It should be noted here that the PS4 tetrahedron shown here is next to the starting position of the labeled Li+ ion.
Simulation Details
Ab-initioMD incorporated with well-tempered metadynamics (MTD) is performed on γ-Li3PS4 at constant cell volume and at a temperature of 400 K using the open source package Quantum ESPRESSO [9–11] . Well-tempered MTD is implemented following the procedure in the previous chapter, with initial tray height of 0.001 Ry, and the bias factor γ = 8.0. The energy barriers for the different pathways of Li+ ion are also calculated using the climbing image nudged elastic band method (CI-NEB) [13] as incorporated in Quantum ESPRESSO.
Results and Discussions
- Migration mechanism
- Energy Landscape
- Polyhedral flexibility
On the other hand, in the case of the MTD simulation, within 15 ps, the applied bias potential on one Li+-ion (labeled Li+) triggered a diffusion mechanism at the same temperature (Figure 5.4b). UNDERSTANDING THE MIGRATION MECHANISM OF Li+-IONS IN γ-LI3PS4: A FIRST PRINCIPLES-BASED STUDY USING METADYNAMICS AND NEB-COUPLED DIFFUSION PATHWAYS OF Li+-IONS ACROSS A SIMULATION FIELD. As shown in Figure 5.6, two adjacent sites V+ and V- are further connected via sites R located between sites G and B of the V-sections.
Again, unlike the MTD trajectory, the NEB pathway does not pass (although very close to the M/C) through any of the interstitial sites. Nevertheless, we were intrigued by the relatively larger fluctuation of the backbone tetrahedra, so we calculated the atomic density distribution of the PS4 tetrahedron from the 750 ps full trajectory data (Figure 5.13) to show the dynamics of the backbone tetrahedra.
Conclusion
This work further investigates this aspect to find that geometrical bottlenecks in the conduction channel are sensitive to approaching Li+ ions. It should be noted that frozen-frame simulations result in a significant drop in Li+ diffusion. In Chapter 5, the applicability of the metadynamics technique was extended to the low diffusivity system γ-Li3PS4 to understand the diffusion mechanism associated with Li+ in the system. Ab-initioMD simulations coupled to well-tempered metadynamics are run for a time length of 750 ps at a temperature of 400 K in the NVT ensemble.
It was observed that the accelerated Li+ diffusion induced by the applied bias potential caused larger oscillations in the framework tetrahedra. We hope that the works presented in this thesis will contribute to the concurrent research in the field of solid state ionics.
Bias Factor Dependence
Site Exploration
To demonstrate the correlated motion of the N a+-ion in Na4Zr2Si3O12, some N a+ are taken out of the system, along with their nearest neighbors. APPENDIX A. APPENDIX FOR CHAPTER 4. unsuccessfully hop to its neighbors at 6.4 Åwithin about 1.6 ns before hoping from the starting site of Na1. In Figure A.3a) and b) there are cases when the labeled ion after scattering to the far regions returns close to the original site, respectively after durations of 80 ns and 60 ns.
Thus, the ion explores the local minima quite extensively before it scatters, thus ensuring the quality of the sampling.
Comparison of Free Energy barrier
