Molecular Dynamics Studies on the Lithium Ion Conduction Behaviors

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C: Energy Conversion and Storage; Energy and Charge Transport

Molecular Dynamics Studies on the Lithium Ion Conduction Behaviors Depending on Tilted Grain Boundaries with various symmetries in Garnet-Type LiLaZrO 7

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Hiromasa Shiiba, Nobuyuki Zettsu, Miho Yamashita, Hitoshi Onodera, Randy Jalem, Masanobu Nakayama, and Katsuya Teshima J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b06275 • Publication Date (Web): 23 Aug 2018 Downloaded from http://pubs.acs.org on August 28, 2018

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The Journal of Physical Chemistry

Molecular Dynamics Studies on the Lithium Ion Conduction Behaviors Depending on Tilted Grain Boundaries with various symmetries in Garnet-Type Li7La3Zr2O12 Hiromasa Shiiba1, Nobuyuki Zettsu* 1,2, Miho Yamashita1, Hitoshi Onodera1, Randy Jalem3,4, Masanobu Nakayama3,4,5, and Katsuya Teshima* 1,2 1

Department of Materials Chemistry, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano 3808553, Japan. 2

Center for Energy and Environmental Science, Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan.

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Global Research Center for Energy Based Nanomaterials Science (GREEN), National Institute for Materials Science.

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2

Materials research by Information Integration" Initiative (Mi i), National Institute for Materials Science.

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Frontier Research Institute for Materials Science (FRIMS), Nagoya Institute of Technology, Gokiso, Showa, Nagoya, Aichi 466-8555, Japan. ABSTRACT: Grain boundary (GB) structure is a critical parameter that significantly affects the macroscopic properties of materials; however, the evaluation of GB characteristics by modern analytical methods remains an extremely challenging task. In this work, Li+ conductivity degradation at the GBs of cubic Li7La3Zr2O12 (LLZO) with a garnet framework (which represents the most promising candidate material for solid electrolytes utilized in all-solid-state batteries) has been investigated by various molecular dynamics approaches combined with newly developed analytical techniques. It was found that the transboundary diffusion of Li ions was generally slower than their diffusion in the bulk regardless of the GB symmetry; however, this effect strongly depended on the concentration of Li-deficient sites (trapping Li vacancies) in the GB layer. Furthermore, the compactness and density of the combined GB regions represent key parameters affecting the overall Li+ conductivity of polycrystalline LLZO films.

INTRODUCTION Grain boundaries (GBs) have been widely studied by various researchers due to their importance in scientific and technological applications. The addition of chemical dopants to GBs significantly affects the local structure and general material properties; for instance, it enhances the critical current density in high-temperature superconductors and facilitates oxygen incorporation into oxide-ion conductors.1-10 Therefore, various phenomenological understandings of the GB-related chemical and physical processes have been achieved based on the results of both experimental and theoretical studies in thermodynamics.11 While these findings can help to elucidate general concepts of the GB-dependent phenomena, the atomistic understanding of movements inside GBs within a reasonable time scale has not been fully clarified, which severely limits our ability to control and design GB structures with specific properties for many technologically important materials. Recently, all-solid-state Li-ion batteries have attracted significant attention due to their high energy densities (originated from the device miniaturization) and high safety caused by their non-flammability.12, 13 Many basic concepts of all-solid-state (oxide) batteries are strongly

related to various characteristics, such as safety and high energy density; however, possible enhancements of their basic properties, which can be evaluated by the currently used analytical techniques, become complicated for various technical reasons. In particular, the Li+ conductivity at interface and/or grain boundary of oxide-based electrolytes is more than three orders of magnitude lower than those of other electrolyte systems, including liquids, gelpolymers, and sulfides, which represents the most important unresolved issue related to the full realization of all-solid-state batteries. Various oxide electrolytes such as perovskite-type Li3xLa2/3-x□1/3-2xTiO3 (LLTO), LISICON, NASICON, and garnet-type Li7-xLa3A2-x4+Bx5+O12 (A = Zr, B = Nb, Ta) have been widely investigated; in particular, garnet-type Li7La3Zr2O12 (LLZO) and its derivatives are considered promising materials due to their high Li+ conductivity of around 10-4 S cm−1 at room temperature (evaluated via electrochemical impedance spectroscopy (EIS)) and wide electrochemical window (they are neither oxidized nor reduced in a wide voltage range).14-18 Furthermore, computational studies have been performed to investigate the Li+ conduction behavior of these materials in the bulk state.19-21 Thus, Jalem et al. reported that Li+ migration is

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driven by the simultaneous cooperative motion characterized by long multiple-site successive hops with a very small time scale for fluctuations at intermediate positions.19 Despite the larger number of detailed studies utilizing various experimental and theoretical approaches1422 , the experimentally determined Li+ conductivity did not precisely match the computationally predicted values. For example, the activation energy of cubic LLZO predicted via ab initio molecular dynamics (MD) simulations was equal to 0.1–0.3 eV19-22, which corresponded to the Li+ conductivity of around 10−2 S cm-1 at 300 K. In contrast, the room-temperature activation energy of Li+ conduction in densely sintered ceramics manufactured from cubic LLZO (evaluated by EIS coupled with a numerical analysis approach using equivalent circuit models) was 0.29–0.4 eV (around 10-4 S cm-1).20, 23-25 It may be difficult to argue about the importance of GB conductivity according to the impedance analysis based on circuit models, the observed discrepancy can be attributed to the ambivalent properties of GBs. Various technological limitations of both the experimental and theoretical approaches utilized for elucidating the Li+ conduction behavior of GBs (which originate from the low electron density of Li and low LLZO stability in the presence of the electron beam of a scanning transmission electron microscope) have restricted our ability of reaching a deeper understanding of the processes occurring at the atomic level. Furthermore, while the recent progress in computational science has considerably clarified the mechanisms of bulk and surface physicochemical processes, the investigation of the events occurring at GBs using ab initio density functional theory remains an extremely challenging task due to the very large computational costs, which can be reduced by performing classical MD simulations.26 Very recently, Yu and Siegel demonstrated Monte Carlo simulations on the lithium ion conduction behaviors at the GB in the LLZO solid electrolyte, and indicated possible presence of an anisotropic diffusion phenomena, appeared within Σ3 boundary.27 In this study, we independently analyzed the Li+ conducting behavior of the LLZO cubic phase with tilted GBs with various symmetries by using classical MD approaches to examine their properties at the atomic level. METHODS MD simulations were performed using a Born-like description of the ionic crystal lattice28. The long-range Coulombic interactions were summed via the Ewald method29, whereas the short-range interactions were described using the parameterized Buckingham pair potentials30. The latter were summed to the cut-off value of 10.5 Å, beyond which the influence of the potential was considered negligible. The lattice energy was defined as

 qi q j  − rij E L = ∑∑  + Aij exp  ρ ij i j >i  4πε 0 rij  

  Cij −   r6   ij

   

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free space. The Buckingham potential parameters, Aij, ρij, and Cij, were specific to the pairs of interacting species (the utilized simulation parameters are listed in Table S331). The DL POLY simulation package32 was used for all MD calculations (the corresponding time step was equal to 1 fs). First, the initial models were equilibrated for 20,000 time steps (20 ps) in the NPT ensemble at a temperature of 300 K. During this initial period, the volume of the cell was allowed to relax with time. The Nosé-Hoover thermostat and barostats33,34 were used to control the temperature and pressure, respectively. Afterwards, the temperature was elevated from 300 to 1700 K at a rate of 10 K per 5 ps, and the NPT dynamic simulations were conducted for 100 ps, during which the cell angles were allowed to relax. After equilibration at 1700 K, the obtained structure was cooled to a desired temperature, which allowed the lattice to fully equilibrate in a shorter simulation time as compared to that required for the direct heating of the system to the target temperature. Cooling was performed via an iterative procedure, which involved decreasing the temperature in 10 K steps accompanied by dynamic simulations with durations of 5 ps. For each studied temperature, MD simulations containing 500,000 time steps (500 ps) were performed using the constant volume and temperature ensemble in order to obtain statistical information about diffusion rates. The bulk LLZO calculations were performed using the 3 × 3 × 3 unit cell superstructure with the cubic symmetry containing 5184 atoms. The initial GB models with Li–Li， La–La， Zr–Zr， and O–O atomic distances shorter than 0.5 Å were eliminated to prevent unreasonable atomic repulsions. The electronic structure of the Σ3 (2-1-1) = (1-21) GBs was evaluated by conducting ab initio DFT calculations (the projected densities of states obtained for the La, Zr, and O atoms in the bulk and GB regions are shown in Figure S1). It was confirmed that the simulated GBs contained La3+, Zr4+, and O2- species. The GB energy γGB was defined as

γ GB =

where rij was the separation between ions i and j, qi and qj were the ion charges, and ε0 was the permittivity of the

) (2)

where A was the GB area, EGB was the lattice energy of the GB model, Ebulk was the lattice energy per atom, and N was the number of atoms in a particular GB model. To investigate the ionic transport properties of a particular structure, the MSD of the ions was monitored as a function of time at different temperatures. For a system with N ions, the MSD of ion i at position ri(t + t0) and time t with respect to its initial position ri(t0) was defined as

r 2 (t ) = (1)

1 (E GB − NE bulk 2A

1 N

N

∑ (r (t + t ) − r (t ))

2

i

i =0

0

i

0

(3)

The Li diffusion coefficient D was calculated from the MSD slope of the following function35:

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ri (t + t0 ) − ri (t0 )

2

= 6Dt + B

(4)

where B is the atomic displacement parameter related to thermal vibrations. The Li ionic conductivity σLi was calculated using the Nernst-Einstein equation36:

σ Li = c Li ( z Li F )

2

DLi RT

(5)

where cLi is the Li carrier density, zLi is the Li charge, F is Faraday’s constant, R is the gas constant, and T is the temperature. The Li ionic conductivity values were calculated in the temperature range of 700–1700 K. Since the obtained Li+ conductivities included both the bulk and GB components, their magnitudes were affected by the distances between neighboring GBs. In order to eliminate the geometry factor, a special conversion formula was used to separate the Li ionic conductivities in the bulk and GB regions and estimate the local Li ionic conductivity at the GBs. It was assumed that the total resistance Rtotal utilized in the GB models contained two different types of resistances: bulk resistance (Rbulk) and GB resistance (RGB). In this case, the total resistance can be expressed as follows:

lRtotal = aRGB + (l − a )Rbulk

(6)

where l is the distance between GBs, and a is the thickness of the GB region. The bulk resistance was obtained from the corresponding bulk calculations, and the thickness of the GB region was estimated from the variations of the Li+ concentration along the axis perpendicular to the GB surface. After that, the RGB values were obtained for all the used GB models. The Li ion conductivities derived from the values of RGB were considered the local conductivities within the GB regions. The fraction of the GB contribution to the total resistance x was computed using the following formula:

x=

aRGB

aRGB + (l − a )Rbulk

(7)

RESULTS AND DISCUSSION Figure 1 summarizes the GB formation energies calculated for eight stoichiometric equilibrium models at various temperatures using Eq. (2) for the evaluation of the GB contribution to the Li+ conductivity of LLZ, including Σ3 (2-1-1) = (1-21), Σ3 (100)×(2-12), Σ3 (1-10) = (0-11), Σ3 (110)×(411), Σ5 (031) = (03-1), Σ7 (3-2-1) = (2-31), Σ9 (1-14) = (-114), and Σ11 (1-13) = (-113). It was found that all the tilted GBs exhibited different formation energies. In particular, the tilted GBs represented by the Σ3 (2-1-1) = (1-21) and Σ3 (1-10) = (0-11) models were characterized by the first and second lowest GB energies of 0.42 J m-2 and 0.74 J m-2 at 700 K, respectively, which were consistent with the thermodynamically stable faces of the LLZO crystal grown from a molten LiOH flux (see Figure S2). In contrast, the Σ3 (100) × (2-12) GB model exhibited the highest GB ener-

gy of 2.06 J m-2 at 700 K. All the obtained GB formation energies were of the same order of magnitude regardless of temperature. In addition, no significant changes in their values were observed below 1200 K, suggesting that the combined GB structures remained the same at the atomic level. However, their magnitudes significantly increased at temperatures greater than 1300 K, suggesting that the LLZ framework near the GBs was randomized via incongruent melting. Figures S3 – S10 displays the trajectories calculated with respect to the available La (light brown), Zr (grey), and O (red) crystallographic sites of the eight stoichiometric equilibrium atomic GB models by performing MD simulations at a temperature of 1300 K and duration of 500 ps. No migrations of La, Zr, and O atoms were observed. Each model consisted of the bulk and GB regions with the disordered atomic arrangements at the center and both ends. The original garnet framework remained intact in the bulk region (without GBs) at temperatures below 1300 K. First, simulations using isothermal-isobaric (NPT) ensembles were performed for both the tetragonal and cubic frameworks as initial structures at durations of 500 ps to determine the most energetically stable bulk structure of stoichiometric LLZ. Their lattice energies and lattice constants as well as the Li occupancies of the 24d and 48g/96h sites were equilibrated at the same parameters of the cubic structure and temperature of 300 K after 20 ps regardless of the initial configurations (see Figure S11 and Table S1). Thus, no criteria for distinguishing between the tetragonal and cubic phases were obtained in this study. Furthermore, the calculated lattice parameters were in good agreement with the experimental values (the detailed results are presented in the Supporting Information section). Figure S12 displays the radial distribution function (RDF) plots obtained for the Li−Li, La−La, Zr−Zr, and O−O interactions in the bulk and at the GBs with different symmetries from the MD simulations with total durations of 500 ps conducted at a temperature of 700 K (the differences between the bulk and GB models are summarized in Figure S13). Remarkable differences were observed for the La, Zr, and O sites, as compared to Li atoms; in particular, the smallest values were obtained for the tilted GBs of the Σ3 (2-1-1) = (1-21) model and the largest ones − for the GBs of the Σ3 (100) = (2-12) GB model. Furthermore, the atomic displacements within the GB regions correlated with the GB formation energies. As shown in Figure 2, the observed trends were consistent with the GB energies of the used models, suggesting that the bulk LLZ framework remained virtually unchanged at the tilted GBs of Σ3 (2-1-1) = (1-21) with the smallest difference. In contrast, the atomic arrangements at the Σ3 (100) × (2-12) GBs with the highest difference was highly randomized, resembling an amorphous phase with a thickness of around 1.8 nm. Figures 3(a−d) describe the computationally predicted apparent Li+ conductivities of the bulk and GB models,



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including the average and separated conductivities along the a, b, and c axes (it should be noted that the Li+ conductivities of the utilized GB models included bulk conductivities because of the presence of bulk regions). The corresponding Li+ diffusion coefficient and Li+ conductivity at a specified temperature can be evaluated from the slopes of mean square displacements (MSDs) plotted against time and the Nernst-Einstein equation, respectively. Li ionic conductivity at room temperature was evaluated by the extrapolation of the results of hightemperature simulations because relatively long MD runs were required for conducting low-temperature simulations (below 700 K). It should be noted that at very short simulation times, the Li ions initially located in the middle of the bulk region are unable reach the GB region.

in Figure 5(a)) on the GB formation energy (obtained for the energetically stable GBs) indicate that the local Li+ conductivity decreases with decreasing GB energy. It should be noted that the most energetically stable tilted GB of the Σ3 (2-1-1) = (1-21) model exhibited relatively small conductivity, which was more than three orders of magnitude lower than the bulk value. Since the total Li+ conductivity along the c-axis demonstrated the trend observed for the GB formation energy (Figure 5(b)), it was concluded that the thermodynamically stable GBs were unable to provide sufficiently fast diffusion of lithium ions in the cubic structure of Li7La3Zr2O12 solid electrolyte (it should be noted that the fastest diffusion of Li ions was observed for the metastable GB structures of the Σ9 (1-14) = (-114) model).

According to the computational results presented in this work, the isotropic ion conducting characteristics of bulk LLZO with a cubic lattice can be reproduced, as shown in Figure S14. In contrast, the results of MSD analysis conducted for the tilted GBs indicate that the anisotropic Li-ion conducting characteristics strongly depend on their symmetry. It was found the bulk conductivities exhibited the highest values (in other words, the GB conductivity was smaller than the bulk one regardless of the GB orientation), whereas the conductivities measured along the c-axis (perpendicular to the GBs plane) were characterized by the lowest values. We found that the local conductivity across the GB plane primary contributes to the reduction in the total ion conductivity of LLZO. It is consistent with the computational analysis for the Σ3 GB in LLZO, analyzed by Yu and Siegel using static GB models.27 In particular, the local c-axis conductivity within the GB layer is more than one order of magnitude lower than the total conductivity of the material (Figure 4(a)). Furthermore, the observed reduction in the Li ion conductivity is highly dependent on the GB structure, which mirrors the trend observed for the activation energy of Li ion diffusion (see Table S2). Although the majority of previously conducted EIS studies for the evaluation of Li ion conductivity in the LLZO ceramics were unable to characterize the contribution of tilted GBs, the computations performed in this work strongly suggest that the observed discrepancy between the experimental and computational Li+ conductivities can be potentially due to the effect of the GB resistances caused by their structural variations at the atomic level.

Calculating time changes for the Li+ ions occupying the crystallographic sites of the studied region allows the prediction of possible ionic conduction routes in the LLZO framework at a specified temperature. The Li ion trajectories in the bulk LLZO depicted in Figure S15 – S22 show that Li+ species diffused along the 24d–48g/96h–24d continuous three-dimensional network pathways inside the bulk region, where each 24d tetrahedron site was connected to its neighbors via four face-sharing bridging 48g/96h octahedral sites. Such pathways are in good agreement with the recent results of the neutron powder diffraction analysis of LLZO37 and ab initio densityfunctional theory (DFT) and empirical computational studies.31,38 The crystallographic distortion resulting from the formation of tilted GBs deteriorated the Li+ diffusion properties.

The values of the fitted total average Li ionic conductivity (including both the bulk and GB components) and local Li ionic conductivity along the c-axis (Li-ion conductivity across the GB plane) calculated from the Arrhenius plots for all GB models at 300 K are summarized in Figure 4(b). The computationally predicted bulk Li+ conductivity was in good agreement with the data presented in previous reports21, 31; however, it was almost two orders of magnitude higher than the experimental value evaluated by EIS for the pelletized LLZO samples.17, 25 The dependences of the local Li ion conductivities (summarized

In order to further elucidate the Li+ diffusion characteristics at the GBs in terms of their chemical compositions, the variations of the Li+ concentrations near the GBs were calculated along the axis perpendicular to the GB surface (see Figure 6). The Li population was constantly modulated along the c-axis in the bulk region; however, it was anomalously dispersed in the vicinity of the GB plane. For example, the average Li contents at the GBs were equal to 6.57 and 6.77 for the stable Σ3 (2-1-1) = (1-21) and metastable Σ9 (1-14) = (-114) models, respectively. This reduction in the Li+ concentration suggests the formation of Lideficient sites (trapping Li vacancies) in the GB region, which potentially represents the primary reason for the degraded ionic conductivity across the GB plane. Similar trends were obtained for all GB models utilized in our computational studies. The decreased number of Lideficient sites in the combined tilted GBs lowered their Li ion conductivities. Despite the different crystal structures examined in our study, a similar decrease in the Li ionic conductivity at the GBs was observed for perovskite LLTO.39 Finally, the contribution of the thickness the GB layer to the total resistance of LLZO was investigated. Since its magnitude utilized in the computational models (around 5 nm) was significantly smaller than the experimentally observed LLZO domain sizes formed in sintered ceramics


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(in the micrometer range), the calculated contribution of GBs was highly overestimated (compared to the experimental data). Ohta et al. found experimentally that the contribution of the GB resistance to the total resistance of the pelletized LLZO ceramics was equal to 12%.17 In order to minimize the difference between the computational and experimental results, the contribution of the GB layer thickness to the total resistance of LLZO was examined as a function of the distance between the nearest GB neighbors (Figure 7). As expected, the GB contribution to the total resistance decreased with increasing distance; however, it was also found for the first time that its value strongly depended on the symmetry of the GB structure. For example, at a neighboring GB distance of 1 μm, the Σ9 (1-14) = (-114) GB model exhibited the smallest contribution of 9.3%; however, its magnitude increased to the maximum of 91% for the tilted Σ3 (2-1-1) = (1-21) GB model. When the thickness of the GB layer increased to 80 μm, the thermodynamically stable Σ3 (2-1-1) = (1-21) GB exhibited a contribution of 11%, which was close to the experimental value. Therefore, the performed simulations for the GB contribution to the Li+ conductivity can be used for predicting the experimental data. The same approach can be applied to evaluating the effect of the thickness of the GB layer on the total average resistance in all directions. Thus, the GB contribution to the degradation of Li+ conductivity was relatively small. For example, the contributions of the GB layers with thicknesses of 1 μm to the total resistance determined using the Σ9 (1-14) = (-114) and Σ3 (2-1-1) = (1-21) GB models were 3.0% and 5.6%, respectively, which suggested that the diffusion of Li+ ions along the GB plane could proceed much faster than that across the GB plane. Further studies on the dopants effects such as Al, Ta, and Nb at Zr site which are often used to stabilize the cubic structure of LLZO in the experiments will be the subjects of a later articles. Briefly, we have reported the analysis of Li+ conductivity at Σ3 (2-1-1) = (1-21) GB models for Li5La3Nb2O12 based on the similar molecular dynamics analogy,40 and we found those doping significantly affected the stability and Li+ conductivity at GBs. In the present study, the energetics, compositions, and Li ionic transport properties of eight symmetrically tilted GBs of garnet LLZO were theoretically examined at the atomic scale to achieve a better understanding of the GBdependent phenomena. The results of classical MD simulations revealed that Li+ conductivity within and across the GB layer was generally reduced to the bulk conductivity; however, this effect was highly dependent on the offstoichiometric Li-ion composition with different GB structures, as indicated by the obtained atomic trajectories. Thus, the presence of Li+ vacancies inhibited the diffusion across the GB plane, leading to a decrease in conductivity by four orders of magnitude with respect to the bulk conductivity. Furthermore the largest decrease in conductivity was observed for the symmetrically tilted GBs with the lowest formation energies, indicating that the latter parameter correlated with the thermal stability

of GBs. Their high symmetry can be maintained by removing the highly mobile lithium ions from the GB region via geometry relaxation for 500 ps. Since the calculations performed in this work were restricted to the stoichiometric composition of the studied system, it is highly probably that excessive lithium ions are present at the interface between the bulk and GB regions. In other words, at the infinite relaxation time (corresponding to a grand canonical ensemble), all excessive lithium atoms would diffuse to the LLZO surface to form a Li2O layer. From the obtained results, it can be concluded that the deterioration of ion conductivity across the GB layer can be minimized by the presence of tilted GBs with a lower symmetry. To achieve this goal, the formation of metastable GB structures through a kinetically-controlled reaction must be investigated under diffusion-limited conditions, such as liquid phase sintering with lower solubility solvent. FIGURES

Figure 1. (a) GB energies calculated for each GB model at different temperatures and (b) its enlarged figure.

Figure 2. Correlations of the GB energies with the RDF differences between the bulk and GB models calculated for the (a) La−La, (b) Zr−Zr, and (c) O−O interactions.



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(110)×(411), (e) Σ5 (031) = (03-1), (f) Σ7 (3-2-1) = (2-31), (g) Σ9 (1-14) = (-114), and (h) Σ11 (1-13) = (-113) GB models.

Figure 7. Contribution of the thickness of the GB layer to the total resistance as a function of the distance between the nearest neighboring GB layers: (a) the local conductivity along c-axis (across the GB layer) and (b) the total threedimensional conductivity. Figure 3. (a) Three-dimensional average and local Li ionic conductivities calculated along the (b) a, (c) b, and (d) c axes using the bulk and GB models, respectively.

Figure 4. (a) Local Li ionic conductivity calculated along the c-axis (across the GB layer). (b) Total and local Li ionic conductivities across the GB layers at 300 K obtained via MSD analysis.

ASSOCIATED CONTENT Supporting Information. Tables of Li occupancies in cubic and tetragonal LLZO, local Li ionic conductivity across the GB layer and activation energies, parameters of the Buckingham interionic potentials. Figures of SEM images of LLZO, trajectories of Li, La, Zr, and O atoms for various GB models, variations of the lattice energies and lattice constants of cubic and tetragonal LLZO, RDF differences between the bulk and GB models, Li ionic conductivity in bulk LLZO along each axes, and PDOS of the GB model. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author * E-mail: [email protected], and [email protected]

Author Contributions H. S., N. Z., R. J., M. N., contributed the computational study of the LLZO crystals and grain boundaries. H. S., N. Z. contributed drafting of this paper. N. Z. M. N., and T. K. contributed to make the concept and design of this study.

ACKNOWLEDGMENT Figure 5. (a) Relationships between the GB energies and (a) the local Li ionic conductivity across the GB layer, and (b) the total Li ionic conductivity along the c-axis.

This work was partially supported by JST-CREST. R. J. acknowledges JST-PRESTO. M. N. and R. J. acknowledges 2 NIMS-Mi i. This work was partially supported by JST-CREST. R. J. acknowledges JST-PRESTO. M. N. and R. J. acknowledg2 es NIMS-Mi i. The crystal structure figures were drawn with 41 42 VESTA and VMD .

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Figure 6. Changes in Li populations along the directions perpendicular to the GB layers calculated from the corresponding Li ion trajectories at 700 K using the (a) Σ3 (2-1-1) = (1-21), (b) Σ3 (100)×(2-12), (c) Σ3 (1-10) = (0-11), (d) Σ3

(2)

(3)

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(4)

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(11) (12)

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(19)

(20)

The Journal of Physical Chemistry ceptor-state formation at ZnO grain boundaries. Phys. Rev. Lett. 2006, 97, 106802. Klie, R. F.; Buban, J. P.; Varela, M.; Franceschetti, A.; Jooss, C.; Zhu, Y.; Browning, N. D.; Pantelides, S. T.; Pennycook, S. J. Enhanced current transport at grain boundaries in high-Tc superconductors. Nature 2005, 435, 475–478. Shibata, N.; Pennycook, S. J.; Gosnell, T. R.; Painter, G. S.; Shelton, W. A.; Becher, P. F. Observation of rare-earth segregation in silicon nitride ceramics at subnanometer dimensions. Nature 2004, 428, 730–733. Guo, X.; Waser, R. Electrical properties of the grain boundaries of oxygen ion conductors: acceptor-doped zirconia and ceria. Prog. Mater. Sci. 2006, 51, 151–210. Lu, K.; Lu, L.; Suresh, S. Strengthening materials by engineering coherent internal boundaries at the nanoscale. Science 2009, 324, 349–352. Lee, W.; Jung, H. J.; Lee, M. H.; Kim, Y.-B.; Park, J. S.; Sinclair, R.; Prinz F. B. Oxygen surface exchange at grain boundaries of oxide ion conductors. Adv. Funct. Mater. 2012, 22, 965–971. Nie, J.; Zhu, Y. M.; Liu, J. Z.; Fang, X. Y. Periodic segregation of solute atoms in fully coherent twin boundaries. Science 2013, 340, 957–960. Feng, B.; Yokoi, T.; Kumamoto, A.; Yoshiya, M.; Ikuhara, Y.; Shibata, N. Atomically ordered solute segregation behaviour in an oxide grain boundary. Nat. Commun. 2016, 7, 11079. Sutton, A. P.; Balluffi, R. W. Interfaces in Crystalline Materials; Oxford University Press: Oxford, U.K., 1995. Thangadurai, V.; Narayanan, S.; Pinzaru, D. Garnet-type solid-state fast Li ion conductors for Li batteries: critical review. Chem. Soc. Rev. 2014, 43, 4714-4727. Takada, K.; Ohta, N.; Tateyama, Y. Recent Progress in Interfacial Nanoarchitectonics in Solid-State Batteries. J. Inorg. Organomet. Polym. 2015, 25, 205-213. Huang, M.; Dumon A. Effect of Si, In and Ge doping on high ionic conductivity of Li7La3Zr2O12. Electrochem. Comm. 2012, 21, 62-64. Narayanan, S.; Epp, V.; Wilkening, M.; Thangadurai, V. Macroscopic and microscopic Li+ transport parameters in cubic garnet-type ‘‘Li6.5La2.5Ba0.5ZrTaO12’’ as probed by impedance spectroscopy and NMR. RSC Advances 2012, 2, 2553-2561. Murugan, R.; Ramakumar, S.; Janani, N. High conductive yttrium doped Li7La3Zr2O12 cubic lithium garnet. Electrochem. Comm. 2011, 13, 1373-1375. Ohta, S.; Kobayashi, T.; Asaoka, T. High lithium ionic conductivity in the garnet-type oxide Li7−XLa3(Zr2−X, NbX)O12 (X = 0–2). J. Power Sources 2011, 196, 3342-3345. Li, Y.; Han, J.-T.; Wang, C.-A.; Vogel, S. C.; Xie, H.; Xu, M.; Goodenough, J. B. Ionic distribution and conductivity in lithium garnet Li7La3Zr2O12. J. Power Sources 2012, 209, 278281. Jalem, R.; Yamamoto, Y.; Shiiba, H.; Nakayama, M.; Munakata, H.; Kasuga, T.; Kanamura, K. Concerted Migration Mechanism in the Li Ion Dynamics of Garnet-Type Li7La3Zr2O12. Chem. Mater. 2013, 25, 425-430. Miara, L. J.; Ong, S. P.; Mo, Y.; Richards, W. D.; Park, Y.; Lee, J.-M.; Lee, H. S.; Ceder, G. Effect of Rb and Ta Doping on the Ionic Conductivity and Stability of the Garnet Li7+2x−y(La3−xRbx)(Zr2−yTay)O12 (0 ≤ x ≤ 0.375, 0 ≤ y ≤ 1) Superionic Conductor: A First Principles Investigation. Chem. Mater. 2013, 25, 3048-3055.

(21) Burbano, M.; Carlier, D.; Boucher, F.; Morgan, B. J.; Salanne, M. Sparse Cyclic Excitations Explain the Low Ionic Conductivity of Stoichiometric Li7La3Zr2O12. Phys. Rev. Lett. 2016, 116, 135901. (22) Meier, K.; Laino, T.; Curioni, A. Solid-State Electrolytes: Revealing the Mechanisms of Li-Ion Conduction in Tetragonal and Cubic LLZO by First-Principles Calculations. J. Phys. Chem. C 2014, 118, 6668-6679. (23) Lee, J.-M.; Kim, T. Y.; Baek, S.-W. Abstracts of Papers, 2012 MRS Fall Meeting & Exhibit, Boston, Nov 25-30, 2012; Materials Research Society: Pennsylvania, 2012; J13.02. (24) Murugan, R.; Thangadurai, V.; Weppner, W. Fast lithium ion conduction in garnet-type Li7La3Zr2O12. Angew. Chem., Int. Ed. 2007, 46, 7778−7781. (25) Shimonishi, H.; Toda, A.; Zhang, T.; Hirano, A.; Imanishi, N.; Yamamoto, N.; Takeda, Y. Synthesis of garnet-type Li7− xLa3Zr2O12−1/2x and its stability in aqueous solutions. Solid State Ionics 2011, 183, 48-53. (26) Kim, S.; Hirayama, M.; Taminato, S.; Kanno, R. Epitaxial growth and lithium ion conductivity of lithium-oxide garnet for an all solid-state battery electrolyte. Dalton Trans. 2013, 42, 13112-13117. (27) Yu, S.; Siegel, D. J. Grain Boundary Contributions to Li-Ion Transport in the Solid Electrolyte Li7La3Zr2O12 (LLZO). Chem. Mater. 2017, 29, 9639-9647. (28) Born, M.; Mayer, J. E. Zur Gittertheorie der lonenkristalle. Z. Phys. 1932, 75, 1-18. (29) Ewald, P. P. Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys. 1921, 64, 253-287. (30) Buckingham, R. A. The Classical Equation of State of Gaseous Helium, Neon and Argon. Proc. R. Soc. London, Ser. A 1938, 168, 264-283. (31) Jalem, R.; Rushton, M. J. D.; Manalastas, W.; Nakayama, M.; Kasuga, T.; Kilner, J. A.; Grimes, R. W. Insights into the Lithium-Ion Conduction Mechanism of Garnet-Type Cubic Li5La3Ta2O12 by ab-Initio Calculations. Chem. Mater. 2015, 27, 2821-2831. (32) Todorov, I. T.; Smith, W. The DLPOLY User Manual, version 4.01.1; Daresbury Laboratory, UK, 2010. (33) Nose, S. A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys. 1984, 81, 511519. (34) Hoover, W. G. Canonical dynamics: Equilibrium phasespace distributions. Phys. Rev. A: At., Mol., Opt. Phys. 1985, 31, 1695-1697. (35) Gillan, M. J. The simulation of superionic materials. Physica B (Amsterdam), 1985, 131, 157-174. (36) Zahn, D. Molecular dynamics simulation of ionic conductors: perspectives and limitations. J. Mol. Model. 2011, 17, 1531−1535. (37) Chen, Y.; Rangasamy, E.; Liang, C.; An, K. Origin of high Li+ conduction in doped Li7La3Zr2O12 garnets. Chem. Mater. 2015, 27, 5491-5494. (38) Adams, S.; Rao, R. P. Ion transport and phase transition in Li7−xLa3(Zr2−xMx)O12 (M = Ta5+, Nb5+, x = 0, 0.25). J. Mater. Chem. 2012, 22, 1426−1434. (39) Ma, C.; Chen, K.; Liang, C.; Nan, C.-W.; Ishikawa, R.; More, K.; Chi, M. Atomic-scale origin of the large grain-boundary resistance in perovskite Li-ion-conducting solid electrolytes. Energy Environ. Sci., 2014, 7, 1638-1642. (40) Zettsu, N.; Shiiba, H.; Onodera, H.; Nemoto, K.; Kimijima. T.; Yubuta, K.; Nakayama, M.; Teshima, K. Thin and dense solid-solid heterojunction formation promoted by crystal growth in flux on a substrate. Sci. Rep., 2018, 8, 96.



Page 8 of 43

(41) Momma, K.; Izumi, F. VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr. 2011, 44, 1272-1276. (42) Humphrey, W.; A. Dalke, A.; Schulten, K. VMD: visual molecular dynamics. J. Mol. Graphics 1996, 14, 33–38.


Page 9 of 43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


TOC Graphic


The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41


Page 10 of 43

Page 11 of 43

(c)

1.0

Σ7 (3-2-1)=(2-31) Σ3 (110)×(411)

0.9

Σ9 (1-14)=(-114)

0.8 Σ5 (031)=(03-1) 0.7

Σ11 (1-13)=(-113)

Σ3 (1-10)=(0-11)

0.6 0.5 0.4 Σ3 (2-1-1)=(1-21)

0.3 0.10

0.15 0.20 0.25 0.30 La-La |gGB(r) - gbulk(r)|

Σ9 (1-14)=(-114)

0.9 Σ7 (3-2-1)=(2-31)

0.8

Σ11 (1-13)=(-113) Σ5 (031)=(03-1)

0.7

1.0

Σ3 (110)×(411)

Σ3 (1-10)=(0-11)

0.6 0.5 0.4

Σ7 (3-2-1)=(2-31) Σ3 (110)×(411)

Grain boundary energy (J m-2)

(b)

1.0



1 2 3 4 5 (a) 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41


0.9 0.8

Σ9 (1-14)=(-114) Σ5 (031)=(03-1)

0.7

0.35

0.15 0.20 0.25 0.30 Zr-Zr |gGB(r) - gbulk(r)|


Σ3 (1-10)=(0-11)

0.6 0.5 0.4 Σ3 (2-1-1)=(1-21)

Σ3 (2-1-1)=(1-21)

0.3 0.10

Σ11 (1-13)=(-113)

0.35

0.3 0.10

0.15 0.20 0.25 0.30 O-O |gGB(r) - gbulk(r)|

0.35

The Journal of 3.6 Physical Chemistry

3.6

Total average

log(T) (S K cm-1)

3.4 3.2 3.0 2.8

Page 12 of 43

(b)

Total a-axis

3.4 3.2 3.0 2.8 2.6 2.4 2.2

2.6

2.0 0.6

0.8

1.0 3

-1

1.2

1.4

0.6

0.8

-1

3.6

Total b-axis

3.4

-1

1.2

1.4

-1

10 T (K ) (d)


(c)

1.0 3

10 T (K )


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41


(a)

3.2 3.0 2.8 2.6 2.4 2.2

3.6

Total Bulk Σ5 (031)=(03-1) Σ7 (3-2-1)=(2-31)

3.4 3.2

Σ3 (110)×(411) Σ9 (1-14)=(-114)

3.0 2.8

Σ3 (100)×(2-12)

2.6

Σ3 (2-1-1)=(1-21)

2.4

Σ3 (1-10)=(0-11)

2.2

2.0

c-axis

Σ11 (1-13)=(-113)

2.0 0.6

0.8

1.0 3

-1

1.2

1.4

0.6

-1

0.8

1.0 3

10 T (K )

-1

1.2 -1

10 T (K )


1.4

Page 13 of 43

σ (S cm-1) (at 300 K)

(b)


1 (a) 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41


3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2

Σ5 (031)=(03-1) Bulk

Σ3 (2-1-1)=(1-21) Σ3 (100)×(2-12) Σ11 (1-13)=(-113) Σ3 (1-10)=(0-11) Σ7 (3-2-1)=(2-31)

0.8

1.0 3

-1

Local c-axis

Bulk

3.3×10-2

-

Σ3 (2-1-1)=(1-21)

1.5×10-2

8.0×10-6

Σ3 (100)×(2-12)

5.3×10-4

5.4×10-6

Σ3 (1-10)=(0-11)

1.2×10-2

1.8×10-4

Σ3 (110)×(411)

1.2×10-2

1.6×10-4

Σ5 (031)=(03-1)

1.7×10-2

5.0×10-4

Σ7 (3-2-1)=(2-31)

2.2×10-2

4.0×10-4

Σ9 (1-14)=(-114)

2.7×10-2

2.0×10-3

Σ11 (1-13)=(-113)

1.7×10-2

1.4×10-3

Local c-axis

Σ3 (110)×(411)

Σ9 (1-14)=(-114)

0.6

Total average

1.2 -1

10 T (K )

1.4



(a)

Σ9 (1-14)=(-114)

Σ7 (3-2-1)=(2-31)

Σ11 (1-13)=(-113)

10-3

Σ9 (1-14)=(-114)

Σ5 (031)=(03-1) Σ3 (1-10)=(0-11)

10-4

Σ11 (1-13)=(-113)

-2

 (S cm-1)

 (S cm-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Page 14 of 43

-1 (b) 10

Σ7 (3-2-1)=(2-31) Σ3 (110)×(411)

10

Σ5 (031)=(03-1) Σ3 (1-10)=(0-11) Σ3 (110)×(411)

10-3

Σ3 (2-1-1)=(1-21) -5

10

Σ3 (2-1-1)=(1-21) -4

0.4

0.5 0.6 0.7 0.8 0.9 Grain boundary energy (J m-2)

1.0

10

0.4

0.5 0.6 0.7 0.8 0.9 Grain boundary energy (J m-2)


1.0

Page 15 of 43


0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.4

0.6

0.8

1.0

0.0

0.4

0.6

0.8

1.0

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

(h)

Population of Li (a.u.) 0.2

Fractional coordinates along c axis

0.0


(g)

Population of Li (a.u.)


0.2

0.2


(f)


0.0



0.2


(d) Population of Li (a.u.)

(c) Population of Li (a.u.)


1 2 3 4 5 6 7 8 0.0 9 10 11 (e) 12 13 14 15 16 17 18 19 200.0 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

(b) Population of Li (a.u.)

(a)

0.2

0.4

0.6

0.8



1.0

0.0

0.2

0.4

0.6

0.8


1.0


(a)

(b) 100 Σ3 (100)×(2-12) Σ3 (2-1-1)=(1-21)

Σ3 (110)×(411) Σ3 (1-10)=(0-11) Σ5 (031)=(03-1) Σ7 (3-2-1)=(2-31)

Σ9 (1-14)=(-114) Σ11 (1-13)=(-113)

GB contributions (%)


1 100 2 3 4 5 10 6 7 8 1 9 10 11 12 0.1 13 14 15 16 0.01 0.1 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Page 16 of 43

Σ3 (100)×(2-12)

10

Σ3 (2-1-1)=(1-21) Σ3 (110)×(411)

1

Σ3 (1-10)=(0-11) Σ5 (031)=(03-1)

0.1

Σ7 (3-2-1)=(2-31) Σ9 (1-14)=(-114) Σ11 (1-13)=(-113)

1 10 100 Distance between GBs (m)

0.01 0.1



Page 17 of 43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41




Li occupancy 24d

48g/96h

Cubic

0.617

0.858

Tetragonal

0.622

0.856


Page 18 of 43

Page 19 of 43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41


σ (S cm-1) (at 300 K)

Activation energy (eV)

Bulk

3.3×10-2

0.18

Σ3 (2-1-1)=(1-21)

8.0×10-6

0.41

Σ3 (100)×(2-12)

5.4×10-6

0.44

Σ3 (1-10)=(0-11)

1.8×10-4

0.32

Σ3 (110)×(411)

1.6×10-4

0.33

Σ5 (031)=(03-1)

5.0×10-4

0.31

Σ7 (3-2-1)=(2-31)

4.0×10-4

0.31

Σ9 (1-14)=(-114)

2.0×10-3

0.26

Σ11 (1-13)=(-113)

1.4×10-3

0.27


The Journal of Physical Chemistry 1 Interaction 2 3 Li0.7+ - O1.44 5 La2.1+ - O1.46 7 Zr2.8+ - O1.48 1.4- - O1.49 O 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Aij (eV)

ρij (Å)

Cij (eV Å6)

876.86

0.2433

0

14509.63

0.2438

30.83

1366.09

0.3181

0

4869.99

0.2402

27.22


Page 20 of 43

Page 21 of 43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41


{211} {110}



GB

Bulk

GB

Bulk

: Li : La : Zr :O

5 nm


Page 22 of 43

Page 23 of 43


GB 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Bulk

GB

Bulk

: Li : La : Zr :O

5 nm


GB


GB

Bulk

GB

Bulk

: Li : La : Zr :O

5 nm


Page 24 of 43

GB

Page 25 of 43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

GB


Bulk

GB

: Li : La : Zr :O

5 nm


Bulk

GB


GB

Bulk

GB

Bulk

: Li : La : Zr :O

5 nm


Page 26 of 43

GB

Page 27 of 43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

GB


Bulk

GB

: Li : La : Zr :O

5 nm


Bulk

GB


GB

Bulk

GB

: Li : La : Zr :O

5 nm


Page 28 of 43

Bulk

GB

Page 29 of 43

GB 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41


Bulk

GB

Bulk

: Li : La : Zr :O

5 nm


GB

(b) -5.40 -5.45 -5.50 -5.55


Page 30 of 43

13.2 Cubic Tetragonal

a-axis

13.1 13.0 12.9 12.8 12.7

-5.60 12.6 5

10 Time (ps)

15

20

13.2 b-axis

(d)

13.2 13.1

5

10 Time (ps)

c-axis

Cubic Tetragonal

13.1

0

Lattice constant (Å)

0


1 2 3 4 5 6 7 8 9 10 11 12 (c) 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Cubic Tetragonal

-5.35


Lattice energy (106 kJ mol-1)

(a)

13.0 12.9 12.8 12.7

15

20

Cubic Tetragonal

13.0 12.9 12.8 12.7

12.6

12.6 0

5

10 Time (ps)

15

20

0

5

10 Time (ps)

15


20

Page 31 of 43


1 (a)

(c)

4

4

3

3

(d) 6

gbulk(r)O-O

gbulk(r)Zr-Zr

gbulk(r)La-La

2

7

2

1

1

0

0

5 4 3 2 1

4

6

8

0

2

r (Å)

4

6

8

0 0

2

r (Å)

(f)

6

8

0

2

4

3

3

2

1

0

0

8

6

8

8 7 6

2

1

6

(h)

(g)

4

4

r (Å)

gGB(r)Zr-Zr

3 (2-1-1)=(1-21) 3 (100)(2-12) 3 (1-10)=(0-11) 3 (110)(411) 5 (031)=(03-1) 7 (3-2-1)=(2-31) 9 (1-14)=(-114) 11 (1-13)=(-113)

4

r (Å)

gGB(r)O-O

2

gGB(r)La-La

gGB(r)Li-Li

(b)

8

gbulk(r)Li-Li

2 3 4 4 5 3 6 7 2 8 9 1 10 11 0 12 0 13 14 (e) 15 16 4 17 18 3 19 20 2 21 22 1 23 0 24 25 0 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

5 4 3 2 1

2

4 r (Å )

6

8

0

2

4 r (Å)

6

8

0 0

2


4 r (Å )

6

8

0

2

4 r (Å)


0.4

(b) 0.4

0.2

0.2

{gGB(r) - gbulk(r)}La-La

{gGB(r) - gbulk(r)}Li-Li

(a)

0.0 3 (2-1-1)=(1-21)

-0.2

3 (1-10)=(0-11) 3 (110)(411)

-0.4

5 (031)=(03-1) 7 (3-2-1)=(2-31)

-0.6

0.0

-0.2

3 (100)(2-12)

-0.4

-0.6

9 (1-14)=(-114) 11 (1-13)=(-113)

-0.8

-0.8 0

2

4

6

8

0

2

r (Å)

4

6

8

6

8

r (Å)

0.4

(d) 0.4

0.2

0.2

{gGB(r) - gbulk(r)}O-O

(c)

{gGB(r) - gbulk(r)}Zr-Zr

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Page 32 of 43

0.0

-0.2

0.0

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8 0

2

4

6

8

0

2

r (Å)

4

r (Å) ACS Paragon Plus Environment

Page 33 of 43

3.6


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41


a b c

3.4

3.2

3.0

2.8 0.6

0.8

1.0 3

-1

1.2

1.4

-1

10 T (K )



GB

Bulk

GB

Bulk

5 nm


Page 34 of 43

Page 35 of 43


GB 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Bulk

GB

Bulk

5 nm


GB


GB

Bulk

GB

Bulk

5 nm


Page 36 of 43

GB

Page 37 of 43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

GB


Bulk

GB

5 nm


Bulk

GB


GB

Bulk

GB

Bulk

5 nm


Page 38 of 43

GB

Page 39 of 43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

GB


Bulk

GB

5 nm


Bulk

GB


GB

Bulk

GB

5 nm


Page 40 of 43

Bulk

GB

Page 41 of 43

GB 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41


Bulk

GB

Bulk

5 nm


GB


Page 42 of 43

La bulk La GB1 La GB2

10 5 0 -5

La bulk

-10

(states atom-1 eV-1)

-8 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0

-6

-4

-2 0 E - Ef (eV)

2

4

-6

-4

La GB2

6

La GB1

Zr Bulk Zr GB1 Zr GB2

-8


Density of states

Density of states Density of states


(a) 1 2 3 4 5 6 7 8 9 10 11 12(b) 13 14 15 16 17 18 19 20 21 22 23 24 25(c) 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

-2 0 E - Ef (eV)

2

4

6

Zr GB1 O bulk

2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0

Zr bulk

O bulk O GB1 O GB2

O GB2 Zr GB2

-8

-6

-4

-2 0 E - Ef (eV)

2

4

6


O GB1

Page 43 of 43

most stable GB Σ3 (2-1-1)=(1-21)

100


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41


Σ3 (100)×(2-12)

10 Σ3 (2-1-1)=(1-21)

Σ3 (110)×(411)

1

Σ3 (1-10)=(0-11) Σ5 (031)=(03-1) Σ7 (3-2-1)=(2-31)

Σ9 (1-14)=(-114)

Σ9 (1-14)=(-114)

0.1

0.01 0.1

Σ11 (1-13)=(-113)



fastest GB

Molecular Dynamics Studies on the Lithium Ion Conduction Behaviors

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