Songge Yang

Guangchen Liu

Yueh-Lin Lee

Jean-Marc Bassat

Jacinthe Gamon

Antoine Villesuzanne

John Pietras

Xiao-Dong Zhou

Yu Zhong

A Systematic Ab Initio Study of Vacancy Formation Energy, Diffusivity and Ionic Conductivity of Ln 2 NiO 4+δ (Ln=La, Nd, Pr)

Keywords: ab initio, diffusivity, ionic conductivity, r-SOCs, activation energy barrier

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Introduction

In the past decade, the Ln2NiO4+δ (Ln=La, Nd, Pr) Ruddlesden-Popper (RP) family with the K2NiF4-type structure has attracted a lot of attention. Ln2NiO4+δ has been proposed as a performant air electrode material in reversible solid oxide cells (r-SOCs), which is capable of working in both solid oxide fuel cell (SOFC) mode and solid oxide electrolyzer cells (SOECs) mode [START_REF] Mogensen | Reversible solid-oxide cells for clean and sustainable energy[END_REF][START_REF] Choi | Performance of La0. 1Sr0. 9Co0. 8Fe0. 2O3-δ and La0. 1Sr0. 9Co0. 8Fe0. 2O3-δ-Ce0. 9Gd0. 1O2 oxygen electrodes with Ce0. 9Gd0. 1O2 barrier layer in reversible solid oxide fuel cells[END_REF][START_REF] Yoo | La2NiO4+ δ as oxygen electrode in reversible solid oxide cells[END_REF][START_REF] Vibhu | Electrochemical ageing study of mixed lanthanum/praseodymium nickelates La2-xPrxNiO4+ δ as oxygen electrodes for solid oxide fuel or electrolysis cells[END_REF][START_REF] Jiang | Bi-doped La1. 5Sr0. 5Ni0. 5Mn0. 5O4+ δ as an efficient air electrode material for SOEC[END_REF][START_REF] Ghamarinia | Electrochemical characterization of La2NiO4-infiltrated La0. 6Sr0. 4Co0. 2Fe0. 8O3-δ by analysis of distribution of relaxation times[END_REF]. Compared with the traditional air electrode materials, the Ln2NiO4+δ RP phase has a higher resistance to Cr-poisoning under SOFC operating conditions, which prevents performance degradation [START_REF] Egger | La2NiO4+ δ as electrode material for solid oxide fuel cells and electrolyzer cells[END_REF]. Additionally, the Ln2NiO4+δ structure has excellent oxygen transport properties [START_REF] Hyodo | Electrical conductivity and oxygen diffusivity in Cuand Ga-doped Pr2NiO4[END_REF][START_REF] Boehm | Oxygen diffusion and transport properties in non-stoichiometric Ln2-xNiO4+ δ oxides[END_REF][START_REF] Miyoshi | Mixed conductivity and oxygen permeability of doped Pr2NiO4-based oxides[END_REF][START_REF] Shaula | Oxygen transport in La2NiO4+ δ: Assessment of surface limitations and multilayer membrane architectures[END_REF][START_REF] Li | Effects of surface coatings on the determination of Dchem and kchem in La2NiO4+ δ by conductivity relaxation[END_REF][START_REF] Čebašek | Determination of inter-diffusion coefficients for the A-and B-site in the A2BO4+ δ (A= La, Nd and B= Ni, Cu) system[END_REF][START_REF] Egger | Enhanced oxygen surface exchange of La2NiO4+ δ by means of a thin surface layer of silver[END_REF][START_REF] Staykov | Oxygen Reduction Reaction and Electronic Properties of LnO-Terminated Surfaces of Pr2NiO4 and La2NiO4[END_REF][START_REF] Burriel | Anisotropic oxygen diffusion properties in epitaxial thin films of La 2 NiO 4+ δ[END_REF][START_REF] Ma | Synthesis of shape-controlled La 2 NiO 4+ δ nanostructures and their anisotropic properties for oxygen diffusion[END_REF][START_REF] Ananyev | Oxygen isotope exchange in La 2 NiO 4±δ[END_REF][START_REF] Han | Perovskite oxides for oxygen transport: Chemistry and material horizons[END_REF]. These structures consist of sheets of NiO6 octahedral sharing corners interleaved by Ln2O2 layers where the additional oxygen is localized [START_REF] Boehm | Oxygen diffusion and transport properties in non-stoichiometric Ln2-xNiO4+ δ oxides[END_REF][START_REF] Murata | Fabrication and characterization of oriented Nd2NiO4 bulk and cathode for low-temperature operating solid oxide fuel cell[END_REF]. Furthermore, it has been demonstrated that the interstitialcy mechanism with the oxygen occupying the interstitial layers is the main contributor to oxygen ion transport [START_REF] Munnings | Oxygen transport in the La2Ni1-xCoxO4+ δ system[END_REF][START_REF] Skinner | Oxygen diffusion and surface exchange in La2-xSrxNiO4+ δ[END_REF]. So far, extensive experimental studies have been conducted on the oxygen transport properties of the Ln2NiO4+δ compounds and their doped counterpart. Bassat et al. [START_REF] Bassat | Anisotropic oxygen diffusion properties in Pr2NiO4+ δ and Nd2NiO4+ δ single crystals[END_REF] measured the oxygen diffusion coefficient for single crystals of both Pr2NiO4+δ and Nd2NiO4+δ along the two main crystallographic orientations. They found the oxygen diffusion along the a-b plane for Pr2NiO4+δ is higher than that of Nd2NiO4+δ due to a much lower diffusion activation energy. Boehm et al. investigated the oxygen diffusion and transport properties in non-stoichiometric Ln2-xNiO4+δ (Ln=La, Nd, Pr) oxides [START_REF] Boehm | Oxygen diffusion and transport properties in non-stoichiometric Ln2-xNiO4+ δ oxides[END_REF]. They concluded that the oxygen diffusion performance of Pr2NiO4+δ is the highest among the studied compounds. Regarding the doped Ln2NiO4+δ compounds, Shen et al. observed that the ionic conductivity of La2NiO4+δ compounds increases significantly with the increasing Ca-doping level (up to x=0.1 of La2-xCaxNiO4+δ) on the La site [START_REF] Shen | Preparation and electrical properties of Ca-doped La 2 NiO 4+ δ cathode materials for IT-SOFC[END_REF]. Meanwhile, Hyodo et al. concluded that the doping of Cu and Ga enhanced the oxygen diffusion properties of the Pr2NiO4+δ [START_REF] Hyodo | Electrical conductivity and oxygen diffusivity in Cuand Ga-doped Pr2NiO4[END_REF].

The oxygen transport properties of the Ln2NiO4+δ compounds have been the object of various theoretical studies; see references [START_REF] Chroneos | Anisotropic oxygen diffusion in tetragonal La 2 NiO 4+ δ: molecular dynamics calculations[END_REF][START_REF] Parfitt | Molecular dynamics study of oxygen diffusion in Pr 2 NiO 4+ δ[END_REF][START_REF] Perrichon | Lattice dynamics modified by excess oxygen in Nd2NiO4+ δ: triggering low-temperature oxygen diffusion[END_REF][START_REF] Xu | Factors Controlling Oxygen Interstitial Diffusion in the Ruddlesden-Popper Oxide La2-x Sr x NiO4+ δ[END_REF][START_REF] Frayret | Application of density functional theory to the modeling of the mixed ionic and electronic conductor La2NiO4+ δ: Lattice relaxation, oxygen mobility, and energetics of frenkel defects[END_REF][START_REF] Minervini | Oxygen migration in La2NiO4+ δ[END_REF][START_REF] Villesuzanne | On the role of lattice dynamics on low-temperature oxygen mobility in solid oxides: a neutron diffraction and first-principles investigation of[END_REF][START_REF] Zhang | Oxygen migration and proton diffusivity in transition-metal (Mn, Fe, Co, and Cu) doped Ruddlesden-Popper oxides[END_REF][START_REF] Szaro | An ab initio study of the oxygen defect formation and oxide ion migration in (Sr1-xPrx) 2FeO4±δ[END_REF][START_REF] Li | Enhancement of ionic transport in complex oxides through soft lattice modes and epitaxial strain[END_REF][START_REF] Xie | Oxygen Point Defect Chemistry in Ruddlesden-Popper Oxides (La1-x Sr x) 2MO4±δ (M= Co, Ni, Cu)[END_REF][START_REF] Flura | Development of an Isotopic Oxygen Exchange Setup Operating Under High Pressure: Preliminary Measurements Performed on Ln2NiO4+ d (Ln= La, Pr, Nd, MIEC) Oxides[END_REF][START_REF] Sadykov | Transport properties of Ca-doped Ln2NiO4, 10TH International Conference on Sustainable Energy and Environmental Protection Hydrogen and Fuel Cells[END_REF] for examples. The interstitialcy diffusion mechanism was revealed by both atomistic [START_REF] Chroneos | Anisotropic oxygen diffusion in tetragonal La 2 NiO 4+ δ: molecular dynamics calculations[END_REF][START_REF] Parfitt | Molecular dynamics study of oxygen diffusion in Pr 2 NiO 4+ δ[END_REF] and first-principles molecular dynamics [START_REF] Perrichon | Lattice dynamics modified by excess oxygen in Nd2NiO4+ δ: triggering low-temperature oxygen diffusion[END_REF],

and the role of the apical oxygen atoms and lattice dynamics (low-energy-phonons) in this mechanism was assessed from first-principles [START_REF] Perrichon | Lattice dynamics modified by excess oxygen in Nd2NiO4+ δ: triggering low-temperature oxygen diffusion[END_REF][START_REF] Villesuzanne | On the role of lattice dynamics on low-temperature oxygen mobility in solid oxides: a neutron diffraction and first-principles investigation of[END_REF][START_REF] Li | Enhancement of ionic transport in complex oxides through soft lattice modes and epitaxial strain[END_REF]. The effects of strain and oxygen overstoichiometry were investigated by molecular dynamics [START_REF] Chroneos | Anisotropic oxygen diffusion in tetragonal La 2 NiO 4+ δ: molecular dynamics calculations[END_REF][START_REF] Parfitt | Molecular dynamics study of oxygen diffusion in Pr 2 NiO 4+ δ[END_REF] and DFT [START_REF] Xu | Factors Controlling Oxygen Interstitial Diffusion in the Ruddlesden-Popper Oxide La2-x Sr x NiO4+ δ[END_REF][START_REF] Frayret | Application of density functional theory to the modeling of the mixed ionic and electronic conductor La2NiO4+ δ: Lattice relaxation, oxygen mobility, and energetics of frenkel defects[END_REF][START_REF] Flura | Development of an Isotopic Oxygen Exchange Setup Operating Under High Pressure: Preliminary Measurements Performed on Ln2NiO4+ d (Ln= La, Pr, Nd, MIEC) Oxides[END_REF].

Additionally, Xu et al. [START_REF] Xu | Factors Controlling Oxygen Interstitial Diffusion in the Ruddlesden-Popper Oxide La2-x Sr x NiO4+ δ[END_REF] and Sadykov et al. [START_REF] Sadykov | Transport properties of Ca-doped Ln2NiO4, 10TH International Conference on Sustainable Energy and Environmental Protection Hydrogen and Fuel Cells[END_REF] discussed the possible existence of peroxide ions as intermediate species in the diffusion process with the DFT simulations. Defect energetics and their influence on diffusion were investigated by classical molecular dynamics [START_REF] Minervini | Oxygen migration in La2NiO4+ δ[END_REF] and DFT [START_REF] Frayret | Application of density functional theory to the modeling of the mixed ionic and electronic conductor La2NiO4+ δ: Lattice relaxation, oxygen mobility, and energetics of frenkel defects[END_REF][START_REF] Xie | Oxygen Point Defect Chemistry in Ruddlesden-Popper Oxides (La1-x Sr x) 2MO4±δ (M= Co, Ni, Cu)[END_REF].

Accordingly, the aim of the present work is to systematically investigate the vacancy formation energy, diffusivity, and ionic conductivity of the Ln2NiO4+δ (δ = 0.125) compounds through the ab initio approach by coupling 0 K DFT volume-dependent properties with the quasiharmonic approximation which gives the temperature-dependent free energy as a function of volume. The nudged elastic band (NEB) method is adopted to predict the energy barrier, diffusion coefficient, and ionic conductivity of the oxygen ions. In order to accurately describe the strong correlation effects of the transition metal Ni, and f-orbital Pr and Nd, the a rotationally invariant GGA+U approach was used to improve the accuracy of the calculations. Furthermore, considering the change of the oxygen transport properties caused by the thermal expansion effect, a combination of the quasi-harmonic approximation (QHA) and a linear regression model is used to study and reproduce the temperature-dependent properties of the Ln2NiO4+δ RP phases. According to QHA, the physical properties such as energy barrier [START_REF] Madsen | Calculating the thermal conductivity of the silicon clathrates using the quasi-harmonic approximation[END_REF], elasticity [START_REF] Wang | A first-principles approach to finite temperature elastic constants[END_REF], and magnetism [START_REF] Ma | Ab initio thermodynamics of the CoCrFeMnNi high entropy alloy: Importance of entropy contributions beyond the configurational one[END_REF] can be estimated as a function of temperature and volume when the thermal expansion is taken into account. This method enables predicting temperature-dependent properties solely caused by thermal expansion without using more fastidious molecular dynamic simulations. Finally, the comparison between the ab initio modeling and available experimental data from the literature is also discussed.

Methodology

Ab initio computational details

The ab initio calculations were applied using the Vienna ab initio simulation package (VASP) [START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF][START_REF] Kresse | Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set[END_REF]. Throughout the study, the generalized gradient approximation (GGA) [START_REF] Perdew | Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation[END_REF] with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation function [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] has been utilized using the projector augmented wave method. The k-point meshes with a density of no less than 5000 pra (per-reciprocal atom) were used to sample the Brillouin zones. The accurate total energy calculations were applied using the linear tetrahedron method with Blӧchl's corrections [START_REF] Blöchl | Improved tetrahedron method for Brillouin-zone integrations[END_REF]. The total energies obtained by relaxation calculations were converged to 10 -6 eV/cell with a 1.75 times plane-wave cutoff energy suggested by the corresponding element pseudopotentials. Meanwhile, the static calculations, which converged to 10 -8 eV/cell with 520 eV cutoff energy, were applied after each relaxation procedure. Finally, the calculated static energies were fitted by the Birch-Murnaghan equation of state [START_REF] Murnaghan | Finite deformations of an elastic solid[END_REF] for each structure to get a more accurate total energy. For the magnetic order, both ferromagnetic (FM) and antiferromagnetic (AFM) were considered for the Ni atoms (See S1 in the supplementary file). The present work adopted the structure with the most stable magnetic configuration.

Although the ab initio calculations provide good agreement with the experiments for metallic systems as well as a reasonable trend for the geometry and reactivity of metal oxides and molecular systems, this method fails to obtain the correct electronic structure for the strongly correlated system due to errors associated with on-site Coulomb and exchange interactions [START_REF] Lee | Ab initio energetics of La B O 3 (001)(B= Mn, Fe, Co, and Ni) for solid oxide fuel cell cathodes[END_REF][START_REF] Anisimov | First-principles calculations of the electronic structure and spectra of strongly correlated systems: the LDA+ U method[END_REF]. These errors limit the applicability of the ab initio calculations in late-transition metal oxides and rare-earth compounds [START_REF] Pacchioni | Modeling doped and defective oxides in catalysis with density functional theory methods: Room for improvements[END_REF]. A possible workaround is using the GGA+U method, which considers local corrections within selected orbitals. The value of U is chosen to match experimentally observed properties or computational results performed with a high level of theory [START_REF] Staykov | Oxygen Reduction Reaction and Electronic Properties of LnO-Terminated Surfaces of Pr2NiO4 and La2NiO4[END_REF]. It has been reported that a reasonably broad range for the Ni cations was selected between 4.5-8.5 eV. Accordingly, in the present study, we only adopt the value of U=6 eV for Ni, based on previous studies in the literature [START_REF] Staykov | Oxygen Reduction Reaction and Electronic Properties of LnO-Terminated Surfaces of Pr2NiO4 and La2NiO4[END_REF][START_REF] Akbay | The interaction of molecular oxygen on LaO terminated surfaces of La 2 NiO 4[END_REF]. The optimal U value for Pr f-orbitals is selected to be 3 eV according to the recent study proposed by Staykov et al. [START_REF] Staykov | Oxygen Reduction Reaction and Electronic Properties of LnO-Terminated Surfaces of Pr2NiO4 and La2NiO4[END_REF].

Meanwhile, the optimization of the U value for Nd f-orbitals has been performed. The results were fitted to the electronic density of states (EDOS) (See S2 in the supplementary file) obtained with the Heyd-Scuseria-Ernzerhof (HSE06) hybrid function. From Figure S2, it is concluded that the EDOS obtained with PBE function using GGA+U with U=6 eV for both Ni 3dand Nd 4f-orbitals reproduces the EDOS and band gap obtained with the HSE06 functional near the fermi energy. Thus, the optimal U value for Nd f-orbitals is selected as 6 eV in the present work.

The oxygen stoichiometric Ln2NiO4.0 (Ln=La, Nd, Pr) compounds were selected for the present work's total energy, vacancy formation energy, and thermal expansion coefficient calculations. The lattice parameters and space group of La2NiO4.0, Nd2NiO4.0, and Pr2NiO4.0 are included in the supplementary file (See S3 in the supplementary file), containing the comparison between the experimental and ab initio value (See Table S1 and S2 in the supplementary file).

The Phonon frequency calculations are carried out by the supercell method, implemented in YPHON code [START_REF] Wang | YPHON: A package for calculating phonons of polar materials[END_REF], from force constants calculated with VASP. The present work used 126-atom supercells for the calculations of Ln2NiO4 and 228-atom supercells for the calculations of Ln2NiO4+δ. In the VASP calculations, a 4×4×4 k-point mesh for the 126-atom supercell and a 3×3×3 k-point mesh for 228-atom supercells were used, respectively. The forces induced by small displacements were calculated.

Finally, the activation barrier was obtained based on the climbing NEB (Cl-NEB) method implemented in the VTST code [START_REF] Henkelman | A climbing image nudged elastic band method for finding saddle points and minimum energy paths[END_REF]. In this section, the diffusion properties of the overstoichiometric Ln2NiO4+δ with δ=0.125 was estimated, i.e., the NEB simulations were performed with the formula: Ln16Ni8O33. Considering the computational expense, only one image between the starting and ending geometries was applied, in which the NEB force converged to 0.05 eV Å - 2 . Throughout this study, the graphical visualization package VESTA [START_REF] Momma | VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data[END_REF] was used to check the ionic positions in each NEB image.

Quasi-harmonic approximations (QHA)

The free energy of Ln2NiO4.0 can be estimated by QHA as [START_REF] Ma | Ab initio thermodynamics of the CoCrFeMnNi high entropy alloy: Importance of entropy contributions beyond the configurational one[END_REF][START_REF] Mei | Density-functional study of the thermodynamic properties and the pressure-temperature phase diagram of Ti[END_REF][START_REF] Wang | Thermodynamic properties of Al, Ni, NiAl, and Ni3Al from first-principles calculations[END_REF][START_REF] Wang | Mean-field potential approach to the quasiharmonic theory of solids[END_REF][START_REF] Körmann | Free energy of bcc iron: Integratedab initioderivation of vibrational, electronic, and magnetic contributions[END_REF]:

𝐹(𝑉, 𝑇) = 𝐸(𝑉) + 𝐹 𝑣𝑖𝑏 (𝑉, 𝑇) + 𝐹 𝑒𝑙𝑒 (𝑉, 𝑇) (1)
where 𝐸(𝑉) is the total energy of Ln2NiO4 at 0K. It can be calculated and fitted by the Murnaghan equation of states (EOSs) [START_REF] Murnaghan | Finite deformations of an elastic solid[END_REF]:

𝐸(𝑉) = 𝑎 + 𝐵 0 𝑉 𝐵 0 ′ (1 + ( 𝑉 0 𝑉 ) 𝐵 0 ′ 𝐵 0 ′ -1 ) (2)
In Eq. ( 2), the fitting parameter 𝑎 can be expressed as 𝐸 0 -

𝐵 0 𝑉 0 𝐵 0 ′ -1
. The parameters 𝑉 0 , 𝐸 0 , 𝐵 0 , and 𝐵 0 ′ represent the equilibrium volume, energy, bulk modulus, and its first derivation with respect to pressure, respectively.

𝐹 𝑣𝑖𝑏 (𝑉, 𝑇) and 𝐹 𝑒𝑙𝑒 (𝑉, 𝑇) are the free energy of Ln2NiO4 contributed by phonon and electron, respectively, which can be estimated by:

𝐹 𝑣𝑖𝑏 (𝑉, 𝑇) = 𝑘 𝐵 𝑇 ∫ 𝑙𝑛 ∞ 0 [2 sinh ( ћ𝜔 2𝑘 𝐵 𝑇 )] 𝑔(𝜔, 𝑉)𝑑𝜔 (3)
𝐹 𝑒𝑙𝑒 (𝑉, 𝑇) = ∫ 𝑛(𝜀, 𝑉)𝑓𝜀 𝑑𝜀 -∫ 𝑛(𝜀, 𝑉)𝜀 𝜀 𝐹 𝑑𝜀 + 𝑘 𝐵 𝑇 ∫ 𝑛(𝜀, 𝑉)[𝑓𝑙𝑛𝑓 + (1 -𝑓) ln(1 -𝑓)] 𝑑𝜀 (4)
where 𝜔 represents the phonon frequency, 𝑔(𝜔, 𝑉) represents the PDOS at frequency 𝜔 volume 𝑉 (See S4 in the supplementary file), 𝑘 𝐵 is Boltzmann's constant, and ћ is the reduced Planck constant. 𝑛(𝜀, 𝑉) is the electronic density of state (EDOS) with 𝑓 being the Fermi distribution.

The equilibrium volume 𝑉 𝑒𝑞 (𝑃, 𝑇) at given T and P can be obtained by finding the root of the following equation:

-( 𝜕𝐹(𝑉,𝑇) 𝜕𝑉 ) 𝑇 = 𝑃 (5)

Result and discussions

Vacancy formation energy in Ln2NiO4 compound

The oxygen point defects are essential for the oxygen reduction reaction (ORR) as they are likely strongly coupled to oxygen dissociation transport and incorporation [START_REF] Lee | Ab initio energetics of La B O 3 (001)(B= Mn, Fe, Co, and Ni) for solid oxide fuel cell cathodes[END_REF]. In this section, the bulk oxygen vacancy formation energy is investigated to examine the possibility of oxygen transport in bulk stoichiometric Ln2NiO4 compound (Ln=La, Nd, Pr). The vacancy formation energy is given by:

∆𝐸 𝑉𝑓 = 𝐸 𝐿𝑛 2 𝑁𝑖𝑂 4-𝛿 𝑎𝑏 𝑖𝑛𝑖𝑡𝑖𝑜 (𝑣𝑎𝑐) -𝐸 𝐿𝑛 2 𝑁𝑖𝑂 4 𝑎𝑏 𝑖𝑛𝑖𝑡𝑖𝑜 + 1 2 (𝐸 𝑂 2 𝑎𝑏 𝑖𝑛𝑖𝑡𝑖𝑜 + ∆ℎ 𝑂 2 0 ) (6)
where 𝐸 𝐿𝑛 2 𝑁𝑖𝑂 4-𝛿 𝑎𝑏 𝑖𝑛𝑖𝑡𝑖𝑜 (𝑣𝑎𝑐) and 𝐸 𝐿𝑛 2 𝑁𝑖𝑂 4 𝑎𝑏 𝑖𝑛𝑖𝑡𝑖𝑜 are the energies of stoichiometric Ln2NiO4 with and without an oxygen vacancy, respectively. 𝐸 𝑂 2 𝑎𝑏 𝑖𝑛𝑖𝑡𝑖𝑜 is the ab initio calculated energy of the O2 molecule and ∆ℎ 𝑂 2 0 is a correction term, which accounts for errors that do not cancel between the treatment of oxygen in the gas and solid phases. It has been reported that the Perdew-Burke-Ernzerhof (PBE)

(O) pseudopotential has an O2 energy correction ∆ℎ 𝑂 2 0 = 1.36𝑒𝑉 [START_REF] Lee | Ab initio energetics of La B O 3 (001)(B= Mn, Fe, Co, and Ni) for solid oxide fuel cell cathodes[END_REF]. Accordingly, this value is selected as a correction term. The ab initio calculated vacancy formation energies of La2NiO4.0, Nd2NiO4.0, and Pr2NiO4.0 are shown in Table 1. According to the space group symmetry, there are two non-equivalent oxygen sites in the Ln2NiO4 compound: 1) the apical site and 2) the equatorial site. The vacancy formation energy of the apical site is around 1 eV higher than the equatorial site. However, even with the equatorial site, the formation energy of vacancy is about 5.03-5.19 eV/vac. It is very high compared with the barrier of the interstitialcy diffusion mechanism, which holds in the range of 0.5 eV-1.5 eV [START_REF] Chroneos | Anisotropic oxygen diffusion in tetragonal La 2 NiO 4+ δ: molecular dynamics calculations[END_REF][START_REF] Parfitt | Molecular dynamics study of oxygen diffusion in Pr 2 NiO 4+ δ[END_REF]. Accordingly, the oxygen vacancy diffusion does not play a critical role for the oxygen stoichiometric La2NiO4, Nd2NiO4, and Pr2NiO4 compounds.

3.2

The energy barrier calculation of oxygen interstitial in Ln2NiO4+δ

The interstitialcy diffusion mechanism in Ln2NiO4+δ

It has been reported that the stoichiometric tetragonal Ln2NiO4 phases have several O 2- interstitialcy migration pathways along the a-b plane and the c-axis [START_REF] Cleave | Atomistic computer simulation of oxygen ion conduction mechanisms in La2NiO4[END_REF]. As the oxygen partial pressure increases, the O 2-will occupy the interstitial site in the Ln2O2 layers and form Ln2NiO4+δ.

It was reported that Ln2NiO4 could incorporate additional oxygen up to δ=0.18 for La2NiO4 [START_REF] Skinner | Oxygen diffusion and surface exchange in La2-xSrxNiO4+ δ[END_REF][START_REF] Frayret | Application of density functional theory to the modeling of the mixed ionic and electronic conductor La2NiO4+ δ: Lattice relaxation, oxygen mobility, and energetics of frenkel defects[END_REF][START_REF] Jorgensen | Structure of the interstitial oxygen defect in La 2 NiO 4+ δ[END_REF], 60] and δ=0.22 for Nd2NiO4 and Pr2NiO4 [START_REF] Sadykov | Transport properties of Ca-doped Ln2NiO4, 10TH International Conference on Sustainable Energy and Environmental Protection Hydrogen and Fuel Cells[END_REF][START_REF] Grenier | A2MO4+ δ oxides: flexible electrode materials for solid oxide cells[END_REF], respectively, and even to larger values if thermal treatments under oxygen pressure are applied [START_REF] Flura | Development of an Isotopic Oxygen Exchange Setup Operating Under High Pressure: Preliminary Measurements Performed on Ln2NiO4+ d (Ln= La, Pr, Nd, MIEC) Oxides[END_REF].

Figure 1 The diffusion mechanism of O 2-in Ln2NiO4+δ

The diffusion mechanism of Ln2NiO4+δ is shown in Meanwhile, it has been reported [START_REF] Minervini | Oxygen migration in La2NiO4+ δ[END_REF] that the oxygen diffusion of La2NiO4+δ is highly anisotropic, with activation energies such as 𝐸 𝑎 (||𝑎, 𝑏)=0.3-0.9 eV and 𝐸 𝑎 (||𝑐)=2.9-3.5 eV. Therefore, it is suggested that the O 2-should migrate parallel to the a-b plane (path #1) instead of the c direction.

Based on the energy barriers of path #1 among these three compounds, it is shown that the activation barriers computed are 0.789 eV for La2NiO4+δ, 0.932 eV for Nd2NiO4+δ, and 0.722 eV for Pr2NiO4+δ, respectively. Furthermore, it indicates that the O 2-migration along path #1 in Pr2NiO4+δ and La2NiO4+δ is much easier than in Nd2NiO4+δ. According to the initial state, it is observed that the insertion of O10 will push away O9, O11, and O16, which provides the space for the residence of O10 on the interstitial sites. In this procedure, the distance (fractional coordinate) between O9 and O11 increases from 0.5 to 0.62.

During the transition, O10 will move along the a-axis and kick the apical O16 out of the NiO6 octahedron. Meanwhile, the migration of O10 will push up the La, Nd, and Pr ions to overcome will take up the residence at an adjacent interstitial site.

Based on these three NEB images in Figure 4, it could be speculated that the energy barriers of these three compounds at the transition state are closely related to how far La, Nd, and Pr ions travel during the migration of O 2-. Compared with the original positions of La, Nd, and Pr ions at the initial state, the migration of O10 and O16 will push La, Nd, and Pr ions up 0.004 (from 0.143 up to 0.139), 0.006 (from 0.144 up to 0.138), and 0.004 (from 0.144 up to 0.140) units to overcome the barrier, respectively. It suggests that less activation energy is required to push the La and Pr ions up for 0.004 units than the Nd ion up for 0.006 units, which might be the reason why the barrier of Nd2NiO4+δ is relatively higher, and the energy barrier of Pr2NiO4+δ and La2NiO4+δ are much lower. This result agrees with experimental data from the literature, where the higher activation energy of the neodymium compound was attributed to its smaller c lattice parameter [START_REF] Bassat | Anisotropic oxygen diffusion properties in Pr2NiO4+ δ and Nd2NiO4+ δ single crystals[END_REF][START_REF] Villesuzanne | On the role of lattice dynamics on low-temperature oxygen mobility in solid oxides: a neutron diffraction and first-principles investigation of[END_REF]. The latter is suggested to be responsible for (i) a shorter inter-octahedral space and (ii) a less pronounced effect of low energy phonon modes, which act as a prerequisite for high oxygen mobility at moderate temperatures.

The temperature-dependent properties of Ln2NiO4+δ

In the previous section, the activation energy barriers of different O 2-migration paths in Ln2NiO4+δ compounds for a particular volume were systematically discussed. The volume expansion caused by the temperature increase plays an essential role in determining the temperature-dependent activation energy barrier of Ln2NiO4+δ and needs to be investigated.

Temperature-dependent equation of states (EOSs) and thermal expansion

The EOSs (based on Eq. ( 1)) of Ln2NiO4 with the change of temperatures were predicted based on the quasi-harmonic approximation (QHA), as shown in Figure S5 (in the supplementary file). The black dashed line marks the evolution of the equilibrium volume at P=0 atm with increasing temperature. The volume is the cell volume divided by the number of atoms (volume per atom). The temperature-dependent properties were obtained by considering the thermal effect, such as the free energy, crystal volume, and thermal expansion.

The thermal expansion of the La2NiO4, Nd2NiO4, and Pr2NiO4 compounds are plotted in Figure 5, including the comparison with experimental data [START_REF] Murata | Fabrication and characterization of oriented Nd2NiO4 bulk and cathode for low-temperature operating solid oxide fuel cell[END_REF][START_REF] Makhnach | High-temperature oxygen non-stoichiometry, conductivity and structure in strontium-rich nickelates La2-xSrxNiO4-δ (x= 1 and 1.4)[END_REF][START_REF] Zhou | Electrochemical performance and stability of the cathode for solid oxide fuel cells: V. high performance and stable Pr2NiO4 as the cathode for solid oxide fuel cells[END_REF] from the literature. It can be calculated based on Eq. [START_REF] Egger | La2NiO4+ δ as electrode material for solid oxide fuel cells and electrolyzer cells[END_REF].

∆𝑉 𝑉 0 = 𝑉 𝑒𝑞 -𝑉 0 𝑉 0 (7)
where 𝑉 𝑒𝑞 and 𝑉 0 represent the temperature-dependent equilibrium volume and equilibrium volume at 0 K, respectively. According to the literature review, Ln2NiO4 shows phase transition [START_REF] Song | Structure, electrical conductivity and oxygen transport properties of Ruddlesden-Popper phases Ln n+ 1 Ni n O 3n+ 1 (Ln= La, Pr and Nd; n= 1, 2 and 3[END_REF][START_REF] Broux | Structure and reactivity with oxygen of Pr 2 NiO 4+ δ: an in situ synchrotron X-ray powder diffraction study[END_REF] from tetragonal (I4/mmm) to orthorhombic (Fmmm) at low temperatures (448K for La2NiO4 [START_REF] Song | Structure, electrical conductivity and oxygen transport properties of Ruddlesden-Popper phases Ln n+ 1 Ni n O 3n+ 1 (Ln= La, Pr and Nd; n= 1, 2 and 3[END_REF], 823K for Nd2NiO4 [START_REF] Song | Structure, electrical conductivity and oxygen transport properties of Ruddlesden-Popper phases Ln n+ 1 Ni n O 3n+ 1 (Ln= La, Pr and Nd; n= 1, 2 and 3[END_REF], and 753K for Pr2NiO4, [START_REF] Broux | Structure and reactivity with oxygen of Pr 2 NiO 4+ δ: an in situ synchrotron X-ray powder diffraction study[END_REF] respectively). However, our calculated structure is the high-temperature tetragonal phase. Therefore, to make an apple-to-apple comparison, the reference state temperatures of these three phases are selected as the phase transition temperature. It should be noted that the original experimental data for these three compounds are based on linear thermal expansion. However, the calculated thermal expansions by QHA are based on the volume thermal expansion. Therefore, the present study approximately converted the linear expansion experimental data to the volume expansion experimental data by using Eq (8) [START_REF] Carvill | Mechanical engineer's data handbook[END_REF] to make a direct comparison:

∆𝑉 𝑉 0 = 3 ∆𝐿 𝐿 0 (8)
Figure 5 (a) shows that the calculated thermal expansion of La2NiO4 is in excellent agreement with the experimental data proposed by Makhnach et al. [START_REF] Makhnach | High-temperature oxygen non-stoichiometry, conductivity and structure in strontium-rich nickelates La2-xSrxNiO4-δ (x= 1 and 1.4)[END_REF]. In contrast, without the consideration of the U value for f-orbital Nd and Pr, the predicted thermal expansion of Nd2NiO4 and Pr2NiO4 (the blue line in Figure 5 (b) and Figure 5 (c)) exhibit significant discrepancies with the experimental data [START_REF] Murata | Fabrication and characterization of oriented Nd2NiO4 bulk and cathode for low-temperature operating solid oxide fuel cell[END_REF][START_REF] Zhou | Electrochemical performance and stability of the cathode for solid oxide fuel cells: V. high performance and stable Pr2NiO4 as the cathode for solid oxide fuel cells[END_REF] with the change of temperature. The maximum differences in volume expansions between ab initio and literature data are 0.43% for Nd2NiO4 and 0.38% for Pr2NiO4 at 300K. After adopting the optimized U parameter for both Nd and Pr, the agreements of thermal expansion between the ab initio prediction and experimental data [START_REF] Murata | Fabrication and characterization of oriented Nd2NiO4 bulk and cathode for low-temperature operating solid oxide fuel cell[END_REF][START_REF] Zhou | Electrochemical performance and stability of the cathode for solid oxide fuel cells: V. high performance and stable Pr2NiO4 as the cathode for solid oxide fuel cells[END_REF] are greatly improved. The slope of the thermal expansion has a reasonable agreement with the experimental data. Based on the experimental data, it was found that the phase transition from tetragonal (I4/mmm) to orthorhombic (Fmmm) does not have a significant effect on the slope of the thermal expansion in Ln2NiO4. However, the discrepancies at low temperatures might cause by the phase transition. The comparison of thermal expansion in these three compounds is plotted in Figure 5 (d). Based on our simulations, the Pr2NiO4 compound exhibits much higher thermal expansion than Nd2NiO4 and La2NiO4. Comparison using the ab initio results.

Extrapolation of temperature-dependent properties of the Ln2NiO4+δ

The temperature-dependent properties of the Ln2NiO4+δ were derived by applying a systematic three-step procedure. The first step in this procedure is calculating the energy barrier, phonon, and electronic density of states of Ln2NiO4+δ at 0 K as a function of volumes. In the present study, the calculated volumes are 13.70Å 3 , 14.11Å 3 , and 14.53Å 3 for La2NiO4+δ, 13.12Å 3 , 13.52Å 3 ,

(a) (b) (c) (d)
and 13.92Å 3 for Nd2NiO4+δ, and 13.39Å 3 , 13.80Å 3 , and 14.21Å 3 Pr2NiO4+δ, respectively. In the second step, the equilibrium 𝑉 𝑒𝑞 (𝑃, 𝑇) at different temperatures is calculated with QHA. In the third step, the calculated physical properties from the first step at the different volumes were approximated by the combination of QHA and the linear regression model. The above procedure assumes that the temperature-dependent properties of the Ln2NiO4+δ are solely caused by thermal expansion.

In particular, the present work associated the calculated migration energy barriers with the diffusivity and ionic conductivity with the transition state theory [START_REF] Kong | Transition state theory of the preexponential factors for self-diffusion on Cu, Ag, and Ni surfaces[END_REF] and Nernst-Einstein relation [START_REF] Liu | Tailoring the Cation Lattice for Chloride Lithium-Ion Conductors[END_REF] as:

D(V,T) = 𝑛 𝑝 2𝑑 𝑙(𝑉, 𝑇) 2 𝑣 0 𝑒𝑥𝑝 (- ∆𝐸 𝑏 (𝑉,𝑇) 𝑘 𝐵 𝑇 ) (9)
𝑣 0 = 𝑘 𝐵 𝑇 ℎ exp (- ∆𝐹 𝑣𝑖𝑏 (𝑉,𝑇) 𝑘 𝐵 𝑇 ) (10)
𝜎(𝑉, 𝑇) = 𝑁𝑞 2 𝐷(𝑉,𝑇) 𝑉(𝑇)𝑘 𝐵 𝑇 [START_REF] Shaula | Oxygen transport in La2NiO4+ δ: Assessment of surface limitations and multilayer membrane architectures[END_REF] where 𝑣 0 represents the attempt frequency (See S6 in the supplementary file), ∆𝐹 𝑣𝑖𝑏 (𝑉, 𝑇) is the vibrational free energy difference (See S7 in the supplementary file) between the transition (saddle-point) state and the equilibrium (binding or stable) state, 𝑙(𝑉, 𝑇) and ∆𝐸 𝑏 (𝑉, 𝑇) denote the hopping distance and energy barrier at different volumes and temperatures, respectively (See S8 in the supplementary file). The hopping distance is approximately equal to the migration distance of the NEB process. 𝑁 is the number of mobile carriers. 𝑛 𝑝 is the number of equivalent diffusion paths (𝑛 𝑝 =4 corresponds to the four available directions around the interstitial oxygen for the pushpull interstitialcy mechanism). 𝑑 is the dimension of the space where the diffusion process occurs (d=3 since we compare our results to experimental data primarily for ceramics, in which a-b planes are randomly oriented).

As 𝑛 𝑝 , d, l, and 𝑉(𝑇) are identical or similar in La2NiO4+δ, Nd2NiO4+δ, and Pr2NiO4+δ, the differences in diffusion coefficients and ionic conductivity should arise from activation energy barriers ∆𝐸 𝑏 and attempt frequencies 𝑣 0 . The calculated attempt frequencies are shown in Figure S6 (in the supplementary file). As seen in the figure, the attempt frequencies of all these three compounds rise with the increasing temperature. Among all these three compounds, the Pr2NiO4+δ exhibits much higher attempt frequencies at high temperatures. The resulting frequency reaches 5.2 THz at 1600K. Overall, the calculated attempt frequencies are in the range of the usually assumed value from the literature (1-10 THz) [START_REF] Koettgen | Ab initio calculation of the attempt frequency of oxygen diffusion in pure and samarium doped ceria[END_REF][START_REF] Koettgen | The Effect of Jump Attempt Frequencies on the Ionic Conductivity of Doped Ceria[END_REF].

The predicted temperature-dependent properties of volume, energy barrier, hopping distance, diffusivity, and ionic conductivity are shown in Figure 6. Generally, the volume of solid material increases linearly with the increase in temperatures. To approximate the temperaturedependent properties, the linear regression model was adopted to fit the ab initio data calculated by QHA. Particularly, the solid red curve on the first sub-graph represents the predicted temperature-dependent EOS based on QHA, as discussed in the previous section. The black circles represent the ab-initio calculation results. The linear regression approach was used to derive the relationship between Volume vs. Temperature, Volume vs. Energy barrier, and Volume vs.

Hopping distance (based on the ab-initio results) because all these properties are linearly related to temperature. The fitting results are shown as black dashed lines in Figure 6. Eventually, the diffusivity and conductivity with the temperature change were predicted.

Based on Figure 6, it is observed that the fitted linear curves exhibit good agreement with the ab initio data, which can reasonably represent the temperature-dependent properties of the Ln2NiO4+δ. Generally, it is observed that the hopping distance, diffusivity, and ionic conductivity gradually increase with the increasing temperature, whereas the energy barrier decreases steadily with the temperature increase.

. Diffusion coefficient

The Arrhenius plots of Ln2NiO4+δ comparing the predicted diffusion coefficient with previous experimental data [START_REF] Skinner | Oxygen diffusion and surface exchange in La2-xSrxNiO4+ δ[END_REF][START_REF] Bassat | Anisotropic oxygen diffusion properties in Pr2NiO4+ δ and Nd2NiO4+ δ single crystals[END_REF][START_REF] Kilner | Mass transport in La2Ni1-xCoxO4+ δ oxides with the K2NiF4 structure[END_REF][START_REF] Burriel | Influence of crystal orientation and annealing on the oxygen diffusion and surface exchange of La2NiO4+ δ[END_REF] and modeling data [START_REF] Chroneos | Anisotropic oxygen diffusion in tetragonal La 2 NiO 4+ δ: molecular dynamics calculations[END_REF] are shown in Figure 7. The green dashed lines represent the diffusion coefficient predicted according to the temperaturedependent energy barrier (see Eq. ( 8)). In contrast, the solid red lines denote the diffusion coefficient calculated from Eq. ( 8), using 0 K values for l and ∆𝐸 𝑏 , and assuming that these quantities are temperature-independent.

(b) (a) (c) (d)
Figure 7 The ab initio predicted diffusion coefficient, including the comparison with experimental data [START_REF] Skinner | Oxygen diffusion and surface exchange in La2-xSrxNiO4+ δ[END_REF][START_REF] Bassat | Anisotropic oxygen diffusion properties in Pr2NiO4+ δ and Nd2NiO4+ δ single crystals[END_REF][START_REF] Song | Structure, electrical conductivity and oxygen transport properties of Ruddlesden-Popper phases Ln n+ 1 Ni n O 3n+ 1 (Ln= La, Pr and Nd; n= 1, 2 and 3[END_REF][START_REF] Kilner | Mass transport in La2Ni1-xCoxO4+ δ oxides with the K2NiF4 structure[END_REF][START_REF] Burriel | Influence of crystal orientation and annealing on the oxygen diffusion and surface exchange of La2NiO4+ δ[END_REF] and modeling data [START_REF] Chroneos | Anisotropic oxygen diffusion in tetragonal La 2 NiO 4+ δ: molecular dynamics calculations[END_REF] (a) La2NiO4+δ (b) Nd2NiO4+δ (c) Pr2NiO4+δ (d)

Comparison

In the case of La2NiO4+δ, as shown in Figure 7 (a), the estimated activation energy barrier at 0 K (considering the contribution from phonon) is 0.759 eV, which is very close to the experimental activation energy by Skinner et al. [START_REF] Skinner | Oxygen diffusion and surface exchange in La2-xSrxNiO4+ δ[END_REF] (∆𝐸 𝑏 =0.85 eV) for the dense ceramic.

Notably, compared with the activation energy barrier (∆𝐸 𝑏 =0.51 eV) predicted by Alexander et al.

[25] using the molecular dynamic simulation (blue marks), the current ab initio calculation yields a much closer value of activation energy barrier with the experimental data [START_REF] Skinner | Oxygen diffusion and surface exchange in La2-xSrxNiO4+ δ[END_REF][START_REF] Kilner | Mass transport in La2Ni1-xCoxO4+ δ oxides with the K2NiF4 structure[END_REF]. For the results of Nd2NiO4+δ and Pr2NiO4+δ (Figure 7 (b) and (c)), the ab initio calculated activation energies at 0 K are 0.917 eV and 0.724 eV, respectively, which are similar to the experimental values [START_REF] Bassat | Anisotropic oxygen diffusion properties in Pr2NiO4+ δ and Nd2NiO4+ δ single crystals[END_REF] for the dense ceramic Nd2NiO4+δ (∆𝐸 𝑏 =1.1 eV) and Pr2NiO4+δ (∆𝐸 𝑏 =0.8 eV). However, while the slope of the diffusion coefficient was successfully reproduced, the estimated pre-exponential factors of La2NiO4+δ and Pr2NiO4+δ by the transition state theory are slightly higher than the experimental data. The reason for this discrepancy will be discussed in Section 3.4.3. In addition, Figure 7 (d) compares the diffusion coefficients in these three over-stoichiometric compounds.

According to our prediction, it can be observed that the ionic diffusivities of the Ln2NiO4+δ rank as Pr2NiO4+δ>La2NiO4+δ >Nd2NiO4+δ.

Overall, our ab initio predicted diffusion coefficients have a reasonable agreement with experimental data of La2NiO4+δ, Nd2NiO4+δ, and Pr2NiO4+δ, especially for the single crystal data [START_REF] Bassat | Anisotropic oxygen diffusion properties in Pr2NiO4+ δ and Nd2NiO4+ δ single crystals[END_REF][START_REF] Burriel | Influence of crystal orientation and annealing on the oxygen diffusion and surface exchange of La2NiO4+ δ[END_REF]. It is because the single crystal has infinite periodicity in which the crystal lattice of the entire sample is continuous with no grain boundaries, defects, and domain walls, which is closer to the ab initio calculated structure. Meanwhile, it was found that the predicted diffusion coefficients based on the 0 K barriers have much better consistency with the experimental data than the results from the temperature-dependent barrier. While the results based on the 0 K barrier yield better agreement with the experimental data, the present work believes that the properties predicted according to the temperature-dependent barrier have more physical meaning because of considering the temperature effect.

Ionic conductivity

The Arrhenius plots for ionic conductivity of Ln2NiO4+δ as a function of temperatures are shown in Figure 8. The comparisons between the ab initio and experimental data [START_REF] Boehm | Oxygen diffusion and transport properties in non-stoichiometric Ln2-xNiO4+ δ oxides[END_REF][START_REF] Shen | Preparation and electrical properties of Ca-doped La 2 NiO 4+ δ cathode materials for IT-SOFC[END_REF][START_REF] Song | Structure, electrical conductivity and oxygen transport properties of Ruddlesden-Popper phases Ln n+ 1 Ni n O 3n+ 1 (Ln= La, Pr and Nd; n= 1, 2 and 3[END_REF][START_REF] Ina | Relationship between Local Structure and Oxide Ionic Diffusion of Nd2NiO4+ δ with K2NiF4 Structure[END_REF][START_REF] Zhao | New cathode materials for ITSOFC: phase stability, oxygen exchange and cathode properties of La2-xNiO4+ δ[END_REF] are also included in the figure. Similarly, the solid red and green dash lines denote the ionic conductivities predicted by the 0 K and temperature-dependent barriers, respectively. It is shown that, according to Figure 8, the predicted ionic conductivity increases linearly with the increasing temperature, which reasonably agrees well with the experimental data [START_REF] Boehm | Oxygen diffusion and transport properties in non-stoichiometric Ln2-xNiO4+ δ oxides[END_REF][START_REF] Shen | Preparation and electrical properties of Ca-doped La 2 NiO 4+ δ cathode materials for IT-SOFC[END_REF][START_REF] Song | Structure, electrical conductivity and oxygen transport properties of Ruddlesden-Popper phases Ln n+ 1 Ni n O 3n+ 1 (Ln= La, Pr and Nd; n= 1, 2 and 3[END_REF][START_REF] Ina | Relationship between Local Structure and Oxide Ionic Diffusion of Nd2NiO4+ δ with K2NiF4 Structure[END_REF][START_REF] Zhao | New cathode materials for ITSOFC: phase stability, oxygen exchange and cathode properties of La2-xNiO4+ δ[END_REF], except for the Nd2NiO4+δ. However, it is hard to conclude the reliability of our prediction on Nd2NiO4+δ because the experimental data measured by the different groups have huge fluctuation. Generally, the ionic conductivity is proportional to the diffusion coefficient, which typically scales with the value of the activation energy barrier. It has been reported that the activation energy barrier of Nd2NiO4+δ (∆𝐸 𝑏 =1.1 eV [START_REF] Bassat | Anisotropic oxygen diffusion properties in Pr2NiO4+ δ and Nd2NiO4+ δ single crystals[END_REF]) is much higher than La2NiO4+δ (∆𝐸 𝑏 =0.85~0.88 eV [START_REF] Skinner | Oxygen diffusion and surface exchange in La2-xSrxNiO4+ δ[END_REF]) and Pr2NiO4+δ (∆𝐸 𝑏 =0.8 eV [START_REF] Bassat | Anisotropic oxygen diffusion properties in Pr2NiO4+ δ and Nd2NiO4+ δ single crystals[END_REF]). Therefore, it can be speculated that the ionic conductivity value for Nd2NiO4+δ should be lower than that of La2NiO4+δ and Pr2NiO4+δ. Accordingly, it is suggested that more experimental work should be carried out to investigate the ionic conductivity of Nd2NiO4+δ.

Finally, our ab initio predictions show that the Pr2NiO4+δ exhibits the highest ionic conductivity (see Figure 8 (d)) among all these three over-stoichiometric compounds due to the lowest activation energy barrier.

Discussion on the discrepancies between ab initio calculations and experimental data

Even though the consistency between our ab initio prediction and experimental value in Ln2NiO4+δ compounds is generally acceptable, there are still some discrepancies, such as the pre-exponential factors. In this section, several reasons for the discrepancies between the ab initio prediction and experimental data are listed:

i. The oxygen hyper-stoichiometry δ in Ln2NiO4+δ compounds will change with the temperature and oxygen partial pressure, which was not considered in the present study.

Therefore, any changes in δ will directly affect the carrier concentration and, in turn, the conductivities.

ii. The reliability of the PBE exchange-correlation functional on predicting transition metal oxides and rare-earth compounds is still in question. While the GGA+U approach is able to make some corrections, it is yet to make an accurate prediction.

iii. The size of the supercell and the number of images adopted for the NEB calculations may also affect the prediction reliabilities.

iv.

The temperature-dependent properties are assumed to be only attributed to the thermal expansion coefficient. However, there may be other factors having an impact on the properties.

v. The discrepancies may also come from the influence of defects in real materials (grain boundaries, domain walls, non-homogeneities, etc.), which could somewhat hinder the diffusivity and ionic conductivity.

vi.

The observed diffusion coefficient of dense ceramics is a weighted average of the diffusion coefficients along the a-b plane and c direction. However, the diffusion coefficient along the c direction was neglected in the present work due to its high energy barrier.

Overall, the activation energy barriers are reasonably reproduced by the ab initio calculations.

Meanwhile, the ranking between the La2NiO4+δ, Nd2NiO4+δ, and Pr2NiO4+δ for both the activation energy barrier and diffusion coefficient is in good agreement with the experimental data. However, the calculated value for pre-exponential factors is far more speculative due to the inherently approximate methods for predicting attempt frequency values in silico (1THz~10THz).

Conclusion

In the present work, the vacancy formation energy, diffusivity, and ionic conductivity of the Ln2NiO4+δ (δ=0.125) RP phase was systematically investigated through the ab initio approach.

Firstly, the NEB method is adopted to predict the oxygen ions' energy barrier, which is then utilized to derive the diffusion coefficient and ionic conductivity using the transition state theory and the iii. Among all the three compounds, the Pr2NiO4+δ exhibits the lowest activation energy barrier and highest thermal expansion, diffusivity, and ionic conductivity.

The predicted energy barrier, diffusion coefficient, and ionic conductivity of Ln2NiO4+δ exhibit reasonable agreement with the experimental data. Meanwhile, the computational method developed in this study is an easy, time-efficient, and reliable tool for accessing oxygen diffusivity properties as a function of temperature in oxide material, in which the temperature dependence is predominantly caused by thermal expansion.

Figure 1 .

 1 In the initial configuration, as shown in the unit cell (a) of Figure 1, the O 2-intercalates in the Ln2O2 layer, resulting in overstoichiometric phases Ln2NiO4+δ. Meanwhile, the apical O 2-from one of these octahedra is able to move upwards towards an unoccupied interstitial site with the corresponding relaxation. In the intermediate state (Figure 1 unit cell (b)), the apical O 2-takes up the residence in an adjacent interstitial site, and the vacancy is formed at the apical site. Meanwhile, the initial interstitial O 2- is ready to move toward the apical site of the NiO6 octahedron. Finally, the original interstitial O 2- reforms at the NiO6 octahedron again, as shown in the unit cell (c) of Figure 1. The procedure will repeat in the next unit cell, and the oxygen ion can move along with the path marked in yellow, as shown in the left figure.To determine if other pathways for the oxygen diffusion of the Ln2NiO4+δ compound can be considered, the present study employed the ab initio simulation to investigate three possible migration pathways and to evaluate the migration energy barriers. The schematic diagram of these three pathways is shown in Figure2.

Figure 2 (

 2 Figure 2 (a) exhibits path #1 of the interstitialcy oxygen migration along with the a-b plane,

Figure 2 2 .

 22 Figure 2 The diffusion paths of O 2-in Ln2NiO4+δ selected in the present study (a) Path #1 (along

Figure 3

 3 Figure 3 The energy barriers of different O 2-migration paths in Ln2NiO4+δ (Path #1: solid lines

Figure 4

 4 Figure 4 The schematic graph of ionic positions of path #1 at different states, (a) La2NiO4+δ, (b)

  the barrier. At the final stage, O10 reforms at the NiO6 octahedron, and the original apical O16

Figure 5

 5 Figure 5 The volume thermal expansion calculated through the combined DFT-QHA method

Figure 6 4

 64 Figure 6 The temperature-dependent properties of Ln2NiO4+δ (a) Thermal expansion (b) Energy

Figure 8

 8 Figure 8 The ab initio predicted ionic conductivity, including the comparison with experimental

  Nernst-Einstein relation. Secondly, a combination of QHA and a linear regression model is used to study and reproduce the temperature-dependent properties of the Ln2NiO4+δ materials. As a result, in agreement with previous theoretical and experimental works in literature, the following conclusions can be reached: i. The vacancy diffusion doe not play a dominant role in the Ln2NiO4 phase due to the high vacancy formation energy. ii. The interstitial O 2-of the Ln2NiO4+δ phase should migrate parallel to the a-b plane (interstitialcy mechanism) instead of the c direction.

Table 1

 1

	La2NiO4-δ	Nd2NiO4-δ	Pr2NiO4-δ

Oxygen vacancy (apical /equatorial) formation energy for stoichiometric bulk Ln2NiO4.0 compounds Vacancy formation Energy (eV/vac)

The fixed oxygen hyper-stoichiometries, DFT exchange-correlation functional, supercell size, defects, etc., might cause the discrepancies observed for the pre-exponential factors.

Therefore, more accurate methods to calculate the attempt frequencies are needed to improve the agreement between the ab initio and the experimental results.

Acknowledgement