Atomistic Simulation Studies of Na₄SiO₄

Abstract

Tetrasodium silicate (Na₄SiO₄) has emerged as a promising candidate for battery applications due to its favorable ionic transport properties. Atomic-scale simulations employing classical pair potentials have elucidated the defect mechanisms and ion migration dynamics in Na₄SiO₄. The Na Frenkel defect, characterized by the creation of a Na vacancy and an interstitial Na⁺ ion, is identified as the most energetically favorable defect process, facilitating efficient vacancy-assisted Na⁺ ion migration. This process results in three-dimensional ion diffusion with a low activation energy of 0.55 eV, indicating rapid ion movement within the material. Among monovalent dopants (Li⁺, K⁺, and Rb⁺), K⁺ was found to be the most advantageous for substitution on the Na site. For trivalent doping, Al is the most favorable on the Si site, generating additional Na⁺ ions and potentially enhancing ionic conductivity. Ge was identified as a promising isovalent dopant for the Si site. These theoretical findings suggest that Na₄SiO₄ could offer high ionic conductivity and stability when optimized through appropriate doping. Experimental validation of these predictions could lead to the development of advanced battery materials with improved performance and durability.

1. Introduction

Creating clean, renewable energy is one of the major challenges of the 21st century, and overcoming this challenge necessitates the advancement of innovative material systems. Advances in materials science are crucial for enhancing the efficiency, cost-effectiveness, and sustainability of various energy technologies.

Batteries have transformed into a fundamental aspect of modern technology, seamlessly integrating into nearly every electronic device [1,2,3,4]. Acting as vital energy reservoirs, they have diverse applications across various industries [5,6,7]. Presently, lithium-ion batteries dominate the market, especially in electric vehicles and electronic gadgets [8,9,10]. However, the diminishing availability of lithium minerals poses a challenge to meeting demand. The world is currently grappling with a shortfall in lithium supply. In response, sodium-ion batteries have emerged as a focal point of extensive research, offering promise as a sustainable alternative to lithium-ion batteries [11,12,13,14,15]. Multiple molecules are currently under consideration as potential materials for anodes and cathodes in sodium-ion batteries [16,17,18].

The selection of electrode or electrolyte materials for Na-ion batteries is critical to achieving high performance, safety, and cost-effectiveness. Each material class has its advantages and challenges, and ongoing research focuses on optimizing these materials through structural engineering, composite formation, and surface modifications. Polyanionic-type materials are gaining significant attention in the development of Na-ion batteries due to their high working potential, excellent structural stability, and safety. These materials include compounds where the active cathode species are coordinated by anionic groups such as phosphates (e.g., NaMPO₄ (M = Fe, Mn, Ni)), sulfates (e.g., Na₂MSiO₄ (M = Fe, Mn, Co, Ni)) and borates (e.g., Na₃MB₅O₁₀ (M = V, Fe, Mn, Co, etc.)) [19,20,21,22].

Na₄SiO₄ holds particular significance in this pursuit [23]. As a mineral, it presents potential as a key component in the development of sodium-ion batteries. Research is actively investigating its potential and effectiveness in meeting the increasing demand for energy storage solutions [24,25,26]. Na₄SiO₄ can be produced through a wet mixing method using NaOH and Na₂SiO₃•9H₂O as raw materials [27]. A key benefit of Na₄SiO₄ is its stability and eco-friendly characteristics. It can be easily prepared using simple, abundant raw materials at low cost. Furthermore, like orthophosphates, the orthosilicate (SiO₄⁴⁻) matrix benefits from strong Si–O bonds. These bonds contribute to the structural stability of the material, which is crucial for maintaining integrity during lithium insertion and extraction cycles. The study of CO₂ chemisorption on Na₄SiO₄ doped with oxysalt surfaces using density functional theory (DFT) calculations highlights the potential of doping Na₄SiO₄ with oxysalts to create more effective materials for CO₂ capture, contributing to efforts in mitigating climate change by reducing atmospheric CO₂ levels [24]. The addition of alkali carbonates to Na₄SiO₄ enhances CO₂ capture performance, especially at low temperatures, through the formation of C₂O₅²⁻ species. This has been confirmed by in situ Raman spectroscopy and supported by DFT calculations, providing a strong basis for developing improved CO₂ capture technologies [28].

The study of defect processes in materials is a crucial aspect of materials science and engineering, as defects can significantly impact the physical, electrical, thermal, and mechanical properties of materials. Doping is a versatile and essential process in materials science, enabling the fine-tuning of material properties for a wide range of applications. Advances in doping techniques continue to drive innovation in electronics, optoelectronics, photovoltaics, and many other fields. Understanding and controlling doping processes is key to developing next-generation materials and devices. Na-ion diffusion is a complex but critical process in the development of Na-ion batteries and other sodium-based technologies. A deep understanding of the mechanisms, pathways, and influencing factors is essential for designing materials with optimal performance. Continued research, combining experimental studies with computational modeling, will be key to overcoming current challenges and advancing the field of Na-ion diffusion in materials.

To delve deeper into assessing the viability of Na₄SiO₄ as a battery material, it is crucial to thoroughly investigate additional properties such as defect energy, dopant energy, and ion migration. Conducting theoretical calculations can be challenging due to several factors including the complexity of the system and experimental validation. To overcome this obstacle, atomistic simulation studies have been previously utilized on {Kumar Prajapati, 2023 #6}to comprehensively ascertain and comprehend these properties [29,30,31,32,33,34].

In this study, we use atomistic simulation studies based on the classical pairwise potential to elucidate the defect, diffusion, and dopant properties of Na₄SiO₄.

2. Computational Methods

All calculations were performed using the classical pairwise potential simulation code GULP (General Utility Lattice Program) [35]. GULP is a computational tool used for modeling and simulating the properties of materials, particularly those with crystalline structures. It enables the calculation of various properties of materials, such as lattice dynamics, defect properties, and thermodynamic properties, using different interaction models and optimization algorithms. Interactions between ions in the crystal structure are modeled using long-range (Coulombic) and short-range (Pauli repulsion and van der Waals attraction) forces. Buckingham potentials (see

Table 1. Buckingham potential parameters used in the atomistic simulations of Na₄SiO₄ [40,41,42]. Two-body (Φ_ij (r_ij) = A_ij exp (−r_ij /ρ_ij) − C_ij/r_ij⁶) where A, ρ, and C are parameters reproducing the experimental data. The values of Y and K are shell charges and spring constants, respectively.

Interaction	A/eV	ρ/Å	C/eV·Å6	Y/e	K/eV·Å–2
Na+–O2−	1497.830598	0.287483	0.00	1.00	99,999
Si4+–O2−	1283.91	0.32052	10.66	4.00	99,999
O2—O2−	22,764.30	0.1490	27.88	−2.86	61.50

) describe short-range repulsive forces. The BFGS algorithm (Broyden–Fletcher–Goldfarb–Shanno) is an iterative method for solving unconstrained nonlinear optimization problems. It is particularly well suited for problems where the function to be minimized (or maximized) is smooth. In the context of GULP, the BFGS algorithm is used to optimize the structure of a material by minimizing its potential energy [36]. In all relaxed configurations, the forces on all atoms are smaller than 0.001 eV/Å. The Mott–Littleton method [37] is used to model point defects and migrating ions. The Mott–Littleton method is a well-established technique used in the modeling of point defects and ion migration in crystalline materials. The Mott–Littleton method divides the crystal into two regions to model defects effectively. Region I (Inner Region) is the immediate vicinity around the defect where atomic positions are significantly displaced due to the presence of the defect. In this region, the interactions between atoms and the defect are calculated explicitly. Typically, this region is spherical, and the defect is located at its center. Region II (Outer Region) extends beyond Region I and is where the defect’s influence diminishes. The atoms in Region II are treated using continuum or semi-continuum approaches, where displacements are assumed to be small and can be described using linear elasticity theory or other approximations. Sodium-ion migration is calculated by considering seven interstitial points with equal intervals between neighboring Sodium sites. Defect energies of migrating ions at seven points along the diffusion path will be determined. The midpoint between two adjacent Na vacancy sites is the defect calculation center to reduce systematic errors. The energy difference between the maximum local energy associated with the saddle point along this diffusion path and the lowest Na vacancy formation energy will be calculated and reported as activation energy. Ions are modeled using this method as spherical objects with a full charge at the diluted limit. Defective energies are therefore likely to be overstated. However, relative energies will continue to trend in the same direction. The core-shell technique was used to model the polarization of ions. In earlier research [38,39], we described the formula for determining migratory paths and detailed activation energies of migration.

3. Results and Discussion

3.1. Modelling of Na₄SiO₄ Crystal Structure

Na₄SiO₄ shows a triclinic structure with the P-1 space group according to the X-ray diffraction pattern derived by Baur et al. [23] (CIF file name: ICSD_CollCode62594). This study further explains that it consists of 24 atoms (8 Na, 2 Si, and 14 O) in a unit cell (2 formula units) with the lattice parameter found in the x, y, and z directions as a = 5.58 Å, b = 5.58 Å, and c = 8.39 Å and α = 80.92°, β = 71.84°, and γ = 67.44°. Si atoms are coordinated with four oxygen atoms, forming a tetrahedral structure (see Figure 1). This tetrahedral coordination is a fundamental building block of many silicate structures, including quartz, feldspar, and various other minerals. To ensure the reliability of the classical potentials used in the study, a critical step is to validate these potentials against experimental data. This involves performing full geometry optimization of the crystal structure and comparing the calculated equilibrium lattice constants with experimentally determined values. The calculated lattice constants closely match the experimental values reported (see

Table 2. Calculated and experimental lattice parameters of Na₄SiO₄.

Parameter	Calculated	Experiment [23]	|Δ|%
a (Å)	5.56	5.58	0.33
b (Å)	5.62	5.58	0.77
c (Å)	8.39	8.39	0.07
α (°)	81.34	80.92	0.52
β (°)	71.87	71.84	0.04
γ (°)	67.02	67.44	0.62

).

3.2. Defect Energetics

In this section, we examine the energetics of important defect processes in Na₄SiO₄. To investigate the defect characteristics in Na₄SiO₄, we initially analyzed point defects, including vacancies and interstitials, separately. These calculations form the basis for understanding more complex defect structures such as Schottky and Frenkel defects, as well as anti-site defects where Na and Si swap their atomic positions. The Schottky defect energy in Na₄SiO₄ was calculated by determining the individual vacancy formation energies for four Na⁺, one Si⁴⁺, and four O²⁻ ions and summing them. This approach ensures that the defect maintains the stoichiometry and charge neutrality of the crystal. A Frenkel defect involves the displacement of an ion from its lattice site to an interstitial site, creating a vacancy–interstitial pair. The energy required to form a Na Frenkel defect is the sum of the vacancy formation energy and the interstitial formation energy for Na⁺. In Na₄SiO₄, anti-site defects involve Na and Si atoms swapping positions. For isolated anti-site defects, a single Na atom occupies a Si site, and a single Si atom occupies a Na site. For clustered anti-site defects, multiple pairs of Na and Si atoms swap positions. The following equations in Kröger–Vink notation [43] describe the Schottky, Frenkel, and anti-site defect processes.

$$\mathrm{N}\mathrm{a} \mathrm{F}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{k}\mathrm{e}\mathrm{l}:{\mathrm{N}\mathrm{a}}_{\mathrm{N}\mathrm{a}}^{\mathrm{X}}\to V_{\mathrm{N}\mathrm{a}}^{\prime }+{\mathrm{N}\mathrm{a}}_{\mathrm{i}}^{•}$$

(1)

$${\mathrm{S}\mathrm{i} \mathrm{F}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{k}\mathrm{e}\mathrm{l}:\mathrm{S}\mathrm{i}}_{\mathrm{S}\mathrm{i}}^{\mathrm{X}}\to V_{\mathrm{S}\mathrm{i}}^{⁗}+{\mathrm{S}\mathrm{i}}_{\mathrm{i}}^{••••}$$

(2)

$${\mathrm{O} \mathrm{F}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{k}\mathrm{e}\mathrm{l}:\mathrm{O}}_{\mathrm{O}}^{\mathrm{X}}\to V_{\mathrm{O}}^{••}+{\mathrm{O}}_{\mathrm{i}}^{″}$$

(3)

$$\mathrm{S}\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{k}\mathrm{y}:{4 \mathrm{N}\mathrm{a}}_{\mathrm{N}\mathrm{a}}^{\mathrm{X}}+{\mathrm{S}\mathrm{i}}_{\mathrm{S}\mathrm{i}}^{\mathrm{X}}+4 {\mathrm{O}}_{\mathrm{O}}^{\mathrm{X}}\to 4 V_{\mathrm{N}\mathrm{a}}^{\prime }+V_{\mathrm{S}\mathrm{i}}^{⁗}+4 V_{\mathrm{O}}^{••}+{\mathrm{N}\mathrm{a}}_4\mathrm{S}\mathrm{i}{\mathrm{O}}_4$$

(4)

$${{\mathrm{N}\mathrm{a}}_2\mathrm{O} \mathrm{S}\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{k}\mathrm{y}:2 \mathrm{N}\mathrm{a}}_{\mathrm{N}\mathrm{a}}^{\mathrm{X}}+{\mathrm{O}}_{\mathrm{O}}^{\mathrm{X}}\to {2 V}_{\mathrm{N}\mathrm{a}}^{\prime }+V_{\mathrm{O}}^{••}+{\mathrm{N}\mathrm{a}}_2\mathrm{O}$$

(5)

$${\mathrm{S}\mathrm{i}{\mathrm{O}}_2 \mathrm{S}\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{k}\mathrm{y}:\mathrm{S}\mathrm{i}}_{\mathrm{S}\mathrm{i}}^{\mathrm{X}}+{2 \mathrm{O}}_{\mathrm{O}}^{\mathrm{X}}\to V_{\mathrm{S}\mathrm{i}}^{⁗}+{2 V}_{\mathrm{O}}^{••}+\mathrm{S}\mathrm{i}{\mathrm{O}}_2$$

(6)

$${\mathrm{N}\mathrm{a}/\mathrm{S}\mathrm{i} \mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}–\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}\left(\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\right):\mathrm{N}\mathrm{a}}_{\mathrm{N}\mathrm{a}}^{\mathrm{X}}+{\mathrm{S}\mathrm{i}}_{\mathrm{S}\mathrm{i}}^{\mathrm{X}}\to {\mathrm{N}\mathrm{a}}_{\mathrm{S}\mathrm{i}}^{‴}+{\mathrm{S}\mathrm{i}}_{\mathrm{N}\mathrm{a}}^{•••}$$

(7)

$${\mathrm{N}\mathrm{a}/\mathrm{S}\mathrm{i} \mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}–\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}\left(\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}\right):\mathrm{N}\mathrm{a}}_{\mathrm{N}\mathrm{a}}^{\mathrm{X}}+{\mathrm{S}\mathrm{i}}_{\mathrm{S}\mathrm{i}}^{\mathrm{X}}\to \{{{\mathrm{N}\mathrm{a}}_{\mathrm{S}\mathrm{i}}^{‴}+{\mathrm{S}\mathrm{i}}_{\mathrm{N}\mathrm{a}}^{•••}\}}^X$$

(8)

Figure 2 demonstrates the individual defect reaction energies. The Na Frenkel defect has the lowest energy among the considered defect processes, with an energy of 1.34 eV per defect. This indicates that the Na ions in Na₄SiO₄ have a relatively low barrier for forming Frenkel defects, facilitating Na-ion migration via vacancy-assisted mechanisms. The next most energetically favorable defect process after the Na Frenkel defect is the Na₂O Schottky defect (2.79 eV/defect). This defect involves the simultaneous creation of vacancies for two Na atoms and one O atom, effectively removing a Na₂O unit from the crystal lattice. This higher energy implies that the formation of such defects, and consequently the loss of Na₂O, is only feasible at high temperatures. Isolated anti-site defects have a high energy cost, making them unstable. Due to the exoergic nature of the clustering process (clustering energy: 4.91 − 8.77 = −3.86 eV), isolated anti-site defects tend to aggregate into clusters to achieve a lower energy state. This aggregation into clusters reflects a significant tendency for the system to favor clustered anti-site defects, which are more stable and energetically favorable compared to their isolated counterparts. Other Frenkel and Schottky defects have even higher formation energies, making them practically non-existent under typical conditions. The results indicate that Na Frenkel defects are the most energetically favorable and significant in Na₄SiO₄. Due to their lower formation energy, they dominate the defect landscape and influence the material’s behavior, particularly its ionic conductivity.

3.3. Na-Ion Migration

Examining sodium-ion diffusion pathways and activation energies is crucial for understanding the ionic conductivity of Na₄SiO₄, especially in applications such as solid-state batteries. The direct experimental characterization of these pathways is challenging due to the complexity of the material’s structure and the small scale of ion movements. Therefore, computational methods are often employed to simulate and analyze these diffusion pathways and activation energies. The lithium-ion migration path calculated in LiFePO₄ using classical pair potentials was later exactly observed in high-temperature powder neutron diffraction and the maximum entropy method. The experiment visualized a curved one-dimensional chain for lithium motion with a Li-Li separation of 3.01 Å along the [010] direction [44]. This finding emphasizes the accuracy of our simulation methods in predicting ion migration pathways.

Using current simulation techniques, we constructed the Na-ion diffusion channels with atomic-level activation energies. Seven local Na hops were identified (A, B, C, D, E, F, and G) (see Figure 3). Hop A has a lower activation energy of 0.23 eV, indicating that it is a favorable and energetically accessible pathway for Na⁺ ion migration. Hop B has a moderate activation energy, suggesting it is less favorable compared to Hop A but still relatively accessible (see

Table 3. Local Na hops and their activation energies.

Na-Na Hop	Distance (Å)	Activation Energy (eV)
A	3.14 Å	0.23
B	3.62Å	0.65
C	3.39 Å	0.03
D	3.89Å	0.73
E	3.29 Å	0.55
F	4.07Å	1.02
G	2.91 Å	0.62

). Hop C, with the lowest activation energy among the hops listed, makes it the most energetically favorable for Na⁺ ion movement. Hop D has a moderate activation energy, making it less favorable than Hop A and C but more favorable than Hops D and F. This hop has a moderate activation energy, making it less favorable than Hop A and C but more favorable than Hops D and F. The Hop E pathway has the highest activation energy among the listed hops, indicating that it is the least favorable for Na⁺ ion migration. Hop G has a moderate activation energy, suggesting that it is less favorable than Hop A and C but more favorable than Hop D and F. The energy profile diagrams (see Figure 4) are essential for understanding the activation energies associated with different sodium-ion (Na⁺) diffusion pathways in Na₄SiO₄. These diagrams illustrate the energy barriers that Na⁺ ions must overcome to move through the material, providing insights into the ease or difficulty of diffusion.

We constructed and evaluated five promising long-range pathways by examining local Na⁺ hops and their associated activation energies (see Figure 4 and

Table 4. Long-range Na ion diffusion pathways and their overall activation energies.

Long-Range Pathway	Hop Activation Energies (eV)	Overall Activation Energy (eV)
A↔C↔E↔A	0.23 ↔0.03↔0.55↔0.23	0.55
D↔E↔G↔D↔G	0.73 ↔0.55↔0.62↔0.73↔0.62	0.73
A↔C↔E↔D↔G	0.23 ↔0.03↔0.55↔0.73↔0.62	0.73
A↔B↔E↔G↔D↔G	0.23 ↔0.65↔0.55↔0.62↔0.73↔0.62	0.73
A↔C↔E↔F↔A	0.23 ↔0.03↔0.55↔1.02↔0.23	1.02

). The most favorable long-range path with the lowest overall activation energy (0.55 eV) is A↔C↔E↔A. In this long-range migration, Na⁺ ions migrate in the ac plane. An overall activation energy of 0.73 eV for three long-range pathways suggests the presence of mechanisms or reactions in materials or processes where the energy barrier for these pathways is also relatively low, facilitating easier transition or movement over longer distances. A long-range diffusion pathway with an overall migration energy of 1.02 eV indicates a significant energy barrier that must be overcome for atoms or ions to move through a material over extended distances. This high barrier can impact the material’s performance in various applications.

3.4. Solution of Dopants

Doping in battery materials is a strategic method to enhance various properties of the materials used in batteries, including their electrical conductivity, thermal stability, capacity, and cycle life. This process involves the intentional introduction of impurities (dopants) into the host material to improve its performance. When considering isovalent and aliovalent dopants in Na₄SiO₄, it is important to understand the impact these dopants have on the material’s structure and properties. Isovalent dopants have the same valence as the ions they replace, whereas aliovalent dopants have a different valence, which can lead to charge imbalances that must be compensated for within the crystal structure. Buckingham potentials used for dopants are provided in the electronic Supplementary Materials (ESM).

3.4.1. Monovalent Dopants

Monovalent dopants (M = Li, K, and Rb) in Na₄SiO₄ can alter its properties, including ionic conductivity and structural stability. Monovalent dopants have a single positive charge and can substitute for Na⁺ in the crystal lattice. The doping process is described by the following equation:

$${\mathrm{M}}_2\mathrm{O}+2 {\mathrm{N}\mathrm{a}}_{\mathrm{N}\mathrm{a}}^{\mathrm{X}}\to 2 {\mathrm{M}}_{\mathrm{N}\mathrm{a}}^{\mathrm{X}}+{\mathrm{N}\mathrm{a}}_2\mathrm{O}$$

(9)

Endoergic (or endothermic) solution enthalpies calculated for all dopants mean that the process absorbs energy from the surroundings. This implies that adding dopants to Na₄SiO₄ requires thermal energy, and the process is more thermodynamically favorable at higher temperatures. Among the dopants considered, K⁺ has the lowest solution enthalpy, meaning it requires the least amount of energy to be incorporated into Na₄SiO₄ (see Figure 5). This makes K⁺ the most thermodynamically favorable dopant. A dopant with a size closer to that of the host ion will cause less distortion in the lattice. K⁺, despite being larger, may fit into the Na₄SiO₄ lattice with less strain compared to other dopants, leading to a more stable structure. This can be advantageous for practical doping processes as it suggests easier incorporation and potentially better stability.

3.4.2. Trivalent Dopants

Incorporating trivalent dopants on the silicon site in Na₄SiO₄ can introduce various structural and electronic modifications that may enhance the material’s properties, such as ionic conductivity and stability. Substituting a Si⁴⁺ ion with a trivalent ion (M³⁺) introduces a negative charge in the lattice. This can be compensated for by creating oxygen vacancies or by incorporating additional sodium ions (Na⁺) to maintain charge neutrality. The creation of oxygen vacancies can enhance ionic conductivity by providing pathways for ion migration. Additionally, extra sodium ions introduced for charge compensation can increase the number of mobile ions according to the following equation. In this study, a diverse array of trivalent dopants (M = Al, Ga, Gd, In, Sc, and Y) were examined for incorporation at the Si site.

$${\mathrm{M}}_2{\mathrm{O}}_3+2 {\mathrm{S}\mathrm{i}}_{\mathrm{S}\mathrm{i}}^{\mathrm{X}}+{\mathrm{N}\mathrm{a}}_2\mathrm{O}\to 2 {\mathrm{M}}_{\mathrm{S}\mathrm{i}}^{\prime }+{2 \mathrm{N}\mathrm{a}}_{\mathrm{i}}^{•}+2 {\mathrm{S}\mathrm{i}\mathrm{O}}_2$$

(10)

Aluminum is the most promising trivalent dopant for improving Na₄SiO₄ due to its low solution energy and relatively small ionic radius (see Figure 6). The ionic radius of Al³⁺ is relatively close to that of Si⁴⁺. This similarity minimizes lattice distortions and strain when Al substitutes for Si. Minimal strain means the crystal structure remains stable and intact, making Al a favorable dopant. In previous simulation studies [45,46], Al has been reported to be the most promising dopant on the Si site in a variety of silicate materials. Gallium is also a promising dopant with the second lowest solution energy, indicating that it can also be incorporated into the Na₄SiO₄ lattice. Indium, despite having a larger ionic radius, has a solution energy slightly lower than Sc, but higher than Al and Ga, making it a moderately feasible dopant. Scandium, Yttrium, and Gadolinium, with higher solution energies and larger ionic radii, are less favorable for doping Na₄SiO₄. They introduce more significant lattice distortions and require more energy to incorporate, which may result in less effective enhancement of ionic conductivity.

3.4.3. Tetravalent Dopants

Tetravalent dopants (M = Ge, Sn, Ti, and Ce) on the Si site in Na₄SiO₄ can be used to modify and enhance the material’s properties while maintaining the charge balance, as they have the same +4 valence as silicon. The following reaction equation describes the following process:

$${\mathrm{M}\mathrm{O}}_2+{\mathrm{S}\mathrm{i}}_{\mathrm{S}\mathrm{i}}^{\mathrm{X}}\to {\mathrm{M}}_{\mathrm{S}\mathrm{i}}^{\mathrm{X}}+{\mathrm{S}\mathrm{i}\mathrm{O}}_2$$

(11)

Germanium has the lowest solution energy, indicating that it is the easiest dopant to incorporate into the Na₄SiO₄ lattice (see Figure 7). The ionic radius of Ge⁴⁺ (0.53 Å) is close to that of Si⁴⁺ (0.40 Å), ensuring minimal lattice distortion. This makes Ge⁴⁺ an excellent choice for doping, enhancing ionic conductivity, and maintaining structural integrity. Tin has a moderate solution energy, suggesting that it is relatively easy to incorporate but not as easy as Ge⁴⁺. Sn⁴⁺ can be beneficial for certain applications, but its larger ionic radius (0.69 Å) compared to Si⁴⁺ could introduce some lattice distortions. Titanium has a higher solution energy, making it more difficult to incorporate into the lattice. Ti⁴⁺ has an ionic radius of 0.61 Å, which is larger than Si⁴⁺, potentially causing lattice distortions. However, Ti⁴⁺ can enhance mechanical properties. Cerium has the highest solution energy, indicating the greatest difficulty in incorporating it into the lattice. Ce⁴⁺ has a larger ionic radius, which can lead to significant lattice distortions. It may provide unique properties but is less favorable due to its high solution energy.

4. Conclusions

In this study, we used atomistic simulation techniques to examine the defect, diffusion, and doping properties of a promising battery material, Na₄SiO₄. The Na Frenkel defect was found to be the most energetically favorable defect process facilitating vacancy-assisted Na-ion migration. Three-dimensional Na⁺ ion diffusion is characterized by an activation energy of 0.55 eV. K and Ge are favorable dopants for the Na and Si sites in Na₄SiO₄ respectively. The doping of the Si site in Na₄SiO₄ with trivalent Al³⁺ ions is favorable, generating additional Na⁺ ions in the material. Experimental validation of these findings will be crucial in confirming their practical implications and optimizing Na₄SiO₄ for high-performance battery applications.