The Characteristics of Structural Properties and Diffusion Pathway of Alkali in Sodium Trisilicate: Nanoarchitectonics and Molecular Dynamic Simulation

Based on nanoarchitectonics and molecular dynamics simulations, we investigate the structural properties and diffusion pathway of Na atoms in sodium trisilicate over a wide temperature range. The structural and dynamics properties are analyzed through the radial distribution function (RDF), the Voronoi Si- and O-polyhedrons, the cluster function fCL(r), and the sets of fastest (SFA) and slowest atoms (SSA). The results indicate that Na atoms are not placed in Si-polyhedrons and bridging oxygen (BO) polyhedrons; instead, Na atoms are mainly placed in non-bridging oxygen (NBO) polyhedrons and free oxygen (FO) polyhedrons. Here BO, NBO, and FO represent O bonded with two, one, and no Si atoms, respectively. The simulation shows that O atoms in sodium trisilicate undergo numerous transformations: NBF0 ↔ NBF1, NBF1 ↔ NBF2, and BO0 ↔ BO1, where NBF is NBO or FO. The dynamics in sodium trisilicate are mainly distributed by the hopping and cooperative motion of Na atoms. We suppose that the diffusion pathway of Na atoms is realized via hopping Na atoms alone in BO-polyhedrons and the cooperative motion of a group of Na atoms in NBO- and FO-polyhedrons.


Introduction
It is well known that the structure of silica (SiO 2 ) is an archetypal network-forming system containing SiO 4 tetrahedra.The addition of doped Na ions generates non-bridging oxygen (NBO) [1,2].Therefore, sodium trisilicate (Na 2 O•3SiO 2 ) has various anomalous properties which are essential for industrial applications, ceramics, metallurgy, and glass technologies, as well as for understanding the fundamentals of minerals [3][4][5].Na 2 O•3SiO 2 has been extensively studied using experimental methods such as photoelectron spectroscopy [6], X-ray diffraction [4,7], in situ Raman spectroscopy, and elastic neutron scattering [8], along with various simulation techniques [9][10][11][12].Nesbitt et al. have characterized two types of network oxygen in sodium silicate: three-fold-coordinated BO-Na and two-fold-coordinated Si-O-Si [6].The doped Na ions function as a network modifier, causing significant alterations in the random network of corner-shared SiO 4 tetrahedra with the formation of FO and NBO.Davidenko et al. reported that Na 2 O-SiO 2 has a microheterogeneous structure which contains noncrystalline micro-groups such as silica, Na disilicate, and Na monosilicate [7].Zhao et al. indicated that, when enough Na 2 O is added into the glass matrix to create two NBO atoms (Q2 species), the silica network loses its three-dimensional connectivity needed to sustain the local transformations between α-like and β-like rings, and glass softens upon heating due to the dominant anharmonic effect [8].
The nanoarchitectonics and molecular dynamics (MDs) simulation can provide more detailed information about both the microstructural properties and diffusion mechanisms at the atom level to develop functional materials (Figure 1).The results of MDS found that in The very fast diffusivity of alkali atoms is one of the important dynamical properties of alkali silicates [13][14][15][16][17][18][19].The addition of doped Na ions to pure SiO2 leads to a decoupling of alkali diffusion and diffusive transport in the Si-O network.Davidenko et al. suggested that a distribution where increasing alkali oxide content causes homogeneous, increasing disruption to the Si-O network of pure SiO2 is in conflict with the highly nonlinear dependence of viscosity on alkali concentration [20].In accordance with other studies [21][22][23][24], the existence of the decoupling of alkali diffusion and diffusive transport in the relatively immobile Si-O network is proposed to interpret the fast diffusivity of alkali ions.The pre-peak at 0.9 Å −1 in the structure factor measured experimentally for alkali silicates is evidence for the diffusion pathway.
Various experimental results have shown that the structure of these silicates is found to comprise micro-regions with high Na concentrations.The two structural samples, the modified random network and the compensated continuous random network, predict some clustering of alkali atoms in the silicate's structure [25][26][27].This indicates that the clustering of alkali atoms marks out their diffusion pathway.
Several models for diffusion mechanisms in alkali silicate are suggested [28][29][30].It is demonstrated that the diffusion pathway occupies a relatively small subspace of the system.According to Horbach et al. and Smith et al., the activated hopping of Na through the Si-O matrix is frozen with respect to the movement of Na atoms [9,31].Based on the position of the first peak of the pair radial distribution function (RDF) for the Na-Na pair, they determine the mean hopping distance of Na atoms.The alkali ions hop into empty sites so that the mechanism owes more in character to their vacancy crystalline counterpart than to their interstitial cousins [32][33][34].Habasaki et al. indicated that the cooperative motion mechanism may be suitable to clarify the increasing diffusivity with increasing alkali content [35].Traps due to defects or impurities can lead to a decrease in charge The very fast diffusivity of alkali atoms is one of the important dynamical properties of alkali silicates [13][14][15][16][17][18][19].The addition of doped Na ions to pure SiO 2 leads to a decoupling of alkali diffusion and diffusive transport in the Si-O network.Davidenko et al. suggested that a distribution where increasing alkali oxide content causes homogeneous, increasing disruption to the Si-O network of pure SiO 2 is in conflict with the highly nonlinear dependence of viscosity on alkali concentration [20].In accordance with other studies [21][22][23][24], the existence of the decoupling of alkali diffusion and diffusive transport in the relatively immobile Si-O network is proposed to interpret the fast diffusivity of alkali ions.The pre-peak at 0.9 Å −1 in the structure factor measured experimentally for alkali silicates is evidence for the diffusion pathway.
Various experimental results have shown that the structure of these silicates is found to comprise micro-regions with high Na concentrations.The two structural samples, the modified random network and the compensated continuous random network, predict some clustering of alkali atoms in the silicate's structure [25][26][27].This indicates that the clustering of alkali atoms marks out their diffusion pathway.
Several models for diffusion mechanisms in alkali silicate are suggested [28][29][30].It is demonstrated that the diffusion pathway occupies a relatively small subspace of the system.According to Horbach et al. and Smith et al., the activated hopping of Na through the Si-O matrix is frozen with respect to the movement of Na atoms [9,31].Based on the position of the first peak of the pair radial distribution function (RDF) for the Na-Na pair, they determine the mean hopping distance of Na atoms.The alkali ions hop into empty sites so that the mechanism owes more in character to their vacancy crystalline counterpart than to their interstitial cousins [32][33][34].Habasaki et al. indicated that the cooperative motion mechanism may be suitable to clarify the increasing diffusivity with increasing alkali content [35].Traps due to defects or impurities can lead to a decrease in charge carrier mobility and an increase in recombination rates in semiconductors [36].However, many fundamental aspects of the fast diffusion of Na in Na 2 O•3SiO 2 remain up for debate.
Therefore, in this present study, we focus on Na 2 O•3SiO 2 at different temperatures.Based on the characteristics of the pair RDF, the Voronoi Si-and O-polyhedrons, the cluster function, f CL (r), and the sets of fastest (SFA) and slowest atoms (SSA), we attempt to gain insight into the spatial distribution as well as the diffusion pathway of Na in the Na 2 O•3SiO 2 .

Results and Discussion
The total RDF for neutron diffraction, G(r), for Na 2 O•3SiO 2 is presented in Figure 2.This RDF exhibits the short-range order (SRO).As shown in Figure 2, the first peak of G(r) is located at 1.60 ± 0.02 Å, which is contributed to by g Si-O (r), exhibiting the Si-O bond length.The second and third peaks of G(r) are located at 2.25 ± 0.02 Å and 2.60 ± 0.02 Å, which are contributed to by g Na-O (r) and g O-O (r), exhibiting the Na-O and O-O bond lengths, respectively.It can be seen that there is good agreement between the experimental and simulated G(r) RDFs for an r of up to 6 Å [1].Using the method in [37], the calculation result for the bond angle distribution indicated that the peaks of the O-Si-O and Si-O-Si bond angle distributions are located at 109.6 • and 149.0 • , respectively.For Si-O-Na, the bond angle distribution is centered around 90 • -125 • , very close to the experiments [1,7].
carrier mobility and an increase in recombination rates in semiconductors [36].However, many fundamental aspects of the fast diffusion of Na in Na2O•3SiO2 remain up for debate.
Therefore, in this present study, we focus on Na2O•3SiO2 at different temperatures.Based on the characteristics of the pair RDF, the Voronoi Si-and O-polyhedrons, the cluster function, fCL(r), and the sets of fastest (SFA) and slowest atoms (SSA), we attempt to gain insight into the spatial distribution as well as the diffusion pathway of Na in the Na2O•3SiO2.

Characteristics of Structural Na2O•3SiO2
The total RDF for neutron diffraction, G(r), for Na2O•3SiO2 is presented in Figure 2.This RDF exhibits the short-range order (SRO).As shown in Figure 2, the first peak of G(r) is located at 1.60 ± 0.02 Å, which is contributed to by gSi-O(r), exhibiting the Si-O bond length.The second and third peaks of G(r) are located at 2.25 ± 0.02 Å and 2.60 ± 0.02 Å, which are contributed to by gNa-O(r) and gO-O(r), exhibiting the Na-O and O-O bond lengths, respectively.It can be seen that there is good agreement between the experimental and simulated G(r) RDFs for an r of up to 6 Å [1].Using the method in [37], the calculation result for the bond angle distribution indicated that the peaks of the O-Si-O and Si-O-Si bond angle distributions are located at 109.6° and 149.0°, respectively.For Si-O-Na, the bond angle distribution is centered around 90°-125°, very close to the experiments [1,7].The total RDF of sodium trisilicate at a temperature of 973 K and a comparison with experimental data [1], which was obtained from this work and reported in an experiment in [1].
Figure 3 shows the RDF for the BO-Na, NBF-Na, O-Na, and Si-Na pairs at temperatures of 300, 973, and 1573 K.A pronounced peak is seen in gNBF-Na(r), which is not unclear for gBO-Na(r).Note that The present work

Fábián et al., 2007
Figure 2. The total RDF of sodium trisilicate at a temperature of 973 K and a comparison with experimental data [1], which was obtained from this work and reported in an experiment in [1].
Figure 3 shows the RDF for the BO-Na, NBF-Na, O-Na, and Si-Na pairs at temperatures of 300, 973, and 1573 K.A pronounced peak is seen in g NBF-Na (r), which is not unclear for g BO-Na (r).Note that where    The fractions of different types of O atoms are listed in Table 1.It can be seen that O-polyhedrons either do not contain or do contain Na; meanwhile, Si-polyhedrons do not contain any Na residues.Our simulation shows that most O-polyhedrons comprise BO0, BO1, NBF0, NBF1, and NBF2.It shows that BO0 and NBF0 represent an empty O-polyhedron which does not contain Na.As temperature increases, the fractions m BO1 /m O and m NBF0 /m O increase.In contrast, m BO0 /m O and m NBF2 /m O slightly change in the opposite direction.From these data, we can suggest a simple diffusion sample in Na 2 O•3SiO 2 .Namely, Na travels from its site to empty sites which are located in BO-and NBF-polyhedrons.The motion of Na between polyhedrons leads to the diffusivity of Na.Next, more detail about the tetrahedral network structure of Na 2 O•3SiO 2 is derived from the characteristics of Voronoi polyhedrons, which are summarized in Table 2.As seen, the average volume per polyhedron increases in the order Si-→ BO-→ NBF-polyhedron.The total volume of BO-and NBF-polyhedrons, which contain Na, is about 87% of the simulation box.Although V BO is equal to 1.6 times V NBF , about 78-90% of total Na is placed in NBF-polyhedrons.As temperature increases, the amount of Na residing in NBFpolyhedrons decreases from 91 to 78%, corresponding with the temperature increasing from 300 to 1573 K.This demonstrates that Na atoms are not uniformly distributed through O-polyhedrons but instead are gathered in NBF-polyhedrons, as can be seen from Figure 4.This supports the idea that the diffusion pathway of Na occurs in NBF-polyhedrons with a total volume of 33.0-33.6% of the simulation box.

Table 2.
Characteristics of Voronoi polyhedrons.<v Si >, <v BO >, and <v NBF > are the average volumes per Si-, BO-, and NBF-polyhedron, respectively, measured in Å 3 ; V Si , V BO , V NBF , and V SB are the volumes occupied by Si-, BO-, and NBF-polyhedrons and the volume of the simulation box, respectively; m NaBO and m NaNBF are the numbers of Na atoms residing in BO-and NBF-polyhedrons, respectively; and m Na is the total number of Na atoms.Figure 5 plots the number of BO1-, NBF1-, and NBF2-polyhedrons versus time at different temperatures.As seen, the number of NBF2-polyhedrons is larger than that of BO1 polyhedrons at low temperatures, and it becomes smaller at high temperatures.In the temperature range of 773-973 K, the number of NBF2-polyhedrons is the same as that of BO1 polyhedrons.This is explained by the fact that more Na spreads on BO-polyhedrons at high temperatures.This is also observed from the spatial distribution of all types of OT 2 linkages (T is Si or Na), as plotted in Figure 6.As seen, the Si-O-Na and Si-O-Si linkages are dominant.There is only a small count of Na-O-Na linkages.These linkages slightly change with increasing temperature.This is explained by the fact that pure SiO 2 is composed of a continuous random network of SiO 4 tetrahedra and that the doped Na ions break the Si-O linkages, which leads to the generation of non-bridging oxygen (NBO) in Na 2 O•3SiO 2 .The generation of NBO lowers the glass melting point.

T (K) <v
BO1 polyhedrons at low temperatures, and it becomes smaller at high temperatures.In the temperature range of 773-973 K, the number of NBF2-polyhedrons is the same as that of BO1 polyhedrons.This is explained by the fact that more Na spreads on BO-polyhedrons at high temperatures.This is also observed from the spatial distribution of all types of OT2 linkages (T is Si or Na), as plotted in Figure 6.As seen, the Si-O-Na and Si-O-Si linkages are dominant.There is only a small count of Na-O-Na linkages.These linkages slightly change with increasing temperature.This is explained by the fact that pure SiO2 is composed of a continuous random network of SiO4 tetrahedra and that the doped Na ions break the Si-O linkages, which leads to the generation of non-bridging oxygen (NBO) in Na2O•3SiO2.The generation of NBO lowers the glass melting point.

The Diffusion Pathway of Na Atoms in Na 2 O•3SiO 2
To demonstrate the clustering of Na atoms, we use the link cluster function, f Cl (r); the calculation algorithm employed here can be found elsewhere [38,39].The sets of fastest, slowest, and random atoms (SFA, SSA, and SRA) comprise about 10% of all the atoms.The SFA has a mean square displacement (MSD) larger than that of the remaining atoms.The SSA has an MSD smaller than that of the remaining atoms.The SRA contains atoms that are randomly chosen from the sample.The atoms in the SFA, SSA, and SRA are determined from the atom position in the configuration at 150 ps. Figure 7 plots f Cl (r) at different temperatures.As seen, in the temperature range of 300-1573 K, the considered sets are mostly unchanged.However, the variations in the SSA, SFA, and SRA are quite different with r; namely, the SSA drops drastically from 1000 to 483 atoms as r varies from 1.3 to 1.75 ± 0.05 Å.The value of f Cl (r) at 1.75 Å is about 1000 and 895, corresponding to the SFA and SRA, respectively.(a1) (b1) (c1)    Clearly, these values are significantly larger than that of the SSA.With further increasing r to 3.00 ± 0.05 Å, the SFA and SRA appear as turning points, and then f Cl (r) gradually decreases.The result demonstrates the heterogeneous spatial distribution of the fastest and slowest atoms in the Na 2 O•3SiO 2 network.The problem here is that we do not know whether the Na atoms are mainly distributed in the SFA.To investigate this, we considered the distribution of sets of the fastest and slowest atoms using a visual technique.Here, we considered two consecutive configurations at moments t and t + 2 ps.Then, the MSD was identified, and the SFA and SSA were therefore found.Figure 8 displays the distribution of SFA and SSA at 773, 1173, and 1573 K.As can be seen, the distributions of both the fastest and slowest atoms are not uniform.The SFA mainly includes Na atoms, while the SSA mainly includes Si and O atoms.This demonstrates that the dynamics of the atoms are heterogeneous, and the hopping of Na atoms is mainly distributed for diffusion pathways in Na 2 O•3SiO 2 .
To investigate the diffusion pathway, we specified the number of BO-and NBFpolyhedrons as a function of <x A >, where <x A > is the mean number of Na atoms in the A-polyhedron for the time t sot .The distributions of the fraction m Ax /m A and the time dependence of the deviation of those distributions, δ Ax , for the case t sot = 150 ps are plotted in Figures 9 and 10.It can be seen that the curve for the NBF-polyhedrons possesses a pronounced peak at <x A > = 0.75 and spreads over a wide range, whereas the major BOpolyhedrons possess a small <x A >.As temperature increases, the curve for NBF spreads over a narrower range.This demonstrates that the diffusion pathway is composed of NBFpolyhedrons.With increasing t sot , the deviation, δ Ax , decreases quickly, and it approaches a smaller value at higher temperatures.This observation is understood as follows: under t sot , the average number of Na atoms in the i th A-polyhedron approaches <x Ai >, which is proportional to where E Si , k B , and T are the site energy, Boltzmann constant, and temperature, respectively.On the other hand, from Figure 9, the <x A > of BO is significantly smaller than that of NBF.This means that the site energy for a Na atom located in an NBF-polyhedron must be significantly smaller than that for a Na atom in a BO-polyhedron.In addition, <x A > varies over a wide range, indicating that Na has various energies, E Si .In fact, Na moves frequently between A-polyhedrons, so the average number of Na atoms in the ith polyhedron quickly approaches <x Ai >, and δ Ax also decreases quickly with t sot .As temperature increases, the values of <x Ai > for different A-polyhedrons are close to each other.This leads to δ Ax approaching a smaller value with increasing temperature (see Figure 10).Figure 11 displays the number of Na atoms staying in an A-polyhedron or moving from an A-polyhedron to another one within 2.0 ps.As expected, the number of Na atoms remaining monotonously decreases with increasing temperature.
Due to the movement of Na between O-polyhedrons, the system undergoes numerous Ax → Ax' transformations.The average number of Ax → Ax' transformations is plotted in Figure 10.As seen in Figure 10, there are a small number of BOx atoms undergoing the transformation BO0 ↔ BO1, which increases with temperature.Therefore, more Na atoms reside in BO-polyhedrons at higher temperatures, and Na atoms perform independent jumping in them.In the case of NBF-polyhedrons, the transformations NBF0 ↔ NBF1 and NBF1 ↔ NBF2 occur in the majority of polyhedrons.However, the system comprises a number of NBF-polyhedrons undergoing the transformation NBFx → NBFx', where |x − x'| > 1.The number of such NBF-polyhedrons also increases with increasing temperature.We conclude that, unlike BO-polyhedrons, Na performs independent jumping and cooperative motion.Cooperative motion is realized in more NBF-polyhedrons at higher temperatures.
From the above explanation, we suggest that the diffusion constant for Na in Na 2 O•3SiO 2 may be written as follows: where < d 2 SS > is the mean square distance between a site and its nearest neighbor, v p is the rate of Na atoms moving between O-polyhedrons, γ is the geometrical factor, and f is the correlation coefficient describing the forward-backward jumps of Na atoms [32,33].On the other hand, from Figure 9, the <xA> of BO is significantly smaller than that of NBF.This means that the site energy for a Na atom located in an NBF-polyhedron must be significantly smaller than that for a Na atom in a BO-polyhedron.In addition, <xA> varies over a wide range, indicating that Na has various energies, ESi.In fact, Na moves frequently between A-polyhedrons, so the average number of Na atoms in the ith polyhedron quickly approaches <xAi>, and δAx also decreases quickly with tsot.As temperature increases, the values of <xAi> for different A-polyhedrons are close to each other.This leads to δAx approaching a smaller value with increasing temperature (see Figure 10).displays the number of Na atoms staying in an A-polyhedron or moving from an A-polyhedron to another one within 2.0 ps.As expected, the number of Na atoms remaining monotonously decreases with increasing temperature.Here, <m NaA > is the average number of Na atoms staying in A-polyhedrons or moving from an A-polyhedron to another one within 2 ps; A is BO or NBF; m Na is the total number of Na atoms; <m Ax-Ax' > is the average number of Ax → Ax' transformations within 2 ps; and m A is the total number of A-polyhedrons.

Materials and Methods
The simulation was carried out for Na 2 O•3SiO 2 at ambient pressure over a temperature range of 300-1573 K.The sample was made of 9996 atoms, including 5831 O atoms, 2499 Si atoms, and 1666 Na atoms.All simulation runs were performed using MXDORTO code [40].We used the interaction potentials, consisting of two-body and three-body terms, which can quite reliably reproduce the structure and dynamics of sodium silicate.The density was adopted from that of a real sodium trisilicate glass of 2.4323 g/cm 3 [41].A complete description of these potentials can be found elsewhere [29,42].The pair potential has the following form: The potential parameters are listed in Table 3.Note that the three-body term relating to the Si-O-Si angle has the following form: where f is the force constant; θ kij is the angle among atoms k-i-j; and θ 0 , g r , and r m are the parameters for adjusting the angular part of covalent bonds.The partial charges for the Si and O atoms are calculated as follows: Z Si .Z O = −2.5 and 2Z Na + 3 Z Si + 7 Z O = 0, where Z Na is fixed at 1.0.The pair radial distribution function (PRDF) for BO and NBF is characteristic of their local microstructure.Here, BO, NBO, and FO are the types of oxygen which are bounded, respectively, with two, one, and no Si atoms; NBF is denoted either as NBO or FO.Overall, the status of O (BO, NBO, and FO) was mostly unchanged during the simulation.Only a few BO ↔ NBF transformations were detected at temperatures of 300, 973, and 1373 K.
In this work, the simple nanoarchitectonics of atoms include O-polyhedrons and Si-polyhedrons; bridging oxygen (BO)-, non-bridging oxygen (NBO)-, and free oxygen (FO)-polyhedrons; the group of Na in NBO-and FO-polyhedrons; and the fastest, slowest, and random atoms.The visualization of MD data was carried out to study the structural properties and diffusion pathway of Na atoms in sodium trisilicate [43].In this context, we suppose that the simulation box is fully filled by O-and Si-polyhedrons, and Na is placed inside these polyhedrons.In this work, the A-polyhedron is denoted as the A-centered polyhedron, where A is the Si, BO, or NBF; we also use the Ax-polyhedron, where x is the number of Na atoms in the A-polyhedron.For instance, NBF1 represents the NBFpolyhedron containing one Na atom, respectively.During the simulation process, we found that Na atoms were not placed in fixed polyhedrons, but they frequently moved from one to other polyhedrons.To clarify this effect, we observed A-polyhedrons in configurations separated by 2 ps.The local Na density in the vicinity of an O or Si atom was quantified by the mean number of Na atoms in the A-polyhedron, which is called <x A >. We determined Ax for configurations within a span of time t sot , and then <x A > was obtained by averaging x over those polyhedrons.The value of <x A > depends on t sot and approaches a finite value as t sot → ∞.Consider two consecutive configurations at moments t and t + 2 ps.We specified the number of Na atoms staying in the A-polyhedron and also the number of Na atoms moving to other polyhedrons.In this way, we detected Ax → Ax' transformations.For instance, BO1 at moment t transforms to BO0 at moment t + 2 ps if one Na atom

Figure 1 .
Figure 1. Outline of nanoarchitectonics and molecular dynamics simulation: meaning and procedure.

Figure 1 .
Figure 1. Outline of nanoarchitectonics and molecular dynamics simulation: meaning and procedure.

Figure 2 .
Figure2.The total RDF of sodium trisilicate at a temperature of 973 K and a comparison with experimental data[1], which was obtained from this work and reported in an experiment in[1].

Figure 5 .Figure 5 .
Figure 5.Time dependence of the number of Ax-polyhedrons at different temperatures.Here, A denotes an NBF or BO atom; x denotes the number of Na atoms in the Ax-polyhedron.

Figure 7 .
Figure 7.The CL function for atoms belonging the SFA, SSA, and SRA at different temperatures.

Figure 7 .
Figure 7.The CL function for atoms belonging the SFA, SSA, and SRA at different temperatures.

Figure 8 .
Figure 8.The 5% distribution of the sets of slowest atoms (SSA) and fastest atoms (SFA): (a) 1573 K, (b) 1173 K, and (c) 773 K; the red ball is Si, the blue ball is O, and the green ball is Na.

Figure 8 .
Figure 8.The 5% distribution of the sets of slowest atoms (SSA) and fastest atoms (SFA): (a) 1573 K, (b) 1173 K, and (c) 773 K; the red ball is Si, the blue ball is O, and the green ball is Na.

Figure 9 .
Figure 9.The fraction mAx/mA as a function of the average number of Na atoms in the Apolyhedron <xA> for 150 ps at different temperatures.Here, A denotes BO or NBF; mAx and mA denote the number of A-polyhedrons with <xA> and the total number of A-polyhedrons, respectively.

Figure 9 .
Figure 9.The fraction m Ax /m A as a function of the average number of Na atoms in the A-polyhedron <x A > for 150 ps at different temperatures.Here, A denotes BO or NBF; m Ax and m A denote the number of A-polyhedrons with <x A > and the total number of A-polyhedrons, respectively.

Figure 10 .
Figure 10.The time dependence of the deviation, Ax, at different temperatures: (a)-BO and (b)-NBF.

Figure 11 .
Figure 11.The temperature dependence of <m NaA >/m Na (a,b) and <m Ax-Ax' >/m A (c).Here, <m NaA > is the average number of Na atoms staying in A-polyhedrons or moving from an A-polyhedron to another one within 2 ps; A is BO or NBF; m Na is the total number of Na atoms; <m Ax-Ax' > is the average number of Ax → Ax' transformations within 2 ps; and m A is the total number of A-polyhedrons.
structure of Na 2 O•3SiO 2 .The major O forms are NBO and BO, and less than 0.05% of the total O is FO.In addition, the relative fraction of O types was almost unchanged with temperature (Table1).The spatial distribution of TOy (T is Si or Na, y = 4.5) in Na 2 O•3SiO 2 samples at different temperatures is shown.As seen, the structure of Na 2 O•3SiO 2 comprises mainly SiO 4 units and a small count of NaO 4 and NaO 5 units, which are distributed over the whole space.
and m NBO , m BO , and m O are the total numbers of NBO, BO and O, respectively.This means that most of the Na is located around the NBF and is rarely present in the vicinity of BO.All the constructed samples consist of SiO 4 along with BO and only fewer SiO 5 particles in the high-temperature samples, indicating the tetrahedral network Int.J. Mol.Sci.2024, 25, x FOR PEER REVIEW 5 of 18

Table 2 .
Characteristics of Voronoi polyhedrons.<vSi>, <vBO>, and <vNBF> are the average volumes per Si-, BO-, and NBF-polyhedron, respectively, measured in Å 3 ; VSi, VBO, VNBF, and VSB are the volumes occupied by Si-, BO-, and NBF-polyhedrons and the volume of the simulation box, respectively; mNaBO and mNaNBF are the numbers of Na atoms residing in BO-and NBF-polyhedrons, respectively; and mNa is the total number of Na atoms.

Table 1 .
Distribution of Na in O-polyhedrons and fraction of different types of O. Here, m Ax is the number of Ax-polyhedrons; A is BO or NBF; x is the number of Na atoms in the Ax-polyhedron; m FO , m NBO , m BO , and m O are the total numbers of FO, NBO, BO, and O atoms, respectively.

Table 3 .
The interatomic potential parameters.