Introduction

Borosilicate glasses have been used as one of the vitrification materials of radioactive wastes in many countries.1 International Simple Glass (ISG) is a simplified model composition for a collaborative study of dissolution/corrosion mechanism of high-level waste (HLW) glasses.2,3,4 ISG has been intensively researched recently with studies focusing on its dissolution behavior under different conditions,3,5,6,7,8,9,10,11,12 its surface quality,13 radiation effect,14,15,16,17,18 waste loading,19 mechanical property,20,21 and especially on the properties of its alteration layers using diverse techniques such as electron microprobe analysis,22 Doppler broadening positron annihilation spectroscopy,23 and atom probe tomography.24

However, due to the lack of long-range periodicity, the structural study of glass materials exhibits many challenges. Detailed information of glass structure provides better understanding of glass physical properties and chemical durability, which helps design glass compositions for different waste types and geological disposals. Molecular dynamics (MD) simulation is an effective method of studying structural information in glass materials, and it provides a direct observation of atomic structure, which is difficult to obtain from experimental methods.25 Many studies have been systematically conducted and revealed short-range and medium-range structural features utilizing MD simulation in various glass systems such as sodium silicate glasses,26 aluminosilicate glasses,27 lithium disilicate glasses,28 cerium aliminophosphate glasses,29 europium-doped silicate glasses,30,31 erbium-doped sodium silicate glasses,32 lithium vanado-phosphate glasses,33 and bioactive glasses.34,35,36 However, MD simulation of nuclear waste glasses has been limited due to the lack of pairwise potential sets for boron. With the recent development of modified composition-dependent pairwise potential sets, it is now possible to simulate complex glass compositions.36,37 ISG was successfully simulated with MD,37 structural related properties such as the percentage of 4-coordinated boron, PDFs, bond angle distributions (BADs) and ring-size analysis, were calculated and found to be in agreement with available experimental data.

Even though zirconia content is low in ISG, it is essential to study the local environment of zirconia in ISG and the effect of ZrO2/SiO2 substitution for the benefit of glass formulation. Indeed, addition of ZrO2 has been found to significantly alter the physical properties of alkali silicate glasses, such as increase the density, viscosity, glass transition temperature, and decrease the thermal expansion coefficient.38,39 ZrO2 has been reported to enhance the chemical durability of glasses not only in neutral pH condition but also in alkali-rich environments.38,39,40 However, even though the initial dissolution rate drops significantly with increasing ZrO2 concentration (up to 8 mol%) in soda-lime borosilicate glasses, recent findings showed that the inhibiting effect of ZrO2 on the dynamics of alteration layer formation ultimately results in a greater degree of corrosion in the long term.41,42,43 Zirconium ions were found to be mostly six-fold coordinated and [ZrO6] octohedra link with other network formers such as [SiO4] tetrahedra through corner-sharing.44,45,46 This feature was also observed in MD simulations of a 42CaO-48SiO2-10ZrO2 glass47 and a sodium alumina borosilicate glass.46 Moreover, MD simulations revealed that amorphization of crystalline zircon (ZrSiO4) leads to polymerization of the silicon-oxygen network.48 Similarly, addition of ZrO2 was found to polymerized glass network in sodium-lithium borosilicate glasses by attracting alkali cations to charge compensate six-fold coordinated zirconium ion.49 However, very few studies investigate the structural changes altered by ZrO2 concentration in boron-containing nuclear waste glasses and its relationship between glass structure and dissolution behaviors by using MD simulations.

Therefore, the purposes of this study are to systematically study with MD simulations the structural and dynamical changes of ISG upon ZrO2/SiO2 substitution. Specifically, the local environment of cations and medium-range structural information are computed, including the Zr K-edge extended X-ray absorption fine structure (EXAFS) spectrum, and the changes caused by ZrO2/SiO2 substitution are discussed in terms of their effect on dissolution.

Results

General structural information

To validate our MD simulation results, a comparison of X-ray structure factors of 2Zr (ISG) obtained from experiment and MD simulation is shown in Fig. 1. The calculated structure factor is in good agreement with the experimental data, indicating an overall reasonable structure model generated from our MD simulation. The peak position of the first four peaks and the shoulder of the second peak are matched between experiment and MD simulation, except that the first two peaks from MD simulation have slightly higher intensities. There are small differences on the third valley and the fifth peak, which provide direction for future potential improvement.

Fig. 1
figure 1

Comparison of X-ray diffraction structure factors of 2Zr (ISG) obtained from experiment57 and MD simulation

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Selected bond lengths of simulated 0Zr, 2Zr (ISG), 3Zr, and 4Zr glasses obtained from the first peak of pair distribution functions (PDFs) and comparison with values found in literature are listed in Table 1, and plots of PDFs of 4Zr glass are shown in Fig. 2. Bond length of Si-O is 1.61 Å regardless of composition, which is in excellent agreement with experimental values for glasses with or without zirconium.26,47,50 The two peaks observed for all compositions in the B-O PDFs are due to [3]B–O and [4]B–O bond lengths centered around ~1.46 Å and ~1.55 Å, respectively. Bond lengths of Al–O and Zr–O are ~1.77 Å and ~2.10 Å, respectively, and are also consistent with experimental values (see Table 2).51,52,53,54,55 Bond lengths of Zr with other cations are also in line with values found in the literature, although slight differences might be caused by different compositions and characterization methods. No noticeable changes were observed for bond lengths of cation–oxygen and cation–cation pairs between compositions. BADs of oxygen–cation–oxygen and coordination number of cation–oxygen pairs are not sensitive to ZrO2/SiO2 substitution, and details can be found in supplementary information.

Table 1 Selected bond lengths (Å) in simulated 0Zr, 2Zr (ISG), 3Zr, and 4Zr glasses and comparison with literature values
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Fig. 2
figure 2

Pair distribution functions (PDFs) of cation–oxygen pairs in simulated 4Zr glass. The inset shows PDF of B–O, [3]B–O and [4]B–O pairs

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Table 2 Comparison of 4-coordinated B (N4, %) with different system sizes for simulated 0Zr, 2Zr, 3Zr, and 4Zr glasses
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Preference of modifiers around glass network formers oxygen polyhedra

To further investigate the distribution of modifiers around each network former, CNs of Na and Ca around Si, Al, Zr, B, [3]B, and [4]B were analyzed and all values can be found in supplementary information. The number of Ca ions around each former is lower than that of Na due to the higher content of Na in each glass composition. In order to eliminate these compositional effects, preferential distributions of modifier cations around glass former cations were calculated according to a study by Tilocca et al.,56

$$R_{Na/Ca}^{Former} = \frac{{CN_{former - Na}}}{{CN_{former - Ca}}} \times \frac{{N_{Ca}}}{{N_{Na}}}$$
(1)

where N is the number of modifier cations present. Preferential distributions for Na and Ca around Si, B, [3]B, [4]B, Al, and Zr in simulated 0Zr, 2Zr, 3Zr, and 4Zr glasses are shown in Table 3. A ratio of R > 1 indicates the preference for Na, and vice versa. As shown in Table 3, Si has higher preference of Na than Ca, while Zr has a slightly higher preference for Ca. No preference for Al in 0Zr was observed considering the standard deviations, while Al prefers Ca in glasses containing Zr. [4]B prefers Ca for all compositions, while more Na ions were found around [3]B.

Table 3 Preferential distribution of Na/Ca around Si, B, [3]B, [4]B, Al, and Zr in simulated 0Zr, 2Zr, 3Zr, and 4Zr glasses
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Medium-range structural information

Network connectivity values calculated from Qn distribution of Si, B, Zr, Al and overall NC in simulated 0Zr, 2Zr, 3Zr, and 4Zr glasses are listed in Table 4. Detailed Qn distribution of Si, B, Zr and Al, and percentages of non-bridging oxygen (NBO), bridging oxygen (BO) and tri-bridging oxygen (TBO) cluster in simulated 0Zr, 2Zr, 3Zr, and 4Zr glasses can be found in supplementary information. Overall, NC increases from 3.76 to 3.82 for 0Zr to 4Zr. Due to the high content of glass formers in these glass compositions, NCs of Si, B, Zr, and Al are similar with their coordination numbers, which indicates a well-connected network structure.

Table 4 Network connectivity (NC) of Si, B, Zr, Al and the overall NC in simulated 0Zr, 2Zr, 3Zr, and 4Zr glasses
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Polyhedral analysis was performed in order to study the linkages between network former polyhedra. Numbers of polyhedral linkages between network formers in simulated glasses can be found in supplementary information. Connections between [SiO4]-[SiO4], [AlO4]-[SiO4], and [BO4]/[BO3]-[SiO4] are mostly (over 99.9%) corner-sharing for all compositions. There exist small amount of edge-sharing linkages (less than 4%) between [AlO4]-[BO4], [AlO4]-[AlO4], and [BO4]-[BO4] but the remaining majority (>96%) of these connections are corner sharing. No noticeable changes were observed between compositions among the linkages between Si, Al, and B polyhedra. Among all the linkages with Zr, [SiO4]-[ZrO6], [BO4]/[BO3]-[ZrO6], and [AlO4]-[ZrO6] linkages change from 64.0 to 60.2%, 28.7 to 29.8% and 6.5 to 7.3% for 2Zr (ISG) to 4Zr, respectively, which is consistent with the contents of network-former atoms in each glass composition. Connections between [ZrO6]-[ZrO6] increases from ~1 to 3% for 2Zr to 4Zr. A few edge-sharing and face-sharing [ZrO6]-[ZrO6] linkages were observed with increasing Zr content. Due to the small amount of [ZrO6]-[ZrO6] connections (less than 0.4%) among all the linkages between network formers, no obvious clustering behavior was considered. Snapshots of the simulated 4Zr glass are presented in Fig. 3.

Fig. 3
figure 3

Snapshots of simulated 4Zr: Zr, blue octahedra; Si, yellow tetrahedra; B, green triangles ([3]B) and green tetrahedral ([4]B); Al, red tetrahedral; O, small grey balls; Na, purple crosses; Ca, green crosses

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Ring-size distributions of simulated 0Zr, 2Zr, 3Zr, and 4Zr glasses are shown in Fig. 4. In the analysis, all the network former cations Si4+, B3+, Al3+, and Zr4+ were counted in the ring size analysis. Ring-size distributions of all glass compositions exhibit a Gaussian type of distribution but slightly tilted toward larger sized rings. The peak is located at seven-membered rings, indicating a well-connected three-dimensional random network. Average ring size for 0Zr, 2Zr, 3Zr, and 4Zr glasses is 7.54 ± 0.07, 7.42 ± 0.02, 7.32 ± 0.02 and 7.28 ± 0.02, respectively. With addition of ZrO2, the peak position of ring size distribution shifts toward smaller ring sizes and peak intensity increases, suggesting that a tightened glass network structure after ZrO2/SiO2 substitution.

Fig. 4
figure 4

Ring-size distributions of simulated 0Zr, 2Zr, 3Zr, and 4Zr glasses

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Boron speciation and environments

Comparison of 4-coordinated B (N4) with different system sizes (3000, 6000, 12,000, 24,000 atoms) for simulated 0Zr, 2Zr, 3Zr, and 4Zr glasses are listed in Table 2. N4 values are consistent between various system sizes, where larger system sizes provide smaller standard deviations. N4 values are in excellent agreement (less than 4% difference) with YDB model. N4 of ISG (~60%) is slightly higher than NMR results (~52% or ~58%) from literature.14,37 Detailed CNs of Si, B, Al, and Zr around B in simulated 0Zr, 2Zr, 3Zr, and 4Zr glasses can be found in supplementary information.

From direct observation of glass structure, linkages of single [3]B/[4]B surrounded by Si, Al or Zr were found and some representatives are shown in Fig. 5. Due to the different hydrolysis rate and reaction energy of various linkages in the glasses (e.g., Si–O–Si, Si–O–[4]B. [3]B–O–[3]B, etc.), these features might have an impact on the dissolution of glass and the small amount of retention of boron in alteration layers observed experimentally.37 Detailed discussion can be found in Discussion section.

Fig. 5
figure 5

Snapshots of simulated 4Zr glass showing different linkages of single [BO3]/[BO4] surrounded by Si, Al or Zr polyhedra. (Top row: [BO3]/[BO4] surrounded by [SiO4], second row: [BO3]/[BO4] surrounded by [SiO4] and [ZrO6], third row: [BO3]/[BO4] surrounded by [SiO4] and [AlO4], bottom row: [BO3]/[BO4] surrounded by [SiO4], [AlO4], and [ZrO6].) Zr, blue octahedra; Si, yellow tetrahedra; B, green triangles ([3]B) and green tetrahedral ([4]B); Al, red tetrahedral; O, small grey balls; Na, purple crosses; Ca, green crosses

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Zirconium environment and EXAFS calculations

The calculated Zr K-edge EXAFS spectrum for 2Zr (ISG) and its Fourier transform are compared to the room-temperature experimental data of Bouty et al.57 in Fig. 6. The calculated spectra for 3Zr and 4Zr glasses were identical to that of 2Zr, as would be expected based on the structural analysis presented above. The agreement is good in terms of both the positions and amplitudes of the spectral features. As a result, a very good match to experiment is obtained for the FT magnitude of the first coordination shell. The peak at 0.86 Å in the FT of the experimental spectrum is assigned to noise as it was not present in the calculations or in the experimental Zr K-edge EXAFS spectra of other glasses with similar compositions,58 was not included by Bouty et al.57 in their fits, and does not correspond to any physically possible Zr−O distance.

Fig. 6
figure 6

Experimental and calculated (ΔE0 = 10 eV) Zr K-edge EXAFS spectra (left) and corresponding Fourier transform magnitudes (right) for Zr in ISG (300 K) or Z4C4 (35 K)

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Although the FT magnitude of the peaks between approximately 2.5 and 4 Å and corresponding to the second coordination shell is much lower than that of the first shell, the calculated peak positions and magnitudes are similar to those obtained from the FT of the experimental spectrum. Jollivet et al.58 reported Zr K-edge EXAFS measurements for a series of four borosilicate glasses with varying amounts of CaO and ZrO2 oxide components. The four glasses showed very similar spectra. The spectrum obtained for the glass labeled Z4C4 (56.7 mol% SiO2, 17.9 mol% B2O3, 18.0 mol% Na2O, 3.7 mol% CaO, 3.9 mol% ZrO2), which had a similar composition to ISG, is also shown in Fig. 6. This spectrum is comparable to the ISG spectra, albeit with greater amplitudes as the measurements were made at 35 K to reduce the dampening effect of thermal disorder. The FT magnitude due to the second shell was therefore higher and its good agreement with those of the ISG spectra confirmed that the second shell peaks were not simply the result of truncation ripples in the FT.

Dynamic properties

The mean square displacements (MSDs) were calculated from NVE trajectories to analyze diffusion behaviors of ions in simulated glasses, and plots of MSDs of Na and Al from 2Zr glass (ISG) at different temperatures are shown in Fig. 7. MSD plot can be classified into three regions: a ballistic region for short time (proportional to t2), a plateau region and a diffusion region for long time (proportional to t).28,51,59 The last diffusion region (80–100 ps) was used to calculate the self-diffusion coefficients based on Einstein’s equation. In Fig. 7, diffusion regions of Na in both high temperature range (2500–3700 K) and low temperature range (940–1750 K) are proportional to t, while MSD of Al for long time in low temperature range is still the plateau region. Therefore, no diffusion behavior was considered for Al in low temperature range and similar behaviors were observed for Si, B, O, Zr, and Ca. Diffusion coefficients of B, Ca, Na, Al, Zr, and Si in 2Zr (ISG) glass calculated at each temperature are plotted in Fig. 8.

Fig. 7
figure 7

Mean square displacements (MSDs) of Na (a) and Al (b) in the simulated 2Zr glass (ISG) at different temperatures

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Fig. 8
figure 8

Self-diffusion coefficients of B, Ca, Na, Al, Zr, and Si in 2Zr glass (ISG) at different temperatures

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The activation energy barriers of ions in simulated glasses in different temperature ranges calculated based on Arrhenius relationship are listed in Table 5. B, Si, Ca, Al, and Zr ions are considered immobile in low temperature range. Interestingly, Ea of Na in 2Zr (ISG) has a maximum value of 0.50 eV among all glasses in low temperature range (940–1450 K), while in high temperature range (2500–3700 K), Ea of Na has a minimum value of 0.93 eV. B, Si, and Al in 3Zr glass have a maximum value of 1.76, 1.98, and 1.80 eV among all compositions in high temperature, respectively. Ea of Ca ions increases from 1.29 to 1.38 eV, while Ea of Zr ions decreases from 1.92 to 1.83 eV with increasing ZrO2/SiO2 substitution; however, the standard deviations make the trend unclear. It was found experimentally that 22Na self-diffusion activation energy ranges between 76 to 96 kJ/mol (0.79–1.0 eV, respectively) at 250 to 500 °C in sodium borosilicate glasses depending on the compositions.52

Table 5 Activation energy of self-diffusion of ions in 0Zr, 2Zr, 3Zr, and 4Zr
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Discussion

Bond lengths of Si–O, [3]B–O, [4]B–O, Al–O, and Zr–O are 1.61, 1.46, 1.55, 1.78, and 2.10 Å, respectively, which are all in good agreement with experimental values. There is no noticeable change on bond distance of each cation–oxygen pair between compositions. N4 of boron for ISG is ~60% from MD simulation and ~52%37 or ~58%14 from NMR. Qn of Si for ISG obtained from NMR has the highest percentage of Q3 with some of Q2 and Q4,6 which is lower than MD results (~85% of Q4) when Si, Al, B, and Zr are all considered as formers. On the other hand, if Si is considered as the only network former, Q2 (~35%) and Q3 (~31%) species are dominant in the glass structure. This observation is consistent with Raman spectra study by Mendoza et al,16 where Q4(2T) species ([SiO4] connects with two [SiO4] and two other network-former polyhedra) contributed to the strongest peak in the Qn band between 850 to 1200 cm-1. However, Qn of Si from NMR is still higher as compared to Qn from MD when only Si is considered as network former. Better deconvolution of 29Si NMR spectrum could improve the comparison with MD results, for the benefit of glass structural interpretation and potential development for MD simulation.

Bond angles of O–Si–O and O–Al–O are at ~108° and ~107° for all compositions, respectively, indicating both Si and Al are in tetrahedral sites. BAD of O–B–O has a peak and a shoulder contributed by O–[3]B--O and O–[4]B–O at 119° and 109°, respectively. BAD of O–Zr–O has a highest peak at ~88° and a secondary peak at ~170°, suggesting a slightly distorted octahedral site. This slightly distorted [ZrO6] octahedral are consistent with experimental observations.44,45,46 BAD of O-Na-O shows a higher/sharper peak at ~60° and a lower/broader peak at ~110° for all compositions, which is consistent with a previous study on soda-lime silicate glasses.53 Interestingly, BADs of O–Ca–O exhibit higher intensity of the peak ~60° as compared to the peak ~90°. This feature is distinctive with previous observations on the glass systems without boron content, where the peak at ~90° of BAD of O-Ca-O has a higher intensity.34 According to a study by Xiang and Du,34 the bonds between BO and Na/Ca (BO-Na/Ca-BO) are responsible for the peak at ~ 60°, where Na/Ca ions are coordinated to the oxygen ions belonging to the same tetrahedron. The bonds between NBO and Na/Ca (NBO-Na/Ca-NBO) are responsible for the peak at higher degree. This observation suggests that Ca ions act more as charge compensators instead of breaking BO in these studied glasses. In this case, less ion exchange sites are available when glasses encounter water molecules, according to a study by Bunker et al.52 where modifiers connected to NBO have a higher tendency to involve in glass dissolution.

Due to the higher cation field strength of Ca2+ as compared to Na+, it will be intriguing to investigate their role as charge compensators to network former groups such as [BO4]-, [AlO4]- and [ZrO6]2-. CNs of former-Na pairs are always higher than the CNs of former-Ca pairs. For instance, ~80% of charge of [ZrO6]2- octahedra are compensated by Na from the comparison of CNs between Zr–Na and Zr-Ca pairs, which is consistent with NMR results.42,54 Raman spectroscopy was also used to study the effect of charge compensation of Ca and Na in alumionsilicate glasses.55 If we take into consideration that there are higher concentration of Na2O (12.7 mol%) than CaO (5.7 mol%) in each glass composition, Zr actually has slightly higher preference for Ca as reflected in the preferential distribution analysis (Table 3). This was also supported by higher intensity of Zr–Ca pair distribution as compared to that of Zr-Na pair distribution.37 This preferential behavior of Ca around Zr was also proposed earlier by Calas et al.45 Similar situation was observed for [4]B and Al. On the other hand, Si prefers Na over Ca even after elimination of compositional effect.

The study of Zr in both simplified and simulant borosilicate glasses using EXAFS44,57,60,61 dates back to the late 1990s and has consistently shown that Zr is in 6-fold coordination, with evidence for 7-fold coordinated Zr only appearing at very high ZrO2 contents.61 In contrast, the composition of the second shell has not been unequivocally resolved from EXAFS due in large part to the large number of degrees of freedom (elements, distances, coordination numbers, Debye–Waller factors) in the traditional shell-by-shell EXAFS fitting approach. Early attempts to fit the second shell either failed61 or had to assume its composition.44 More recently, high-resolution data obtained by Jollivet et al.58 suggested that the second shell was composed of 4 Si and 2 B with 2.4 Na as the charge compensating species, but other fits were statistically equivalent. Similar results were obtained for ISG by Bouty et al.,57 who used empirical potential structural refinement to constrain their fits, but with a greater amount of Na (3.1) as the charge compensating species.

The good agreement between the calculated and experimental Zr K-edge EXAFS spectra not only serves to validate the model employed in this work but also helps resolve the composition of the second shell. The MD simulations show that the second coordination shell around Zr is composed on average of 3.6 Si, 1.9 B, and 0.4 Al with 3.4 Na and 0.9 Ca as the charge compensating species. Because this composition results from MD simulations that are independent of the EXAFS measurements and is therefore not a fit, it does not suffer from the issue encountered in previous work with non-unique solutions.

Network connectivity was successfully used to predict degradation and bioactivity of glasses containing mainly single network former, where the higher NC leads to lower dissolution and bioactivity.62,63 However, complications exist for prediction of dissolution for glass systems with multiple network formers such as ISG. Although overall NC from MD increases from 3.76 to 3.83 (by 1.86%) for 0Zr to 4Zr, this solely could not explain the dramatic decrease (higher than 50%) of initial dissolution rate after addition of 1 mol% ZrO2 from a study by Cailleteau et al.43 Due to different hydrolysis rates64 and reaction energies65 of different linkages between network formers, many factors such as [3]B to [4]B ratio and polyhedra linkages impact the glass dissolution under different environments. Therefore, systematical and statistical analyses from MD provide a better prediction of glass dissolution from compositional and structural information. For instance, hydrolysis rate of [4]B bonds is smaller than [3]B bonds according to a study by Yu and Edén.64 From our simulation results, even though overall CN of B–Si is decreasing due to the substitution of ZrO2/SiO2, CN of [4]B–Si decreases slower as compared to [3]B–Si, resulting in less hydrolysis sites available during glass dissolution. Future systematically study on the reaction energies of different linkages under various environments would greatly benefit the prediction of glass dissolution in combination with MD structural study.

In contrast to the generally accepted assumption that boron is not retained at all in alteration layers, Gin et al.37 detected small amounts of boron (~0.4 at%) in the “gel” layer. From the direct observation of glass structure generated by MD simulation, [3]B/[4]B surrounded by Si, Al or Zr without connecting to another [3]B/[4]B, as shown in Fig. 5, might contribute to this experimental observation. Further statistical analysis is needed to validity this hypothesis.

From our diffusion analysis, two distinctive trends are observed for Na ions between high and low temperature range, which are divided by glass transition temperature (Tg). Note that Tg obtained from MD simulation is generally higher than the experimental values owing to the faster cooling rate used in simulation. From a recent study on cooling rate effect, glass transition temperature decreases with decreasing cooling rate.66 After fitting the Tg versus cooling rate curve using Vogel–Fulcher dependence of relaxation time, it was found that extrapolation to the experimental cooling rate from the MD simulations can give a reasonable prediction of the Tg measured experimentally.66 Ea of Na in 2Zr (ISG) has a maximum value of 0.50 eV among all glasses in low temperature range, while Ea has a minimum value of 0.93 eV in high temperature range. From a study by Bunker et al.,52 the results showed that self-diffusion has no effect on the leaching of Na. In high temperature range, B, Si, and Al in 3Zr glass have a maximum value of 1.76, 1.98 and 1.80 eV among all compositions, respectively. Further investigations are required for whether mixed-former effect exists and the relationship between self-diffusion and dissolution behavior.

Radiation would modify nuclear waste glass structure and properties. It has been shown experimentally and from simulations that radiation induces swelling of the samples and change of mechanical properties of nuclear waste glasses.67 These modifications of macroscopic properties are originated from changes of medium range glass structures as a result of increasing disorder and increase of internal energy due to irradiation.67 These structural changes would also impact the reaction of glass water and dissolution behaviors that demand further simulation and experimental studies.

In summary, ZrO2/SiO2 substitution increases the glass network connectivity due to the 6-coordinated [ZrO6] octahedra, which is consistent with the experimental observations on increased glass transition temperature and lowered thermal expansion coefficient. Basic glass structural features such as bond lengths of cation–oxygen and cation–cation pairs, and BAD of oxygen–cation–oxygen are not sensitive to ZrO2/SiO2 substitution. Thus, the polyhedra linkages and glass modifiers seem to have great impacts on the initial dissolution rate here. For a constant CaO and Na2O content, ZrO2/SiO2 substitution leads to a greater proportion of Ca2+ and Na+ ions serving as charge compensating species, resulting in fewer modifiers available to create NBOs. Modifiers act as charge compensators have a lower tendency to be involved in ion-exchange reaction compared to the modifiers connected with NBOs. Fewer ion-exchange and hydrolysis sites are therefore available during glass dissolution as a result of ZrO2/SiO2 substitution. Importantly, ring size analysis revealed that the addition of ZrO2 tightens the glass network by forming smaller rings, which restrain the water access to both ion-exchange and hydrolysis sites. The calculated EXAFS spectra provide more information than traditional shell-by-shell fitting, which opens the door to characterization of local structural environment around Zr and other glass components.

Methodology

Glass compositions and MD simulation details

Compositions studied and density calculated by a method developed by Fluegel68 are listed in Table 6, where 0Zr, 2Zr (ISG), 3Zr, and 4Zr represent different molar percentages of ZrO2 substitution of SiO2 in each glass composition.

Table 6 Composition and density of glasses studied
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The interatomic interactions are described by the Born model of solids using partial charge pairwise potentials and the covalent character of the bond (e.g. Si–O) is described by the partial charge on the ions. The interatomic potential energy \({\boldsymbol{\Phi }}\left( {{\boldsymbol{r}}_{{\boldsymbol{ij}}}} \right)\) consists of long-range Columbic interactions and short-range interactions in the Buckingham form:

$${\mathrm{\Phi }}\left( {r_{ij}} \right) = \frac{{q_iq_j}}{{r_{ij}}} + A_{ij}e^{\frac{{ - r_{ij}}}{{\rho _{ij}}}} - \frac{{C_{ij}}}{{r_{ij}^6}}$$
(2)

where r ij is the interatomic distance between atom i and j; q i and q j are the effective charge for atom i and j, respectively, A ij , ρ ij , and C ij are the parameters for the Buckingham term. Detailed description of the potential for borosilicate can be found in our previous study.36 Atomic charges and Buckingham potential parameters used in this work are shown in Table 7. All simulation cells were in cubic shape. Total numbers of atoms in the simulation cells for structural analysis and diffusion analysis were ~24000 and ~12000, respectively.

Table 7 Atomic charges and Buckingham potential parameters
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MD simulations were performed with DL_POLY 2.20 program developed at Daresbury Laboratory in the UK.69 Three random configurations for each composition were generated and simulated through a melt-quench process. The initial atom positions were generated randomly with calculated density in a cubic simulation box. After initial relaxation at 300 K under zero pressure, the glasses were melted at 6000 K and the temperature was gradually reduced with a cooling rate of ~1.4 K/ps. At each temperature, the system was run under canonical ensemble (constant number, volume and temperature (NVT)) for 200 picosecond (ps), which was followed by equilibration under microcanonical ensemble (constant number, volume, and energy (NVE)) for another 200 ps. At 300 K, the trajectory was recorded every 50 steps of the last 40,000 steps under NVE ensemble for final structural analysis. The cutoff distance used for the short-range interactions was 8 Å, and 10 Å for the real-space sum of the electrostatic interactions, which were calculated using the Ewald sum method with a relative precision of 1 × 10−6. Integration of motion equations was carried out using the Verlet Leapfrog algorithm with a time step of 1 femtosecond (fs).

Structural and dynamical analyses

All the cutoff distances of first/second coordination shell for coordination calculations and BAD analysis are presented in Table 8. The cutoff values were obtained as the first minimum from partial correlation functions.

Table 8 Cutoff distances (Å) of first/second coordination shell for coordination calculations and bond angle distribution analysis in the simulated glasses
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In the network structure analysis, SiO2, B2O3, Al2O3, and ZrO2 are all considered as glass network formers. Network connectivity (NC) was calculated based on the Qn distribution of Si4+, B3+, Al3+, and Zr4+ species analyzed from the structures calculated from MD simulations through the following equation (3):

$$NC = {\textstyle{{\mathop {\sum }\nolimits_{n = 0}^4 n \times Q_n^{Si} \times \left[ {SiO_2} \right] + \mathop {\sum }\nolimits_{n = 0}^4 n \times Q_n^B \times \left[ {BO_{3/2}} \right] + \mathop {\sum }\nolimits_{n = 0}^4 n \times Q_n^{Al} \times \left[ {AlO_{3/2}} \right] + \mathop {\sum }\nolimits_{n = 0}^6 n \times Q_n^{Zr} \times \left[ {ZrO_2} \right]} \over {\left[ {SiO_2} \right] + \left[ {BO_{3/2}} \right] + \left[ {AlO_{3/2}} \right] + \left[ {ZrO_2} \right]}}}$$
(3)

where n is the number of the bridging-oxygen atoms per unit and \(Q_n^{Si}\) is the fraction of Si atoms in Si–O–X linkages where X is one of the four network former cations, \(\mathop {\sum }\limits_{n = 0}^4 n \times Q_n^{Si}\) is then the network connectivity of Si (NCSi) and [SiO2] is the mol% of SiO2 in the glass composition, as well as for B, Al, and Zr.

The glass structures generated from the melt-quench process mentioned above were used for diffusion calculations. Six random configurations for each composition were simulated in order to obtain a reasonable standard deviation. The glasses were gradually heated from 300 to 3700 K with intervals of 800, 940, 1040, 1140, 1240, 1340, 1450, 1750, 2000, 2500, 2700, 3000, and 3300 K. At each temperature, the glasses were first equilibrated for 60 ps under NVT ensemble and then run under NVE for 200 ps, within which the trajectory was recorded in the last 180 ps every 10 fs for diffusion analysis. The MSDs were calculated from these trajectories, based on which the self-diffusion coefficients were obtained using Einstein’s equation,

$$D = \frac{1}{6}\mathop {{\lim }}\limits_{t \to \infty } \frac{{ < r^2 > }}{t}$$
(4)

where <r2>is MSD and t is time. The diffusion activation energy barriers were then calculated from diffusion coefficient at different temperatures by taking the logarithm on Arrhenius equation,

$$D = D_0exp(\frac{{ - E_a}}{{RT}})$$
(5)

where Ea is the diffusion energy barrier, T is the temperature and R is the gas constant. Details of calculations can be found in a study by Ren and Du.59

EXAFS spectra calculations

A configuration was collected every 0.5 ps for the last 11 ps of the 300 K NVE production run. For each configuration, a cluster with a radius of 10 Å centered around each of the Zr ions in the configuration was excised to calculate all the scattering paths with effective distances less than the cluster radius for a Zr K-shell core hole using FEFF9.70,71,72 The S02 parameter calculated by FEFF9 was 0.928 and was used in all calculations. The individual EXAFS spectra thus calculated (1364, 2376, 3168 for 2Zr, 3Zr, and 4Zr, respectively) were averaged for comparison with experimental spectra. The Fourier transform (FT) was applied to the averaged EXAFS spectra using IFEFFIT73 with k3 weight in the range 2.6 ≤ k ≤ 12.2 Å−1 and truncated using a Hanning window with dk = 1.0 Å−1 to match the FT parameters used by Bouty et al.57

Data availability

The authors declare that all data supporting the findings of this study are available within the paper and its supplementary information files.