Sub-Nanometer-Range Structural Effects From Mg²⁺ Incorporation in Na-Based Borosilicate Glasses Revealed by Heteronuclear NMR and MD Simulations

Abstract

Magic-angle-spinning (MAS) nuclear magnetic resonance (NMR) experiments and molecular dynamics (MD) simulations were employed to investigate Na₂O–B₂O₃–SiO₂ and MgO–Na₂O–B₂O₃–SiO₂ glass structures up to ≈0.3 nm. This encompassed the {Na^[p]}, {Mg^[p]}, and {B^[3], B^[4]} speciations and the {Si, B^[p], M^[p]}–BO and {Si, B^[p], M^[p]}–NBO interatomic distances to the bridging oxygen (BO) and nonbridging oxygen (NBO) species, where the superscript indicates the coordination number. The MD simulations revealed the dominance of Mg^[5] coordinations, as mirrored in average Mg²⁺ coordination numbers in the 5.2–5.5 range, which are slightly lower than those of the larger Na⁺ cation but with a narrower coordination distribution stemming from the higher cation field strength (CFS) of the smaller divalent Mg²⁺ ion. We particularly aimed to elucidate such Na⁺/Mg²⁺ CFS effects, which primarily govern the short-range structure but also the borosilicate (BS) glass network order, where both MD simulations and heteronuclear double-resonance ¹¹B/²⁹Si NMR experiments revealed a reduction of B^[4]–O–Si linkages relative to B^[3]–O–Si upon Mg²⁺-for-Na⁺ substitution. These effects were quantified and discussed in relation to previous literature on BS glasses, encompassing the implications for simplified structural models and descriptions thereof.

1. Introduction

Incorporating highly charged and/or small electropositive cations in oxide glasses, in particular rare-earth (RE³⁺) ions, often improves their thermal and mechanical properties.¹⁻⁸ However, the high cost and toxicity of RE³⁺ species make Mg²⁺ an inexpensive and environmental-friendly alternative for glasses in a sustainable future society, as the cation field strength (CFS) of the small Mg²⁺ cation is only marginally lower than that of La³⁺. (The CFS scales as the charge divided by the square of the cation radius).⁹ There has been a decades-long and recently growing interest in exploring the structures of Mg²⁺-bearing aluminosilicate (AS) glasses¹⁰⁻¹⁵ and their correlation with physical properties.¹⁶⁻¹⁹ In contrast, there are comparatively few studies of incorporating Mg²⁺ into the other main glass family of ubiquitous technological and geological importance, i.e., borosilicate (BS) glasses.¹⁹⁻²³ It also remains unclear as to whether the physical-property boosts observed for Mg²⁺-bearing AS glasses (relative to those with lower-CFS alkali/alkaline earth metal ions) also apply to the BS glass context. Results from MO–Na₂O–B₂O₃–SiO₂ glasses merely suggest that the larger Sr²⁺ and Ba²⁺ cations offer higher hardness and better elastic properties than their Mg²⁺ counterpart.²²

These differences may be traced to the distinct effects from high-CFS Mg²⁺ and RE³⁺ incorporation in AS versus BS structures. While the physical glass properties are enhanced slightly from stronger Mg²⁺/RE³⁺–O bonds relative to those of lower-CFS M⁺/M²⁺ glass-network modifiers, all M–O bonds remain markedly weaker than their F–O counterparts, where F denotes a network forming species, F = {Si, Al, B}. Rather, the structure-strengthening effects and accompanying improved thermal/mechanical properties upon high-CFS M^z+ inclusion in AS glasses stem from the effects on the Al speciation, where the dominant AlO₄ groups partially convert into higher-coordination yet network-forming AlO₅ and AlO₆ polyhedra,^24,25 whose higher network cross-linking enhances the physical glass properties.^5,7,25,26 Including high-CFS M^z+ cations in BS glasses, on the other hand, strongly alters their B speciations. B-bearing glass networks comprise both BO₃ (B^[3] coordinations) and BO₄ (B^[4]) groups,^24,27 where large B^[4] populations increase the network connectivity and boost many glass properties, such as the hardness and the glass transition temperature.^{22,26,28−31} However, high-CFS cations tend to promote B^[3] formation at the expense of B^[4],^{19,22,32−37} which lowers the overall glass-network connectivity and may thereby degrade physical properties. Yet, little is known about the precise structure–property relationships of high-CFS M^z+-bearing BS glasses, which are strongly dependent on the precise glass stoichiometry and likely also on the M^z+ size (alone);²² for example, incorporating Mg²⁺ into a few Na₂O–B₂O₃–SiO₂ glass compositions reduced their hardness and elastic properties, whereas these properties improved when instead introducing the larger high-CFS La³⁺ cation.²²

By utilizing atomistic molecular dynamics (MD) simulations and magic-angle-spinning (MAS) nuclear magnetic resonance (NMR) spectroscopy, we report here on the structural alterations from Mg²⁺-for-Na⁺ substitutions in four Na₂O–B₂O₃–SiO₂ base-glass compositions. Besides the {B^[p]} and {O^[q]} speciations, we examine the local Na and Mg coordination environments and their distinctly different propensities for coordinating the bridging oxygen (BO; O^[2] coordinations) and nonbridging oxygen (NBO; O^[1]) species. We particularly aim to understand the dependence of the local glass structure on the M^z+ CFS, which besides Na⁺ and Mg²⁺ also involves results from Ca²⁺-bearing BS glass models along with previously published data from RE³⁺ cations in AS glasses.

We then move the spotlight from the first F^[p] and M^[p] coordination shells onto the F–O–F′ glass-network linkages, where the silicate and borate group intermixing was probed by computational modeling and heteronuclear double-resonance ¹¹B/²⁹Si MAS NMR experiments. Current B/Si interconnectivity insights partially stem from ¹⁷O triple-quantum MAS (3QMAS)³⁸ experimentation.^{24,36,37,39,40} Although widely applied, it may not unambiguously discriminate between the B^[3] and B^[4] coordination numbers of the B–O–Si and B–O–B linkages (let alone quantify them, although such claims have been made^36,37,40). The heteronuclear magnetic ¹¹B–²⁹Si dipolar interaction, which is mediated directly through space and scales as the inverse cube of the ¹¹B–²⁹Si distance,^24,25,41,42 offers a more direct probing of the ¹¹B^[p]/²⁹Si intermixing in BS glasses. However, its application is relatively sparse,⁴³⁻⁴⁹ mainly stemming from the requirement of dedicated glass syntheses from costly ²⁹Si-enriched silica to enable high-quality experimental data for quantitative analyses, which is otherwise severely hampered or even precluded by the low natural abundance (4.7%) of the NMR-active ²⁹Si isotope. The impact of Mg²⁺ incorporation on the relative degrees of B^[3]–O–Si and B^[4]–O–Si bonding in the Mg/Na-bearing BS glass networks is discussed in relation to current literature, as well as the implications for existing simplified BS-glass structure descriptions/models.

2. Materials and Methods

2.1. Borosilicate Glasses

Our study involved the eight BS glass compositions listed in Table 1, encompassing four ternary RNa₂O–B₂O₃–KSiO₂ glasses and four quaternary R[0.5MgO–0.5Na₂O]–B₂O₃–KSiO₂ analogs. They constitute a subset of a large BS glass ensemble examined previously.^22,50,51 We adopt the glass nomenclature of ref (50), where each Na- and Mg/Na-based glass is denoted by NaK–R and MgNaK–R, respectively, with the {K, R} parameters defined by^52,53

1

2

Each nominal glass composition in Table 1 is expressed in terms of its oxide equivalents and the atomic fraction of each element E in the (Mg)–Na–B–Si–O glass,

3

where n_E is the corresponding stoichiometric amount.

Table 1. Borosilicate Glass Compositions^a.

	oxide equivalents (mol %)				atomic fractions					BO4 fractionsb		NBO fractionsc
glass	MgO	Na2O	B2O3	SiO2	xMg	xNa	xB	xSi	xO	NMR	MD	NMR	MD
R = 0.75
Na2.0–0.75		20.0	26.7	53.3		0.114	0.151	0.151	0.584	0.604	0.547	0.038	0.054
MgNa2.0–0.75	10.0	10.0	26.7	53.3	0.029	0.058	0.155	0.155	0.603	0.361	0.353	0.100	0.104
Na4.0–0.75d		13.0	17.4	69.6		0.078	0.104	0.208	0.610	0.623; 0.618	0.527	0.022; 0.022	0.039
MgNa4.0–0.75d	6.5	6.5	17.4	69.6	0.020	0.040	0.106	0.212	0.622	0.313; 0.310	0.301	0.074; 0.074	0.077
R = 2.1
Na2.0–2.1		41.2	19.6	39.2		0.242	0.116	0.116	0.526	0.487	0.447	0.354	0.364
MgNa2.0–2.1	20.6	20.6	19.6	39.2	0.065	0.129	0.123	0.123	0.560	0.402	0.364	0.373	0.382
Na4.0–2.1		29.6	14.1	56.3		0.180	0.086	0.172	0.562	0.697	0.608	0.214	0.228
MgNa4.0–2.1d	14.8	14.8	14.1	56.3	0.094	0.047	0.090	0.180	0.589	0.459;0.445	0.444	0.249;0.251	0.253

^aNominal ternary RNa₂O–B₂O₃–KSiO₂ (“NaK–R”) or quaternary R[0.5MgO–0.5Na₂O]–B₂O₃–KSiO₂ (“MgNaK–R”) glass compositions and their corresponding atomic fractions {x_Mg, x_Na, x_B, x_Si, x_O} defined by x_E = n_E/(n_Mg + n_Na + n_B + n_Si + n_O).

^bFractional populations of B^[4] coordinations, , as obtained either by ¹¹B NMR or by MD simulations, with the respective uncertainties of ±0.01 and ±0.005. The experimental values listed to the left are reproduced from Lv et al.,²² while those to the right (in bold) are results from the present ²⁹Si-enriched glasses; the latter data are employed throughout the experimental analyses of those glass specimens.

^cNBO fraction, x_NBO, out of all BO and NBO species, as either obtained by MD simulations (uncertainty ±0.001) or calculated from eq 9 (±0.01) by using the NMR-derived {} data. The experimental x_NBO values listed to the left and right (in bold) correspond to the results presented by Lv et al.²² and those of the present ²⁹Si-enriched glasses, respectively; the latter data are assumed throughout this work.

^dPrepared with ²⁹Si isotopic enrichment (section 2.2).

MD simulations were performed for all glass compositions in Table 1. The need for dedicated glasses synthesized from costly ²⁹SiO₂ to enable the heteronuclear ¹¹B/²⁹Si NMR experiments, however, limited them to three specimens: Na4.0–0.75, MgNa4.0–0.75, and MgNa4.0–2.1.

The CFS of an M^z+ cation is defined according to⁹

4

where r_M is the cation radius and r_O = 1.36 Å. Table S1 lists the CFS values of the F = {Si, B^[3], B^[4]} network formers and the M^z+ = {Na⁺, Mg²⁺} network modifiers primarily targeted here, along with a few other M^z+ cations discussed in section 3. For consistency, we employed r_M values for sixfold-coordinated M^z+ species (M^[6]) throughout. The coordination number is not indicated for exclusively tetrahedrally coordinated Si ≡ Si^[4] atoms.

2.2. Glass Preparation

The ²⁹Si-enriched Na4.0–0.75, MgNa4.0–0.75, and MgNa4.0–2.1 glasses were prepared in 250 mg batches from analytical grade ²⁹SiO₂ (99.8% ²⁹Si), H₃BO₃, Na₂CO₃, and MgO precursors. After removing potential OH/H₂O contaminations by preheating the ²⁹SiO₂ powder at 950 °C for 24 h, the precursors were mixed thoroughly in a mortar, transferred to a Pt crucible, and decarbonated at 950 °C for 2 h before being heated to final melt temperatures of 1000, 1100, and 1200 °C for the MgNa4.0–2.1, Na4.0–0.75, and MgNa4.0–0.75 batches, respectively. Those temperatures were deduced from multiple preparations using regular (nonenriched) SiO₂ so as to ensure complete melting but minimal evaporation losses and close fictive temperatures to previous BS glass specimens of identical nominal compositions but prepared in larger batches (6 g) at ≈200 °C higher melting temperatures.²² The melt was held for 20 min and then quenched by immersing the crucible bottom in cold water.

All glass specimens were free of crystalline impurities. Their compositions are expected to be close to their batched/nominal counterparts listed in Table 1, as corroborated by the minute evaporation losses during heating of 1.0 wt % and 1.6 wt % for MgNa4.0–2.1 and MgNa4.0–0.75, respectively. Although the Na4.0–0.75 glass revealed a markedly higher loss (5.6% wt %) than the 1–2 wt % we normally observe,^22,34 it mainly reflects accidental melt-loss prior to quenching. Indeed, relative to the batched B₂O₃ contents, the B₂O₃ masses determined from ¹¹B MAS NMR experiments calibrated to H₃BO₃ as the standard (see refs (22) amd (34)) revealed marginal relative deviations from 0.6% for Na4.0–0.75 to 1.7% for MgNa4.0–0.75.

2.3. Molecular Dynamics Simulations

Atomistic MD simulations mimicking a melt-quench process were utilized to produce BS glass models with the stoichiometries from Table 1. The computations utilized the DLPOLY4.09 program,⁵⁴ where NVT ensembles in a cubic box with periodic boundary conditions were simulated using a box size and number of atoms (6600–11600) to match the nominal chemical glass composition and the experimental density (Table S2). Each melt-quench protocol started from randomly positioned atoms, which were equilibrated for 100 ps at 3500 K, followed by a stepwise temperature reduction (5 K/ps) to 300 K. The equations of motion were integrated in steps of 0.2 fs by using the velocity Verlet integrator, while the temperature was controlled by a gentle stochastic thermostat with a 1.0 ps time constant and a 1.0 ps^–1 Langevin friction constant. The structural data were sampled and averaged over the last 150 ps of a final 200 ps equilibration stage. The average value and uncertainty of each reported structural parameter were obtained by performing the melt-quench protocol four times. The selected system sizes, equilibration stages, and other simulation conditions provide well-converged and reliable structural parameters.^34,55−57

All simulations utilized a polarizable shell-model potential.⁵⁸⁻⁶⁰ Every cation carries its full formal charge,⁵⁸ but the O^2– species are represented by core (O_C) and shell (O_S) portions with masses m_C = 15.7994 u and m_S = 0.2000 u, respectively, and corresponding charges z_C = +0.8482e and z_S = −2.8482e (obeying z_C + z_S = −2), where “u” is the atomic mass unit and e is the elementary charge. Each core–shell unit is connected by a harmonic potential with a force constant of 74.92 eV/Å ².⁵⁸ The interaction energy of two atom/ion species α and β separated by a distance r_αβ was modeled by a modified Buckingham potential

5

that accounted for all short-range O_S–O_S and cation–O_S pair interactions. It was evaluated out to r_αβ = 0.8 nm. The B–O force field conforms to eq 5 with C_αβ ≡ 0 but also includes a repulsive term.^55,56Table S3 compiles all parameters, encompassing a three-atom potential for constraining the O–Si–O intratetrahedral angles. Long-range Coulombic interactions were calculated by a smoothed particle-mesh Ewald summation⁵⁴ with a 1.2 nm real-space cutoff and an accuracy of 10^–6.

2.4. Solid-State NMR Experiments

All NMR experiments were performed with Bruker Avance-III spectrometers. The ²⁹Si MAS NMR spectra were acquired at a magnetic field (B₀) of 9.4 T (79.47 MHz ²⁹Si Larmor frequency) using 4 mm zirconia rotors undergoing MAS at ν_r = 14.00 kHz and radio frequency (rf) pulses with a ≈70° flip angle (ν_Si = 84 kHz nutation frequency), relaxation delays of 3600 s, and 16 accumulated NMR signal transients. The ¹¹B (spin-3/2) NMR spectra were recorded at B₀ = 14.1 T (−192.5 MHz ¹¹B Larmor frequency) and ν_r = 24.00 kHz using full 3.2 mm zirconia rotors and strong/short rf pulses (0.33 μs, 13° flip angle, ν_B = 105 kHz). The {B^[3], B^[4]} populations of each glass were determined from the integrated central-transition (CT) NMR-signal intensities,^22,50 which were corrected for the satellite-transition centerband peak that overlaps with the main CT ¹¹B^[4] signal by using standard procedures.⁶¹ Every ¹¹B NMR spectrum was corrected for probehead “background” signals by subtracting the result from the empty rotor recorded under otherwise identical experimental conditions. ²⁹Si (δ_Si) and ¹¹B (δ_B) shifts were referenced relative to neat tetramethylsilane (TMS) and BF₃·OEt₂, respectively.

The Van Vleck dipolar second moment(62) is proportional to the sum over the inverse sixth power of the interatomic distance of each heteronuclear S_j–I_k spin-pair in a structure:

6

Here, μ₀ is the permeability of vacuum, γ_I (γ_S) is the magnetogyric ratio of spin species I (S), and N_I and N_S are the respective total numbers of I and S nuclei in the glass obtained from its stoichiometry with , where N_A is Avogadro’s number and the natural isotopic abundance. I denotes the spin quantum number of the nuclide I, i.e., I = 1/2 (²⁹Si) and I = 3/2 (¹¹B) for the respective M₂(B^[p]–Si) and M₂(Si–B^[p]) entities. We stress the unit of s^–2 in eq 6, which is consistent with our previous work^57,63,64 but differs from the original M₂ definition with units of rad²/s²,⁶² which is most frequently encountered in the literature (e.g., refs (19), (41), and (42)). The two dipolar second-moment definitions are related by 4π²M₂[s^–2] = M₂[rad²/s²]. Units aside, we comment that incorrect M₂(S–I) expressions are stated in two recent review articles,^24,25 which should appear as in ref (64) and eq 6.

The M₂(B^[p]–Si) value of a given BS glass may be estimated from a ¹¹B{²⁹Si} REDOR NMR experiment⁶⁵ that restores/“recouples” the MAS-averaged ¹¹B^[p]–²⁹Si dipolar-interaction effects during a recoupling period (τ_rec) by a series of rotor-synchronized 180° rf pulses applied to ²⁹Si, during which the ¹¹B^[p] NMR signal [S(τ_rec)] becomes attenuated (“dephased”) relative to that obtained by a spin–echo experiment [S₀(τ_rec)]⁶⁵ obtained by CT-selective ¹¹B rf pulses.

All double-resonance ¹¹B{²⁹Si} REDOR NMR experiments were performed at B₀ = 14.1 T and ν_r = 9.00 kHz with each glass powder centered to the 1/3 volume of a 4 mm zirconia rotor to reduce the impact from rf inhomogeneity.^41,66,67 All experiments were started by a saturation-recovery rf-pulse comb followed by a 1.5 s relaxation delay. The 180° dipolar recoupling pulses operated at ν_Si = 46 kHz with the XY8 phase-cycling scheme to minimize rf-pulse errors.⁶⁸ The ¹¹B 90° and 180° spin–echo rf pulses were 17.0 μs and 34.0 μs, respectively. The dipolar recoupling period was sampled out to several ms at even integer multiples n of the rotor period, . Each M₂(B^[3]–Si) and M₂(B^[4]–Si) value was extracted from the respective integrated ¹¹B^[3] and ¹¹B^[4] NMR intensities of the S₀(τ_rec) and S(τ_rec) spectra, whereupon the resulting {τ_rec, ΔS/S₀} data (restricted to ΔS/S₀ ⩽ 0.2^24,41,42) were fitted to the expression^41,42,63

7

2–4 independent NMR-data blocks with 256–512 accumulated signal transients per block were acquired for each glass specimen. The average value of each M₂(B^[p]–Si) estimate and its uncertainty were extracted from these independent {M₂(B^[p]–Si)} best-fit results.

3. Results and Discussion

3.1. Boron and Oxygen Speciations

All BS glasses considered herein comprise networks of interconnected SiO₄, BO₃ and [BO₄]⁻ groups, where and denote the respective fractional populations of B^[3] and B^[4]. Table 1 lists each value obtained from either the ¹¹B MAS NMR spectrum or MD simulations, whereas the corresponding BO₃ fraction is given from the following normalization:

8

The experimental {} data were reproduced from ref (22) except for the three ²⁹Si-enriched Na4.0–0.75, MgNa4.0–0.75, and MgNa4.0–2.1 specimens that were prepared specifically for the present study. Their borate speciations match very well those of glasses prepared from regular (non-²⁹Si-enriched) SiO₂,²² suggesting very close stoichiometries and fictive temperatures for both specimens of each Na4.0–0.75, MgNa4.0–0.75, and MgNa4.0–2.1 stoichiometry (see Table 1 and Figure 1). The MD simulations revealed O speciations solely comprising NBO and BO sites, whereas “free O^2– anions” (O^[0]) are absent throughout and the “O tricluster” (O^[3]) populations remain <0.02% out of each {O^[q]} ensemble. Hence, the fractional populations of NBO (x_NBO) and BO (x_BO) species obey x_NBO + x_BO = 1.

Figure 1.

¹¹B MAS NMR spectra recorded at 14.1 T and 24.00 kHz MAS from (a) Na4.0–0.75, (b) MgNa4.0–0.75, and (c) MgNa4.0–2.1 glass specimens prepared from ²⁹SiO₂ (black traces) or from SiO₂ with ²⁹Si at natural abundance (red traces; reproduced from Lv et al.²²). The glasses were prepared under similar conditions and reveal very similar ¹¹B NMR spectra and borate speciations (Table 1).

The Na⁺/Mg²⁺ cations compensate the negative charges of the [BO₄]⁻ and NBO (O^–) moieties, implying that the fractional populations of B^[4] and NBO anions are coupled according to . Hence, x_NBO is readily deduced from the glass stoichiometry and the ¹¹B NMR-derived value as follows:

9

Equation 9 reflects the well-known dual “network modifier” and “charge compensator” structural properties of the M^z+ cations in borate-based glasses,^{24,27,52,53,69,70} which result in M^z+···O^– and M^z+···[BO₄]⁻ structural motifs from the respective BO → NBO and B^[3] → B^[4] conversions. The borate speciation of a BS-based glass depends strongly on both its stoichiometry and CFS_M.^{19,22,32−37} As expected from the higher CFS_Mg = 0.46 Å^–2 than CFS_Na = 0.18 Å^–2,^19,22,32,33 introducing Mg²⁺ to a NaK–R glass boosts the NBO content for fixed {K, R} parameters, while the B^[4] population is markedly reduced (Table 1). This effect is particularly drastic for the NaK–0.75 and Na4.0–2.1 glasses with : when half of the Na⁺ ensemble is replaced by Mg²⁺, a significant B^[4] → B^[3] conversion occurs, rendering BO₃ groups most abundant in all MgNaK–R specimens.

For a clear-cut discrimination between the M^z+ and F species, we employ the ancient “network modifier” terminology when referring to the M⁺/M²⁺ cations, although the categorical “modifier” and “charge-compensator” classification is oversimplified.^56,57 For instance, both MD simulations and NMR experiments reveal that a significant/major portion of the {Na⁺} ensemble associates with the formally charge-neutral Si(O^[2])₄ and B(O^[2])₃ moieties via Na⁺···O–Si/B^[3] fragments (rather than with the negatively charged [BO₄]⁻ and F–NBO moieties), rendering Na–BO contacts prevalent for all but very NBO-rich glasses.^56,57

A long-standing problem of classical MD simulations of B-bearing glasses is their (in)accurate {, } predictions, which not only depend strongly on the precise glass composition but also on thermal history of the glass due to the temperature-dependent BO₃ ⇄ BO₄ equilibrium.⁷¹⁻⁷³ The engineering of reliable B–O force fields has received significant recent attention.⁷⁴⁻⁸¹ The modeled BO₄ and NBO populations listed in Table 1 were obtained with the B–O interatomic potential parameters of refs (55) and (56), which have been validated for Na- and Na/Ca-bearing borate and boro(phospho)silicate glasses over large composition domains.^34,55,56,82 Contrasting the modeled and experimental BO₄ populations of Table 1, however, reveals a highly variable performance. The agreement is excellent for all MgNaK–R glasses (with relative deviations within 3%) but MgNa2.0–2.1, which along with all NaK–R models reveal markedly larger discrepancies to experiments, typically by ≈10% but with a substantial deviation of 17% for Na4.0–0.75. These errors stem from the distinctly different ranges between the NaK–R and MgNaK–R glasses. As noted in refs (34), (55), and (56) but more clearly shown by Pedone and co-workers,^82,83 MD simulations with our B–O force field are prone to underestimating the BO₄ population: although the deviations are insignificant for glasses with , they grow progressively for increasing values of (Table 1).

The underestimated (overestimated) modeled BO₄ (BO₃) populations are reasons for concern regarding reliable predictions of some medium-range (0.3–1 nm) glass-structure features. Gratifying, however, is the typically (very) good agreement observed consistently relative to experimental data on several interatomic-distance-related structural parameters, such as the relative B^[3]/B^[4]···Na⁺ and P–O–B^[p] contacts in boro(phospho)silicate glasses,^55,57 as well as their B^[p]–O–B^[q] linkage statistics,^34,50,55 which constitute the most sensitive medium-range structural parameters on the precise {, } fractions; notwithstanding that the NMR/MD-derived B^[p]–O–B^[q] populations do differ, all experimental findings were reproduced qualitatively/semiquantitatively by the glass models.^34,50,55 Moreover, the relative B^[3]/B^[4]–O–Si contacts predicted by the glass models in section 3.5 agree very well with our experiments, encompassing the Na4.0–0.75 glass. Indeed, out of the plethora of B–O force fields proposed to date (e.g., refs (74−81)), solely the herein utilized option^55,56 has been assessed extensively against experimental interatomic-distance-related parameters directly reflecting the medium-range glass organization.

3.2. MD-Derived Average F–O and M–O Distances

Here and in sections 3.3 and 3.4, we examine the MD-derived first coordination shells of the network formers and modifiers, paying particular attention to the structural bearings from Mg²⁺, thereby complementing our previous structural reports on large sets of Na⁺ and Na⁺/Ca²⁺ bearing borate and boro(phospho)silicate glasses with variable B, Si, and NBO contents.^56,57

For each {Si, B^[3], B^[4]} network former, Table 2 compiles the average F^[p]–O^[q] interatomic distance, (F^[p]–O^[q]), for each of O^[1] and O^[2], along with the corresponding {(F^[p]–O)} result that represents the weighted average of the F^[p]–O^[1]/O^[2] distances over all {FO_p} polyhedra. We stress that each F^[p]–O/O^[q] and M^[p]–O/O^[q] interatomic distance (“bond-length”) discussed herein constitutes the arithmetic average across the entire F–O^[q] (M–O^[q]) bond ensemble in the structure. was not extracted from the respective pair-distribution maximum, which constitutes the most probable F–O distance but is frequently reported as (F–O) (e.g., see refs (60), (76), and (83−86)). Because the various Si/B^[p]–BO/NBO distances are governed by the high-CFS Si⁴⁺ and B³⁺ cations (Table S1), they remain similar for all BS glasses regardless of the precise network modifier species. All F^[p]–O^[q] distances in Table 2 conform well with those discussed by Stevensson et al.⁵⁶ for Na–(Ca)–B–Si–O glasses, which agreed well with the sparsely available experimental data. The direct dependence of each F–O distance on the NBO content of the glass is evident when contrasting the (Si–O) and (B^[3]–O) data from each NaK–R glass with its MgNaK–R analog (in contrast with (B^[4]–O); vide infra): the distances in the Mg-bearing glass are marginally but consistently shorter by ≈0.3 pm due to the slightly higher NBO contents in those glasses coupled with the shorter F–NBO distances relative to F–BO (Table 2).

Table 2. MD-Derived Average F/M–O Bond Lengths^a.

		(B[3]–O[q]) (pm)			(B[4]–O[q]) (pm)			(Si–O[q]) (pm)			(Na–O[q]) (pm)			(Mg–O[q]) (pm)
glass	xNBO	O	O[2]	O[1]	O	O[2]	O[1]	O	O[2]	O[1]	O	O[2]	O[1]	O	O[2]	O[1]
R = 0.75
Na2.0–0.75	0.054	135.33	135.45	134.00	142.53	142.59	134.69	163.96	164.03	160.29	257.9	260.8	238.4
MgNa2.0–0.75	0.104	135.10	135.29	133.85	143.07	143.31	134.84	163.65	163.73	160.96	258.6	261.0	245.5	213.0	225.8	203.2
Na4.0–0.75	0.039	134.90	134.97	133.88	142.09	142.13	134.71	163.58	163.64	160.09	258.5	261.7	236.4
MgNa4.0–0.75	0.077	134.72	134.83	133.72	142.78	142.99	134.93	163.33	163.40	160.71	258.9	261.9	244.0	211.4	227.7	201.6
R = 2.1
Na2.0–2.1	0.364	136.72	138.37	134.57	144.20	145.43	134.83	164.69	165.46	161.26	252.4	262.5	243.7
MgNa2.0–2.1	0.382	136.26	137.56	134.54	144.60	146.17	135.21	164.49	165.09	161.69	255.9	262.5	248.6	210.3	228.3	206.0
Na4.0–2.1	0.228	136.25	137.20	134.27	143.08	143.68	134.79	164.30	164.78	160.73	254.0	263.1	241.3
MgNa4.0–2.1	0.253	135.85	136.63	134.25	143.66	144.59	135.13	164.02	164.37	161.31	257.3	263.4	246.9	209.6	228.7	204.3
σb	0.001	0.02	0.03	0.02	0.04	0.05	0.08	0.02	0.02	0.04	0.1	0.1	0.2	0.3	0.4	0.1

^aAverage F^[p]–O^[q] and M–O^[q] distances between the NBO (O^[1] coordination) and BO (O^[2]) species for F = {Si^[4], B^[3], B^[4]} and M = {Na, Mg}, where the latter involve the entire {Na^[p]} and {Mg^[p]} ensembles. The (B^[p]–O), (Si–O), and (M–O) values constitute averages across all {O^[1], O^[2]} sites.

^bThe data uncertainties are ±1σ, with σ given for each entity.

We next consider the F–O^[1] and F–O^[2] bond-length variations. The large CFS of the very small B³⁺ cation manifests as tightly confined B^[3]–O^[1] and B^[4]–O^[1] bond lengths (134–135 pm throughout), which are markedly shorter than those of Si–O^[1] (161–162 pm; see Table 2). Nonetheless, whereas the average B^[3]–O^[2] distances are only a few pm longer than those of B^[3]–O^[1], the B^[4]–O^[2] bond lengths are 7–8 pm longer (142–146 pm) than their B^[4]–O^[1] counterparts, which is consistent with previous findings.^75−77,84Table 2 confirms the anticipated feature of nearly constant (F^[p]–O^[1]) values regardless of the n_Si/n_B molar ratio of the glass (related to K) or its NBO content (related to R).⁵⁶ In contrast, all {(F–O^[2])} values increase slightly for increasing x_NBO. For instance, contrast the bond lengths from each NaK–0.75 and MgNaK–0.75 structure with those of its NBO-richer NaK–2.1 and MgNaK–2.1 counterpart: (F^[p]–O^[2]) is increased by 1–2 pm for Si and slightly more for the two B^[3] and B^[4] coordinations (2–3 pm; Table 2). However, for the Mg²⁺/NBO-richer MgNaK–2.1 glasses, for which any bond-length effect from Mg²⁺ is expected to be most pronounced, it is notable that (B^[3]–O^[2]) is shorter by ≈1 pm than that for their NaK–2.1 analogs, whereas the reverse trend of a ≈1 pm longer(B^[4]–O^[2]) bond length is observed.

We onward focus on the average M–O^[1], M–O^[2], and M–O distances listed in Table 2 for the Na⁺ and Mg²⁺ species, which are averages over all M^[p] coordinations in the structure. As expected, the Na/Mg–O^[1] bond lengths are significantly shorter than their Na/Mg–O^[2] counterparts. Marginal variations of the average Mg–O^[1] and Mg–O^[2] distances are observed throughout, regardless of the precise B, Si, or NBO content of the glass (Table 2). The mean Na–{O, O^[1], O^[2]} bond lengths predicted from the NaK–R glass models conform well to those discussed for Na- and Na/Ca-bearing borate/BS glasses in ref (56). Notwithstanding that (Na–O^[2]) only varies marginally among the eight glass structures (Table 2), the Na–O^[1] distances are markedly longer (by 5–8 pm) in each MgNaK–R glass relative to its NaK–R counterpart (Table 2). This is attributed primarily to the sharing of many NBO sites in the MgNaK–R structure between Na⁺ and Mg²⁺, coupled with the tighter control of Mg²⁺ to maintain a short Mg–O^[1] bond (≈203 pm; Table 2), which lengthens (Na–O^[1]) by 5–8 pm relative to the bond length of ≈240 pm of the Mg-free glasses.

3.3. MD-Derived {NaO_p} and {MgO_p} Speciations

Figure 2 plots the distributions of {Na^[p]} and {Mg^[p]} coordinations observed from the glass models. As expected, the larger Na⁺ cation exhibits higher coordination numbers than Mg²⁺, which is mirrored in average coordination numbers ranging over and and corresponding distributions peaking at Na^[6] and Mg^[5]. Along previous findings from (boro)phosphosilicate glasses,^56,64,87 substantial Na^[5] and Na^[7] populations are also present throughout all BS glass models (Figure 2). In contrast, the {Mg^[p]} ensemble is more strongly peaked at p = 5, although significant contributions from MgO₄ and MgO₆ polyhedra are encountered in all structures, with the NaMg2.0–2.1 glass revealing nearly equal Mg^[5] and Mg^[6] populations. The herein modeled values accord well with those observed by Pedone and co-workers from NBO-rich phosphosilicate⁸⁶ and aluminoborosilicate⁸³ glasses, all of which are higher than those reported in ref (85). Potential relationships between / and the glass compositions are discussed in section S1. The less dispersed distribution of {Mg^[p]} populations relative to {Na^[p]} stems from the larger CFS_Mg and the higher capacity of Mg²⁺ to control its coordination shell, as also mirrored in the respective standard deviation of the p-distribution around its average value : it spans for Na⁺ but is consistently smaller for Mg²⁺ () in the MgNaK–0.75 glasses and yet lower in their NBO-rich MgNaK–2.1 counterparts: (Table 3).

Figure 2.

MD-derived distributions of Na^[p] (black and red bars) and Mg^[p] (green bars) coordination numbers in each NaK–R and MgNaK–R glass with (a) {K, R} = {2.0, 0.75}, (b) {K, R} = {4.0, 0.75}, (c) {K, R} = {2.0, 2.1}, and (d) {K, R} = {4.0, 2.1}. The black and red bars represent the coordination numbers of Na in the NaK–R and MgNaK–R (“MgNa”) glasses, respectively. The legends specify the average coordination numbers and of Na⁺ and Mg²⁺, respectively.

Table 3. Distributions of Na^[p] and Mg^[p] Coordinations^a.

			100x[p]Na (%)								100x[p]Mg (%)
glass	Z̅Na	σ[p]Na	4	5	6	7	8	9	Z̅Mg	σ[p]Mg	3	4	5	6	7	8
R = 0.75
Na2.0–0.75	6.38	1.30	5.9	18.8	29.4	26.4	13.9	4.1
MgNa2.0–0.75	6.22	1.27	7.0	21.0	30.5	25.6	11.4	3.1	5.52	0.93	0.5	12.1	37.8	35.7	12.4	1.5
Na4.0–0.75	6.11	1.27	8.0	22.6	31.6	23.0	10.5	2.6
MgNa4.0–0.75	5.90	1.26	9.9	26.0	32.0	20.0	7.8	1.8	5.17	0.91	1.4	21.7	43.2	27.2	5.6	0.9
R = 2.1
Na2.0–2.1	5.81	1.11	9.8	30.6	34.3	18.1	5.5	1.0
MgNa2.0–2.1	6.03	1.18	8.0	24.9	33.0	23.0	8.4	1.8	5.33	0.79	0.0	13.0	48.2	32.3	6.2	0.4
Na4.0–2.1	5.82	1.20	11.5	28.8	31.7	18.4	6.8	1.5
MgNa4.0–2.1	5.98	1.20	8.5	24.8	33.3	22.3	7.9	1.7	5.16	0.80	0.2	19.6	48.6	27.3	4.1	0.2
σb	0.02	0.01	0.3	0.4	0.4	0.5	0.2	0.2	0.02	0.02	0.2	1.1	1.1	0.9	0.7	0.2

^aMD-derived average coordination number of Na⁺ and Mg²⁺ () across the respective {Na^[p]} and {Mg^[p]} distributions, where and denote the corresponding fractional populations of the Na^[p] and Mg^[p] coordination species and is the distribution width.

^bThe data uncertainties are ±1σ, with σ given for each entity.

The herein observed values from a rather modest Na–(Mg)–B–Si–O glass ensemble agree well with those of previous MD-derived results gathered from large sets of Na-bearing borate, boro(phospho)silicate, and phosphosilicate glasses.^56,64 Notably, the Ca²⁺ cation with an intermediate CFS between Na⁺ and Mg²⁺ (Table S1) reveals values close to and only marginally lower than those of Na⁺. However, the range is closer to that of Mg²⁺, spanning 0.8–0.9 in Na–Ca–Si–P–O glasses⁶⁴ and 0.8–1.1 in their B-bearing counterparts.⁵⁶ A general tendency is that both and values increase concomitantly with the B content of the glass (see Table 3 and ref (56)), whereas our rather sparse data available for Mg²⁺ merely suggests that remains nearly constant for fixed-K MgNaK–R glasses but depends on R, i.e., on the NBO content.

In oxide-based glasses, decreases for increasing M^z+ CFS values. For instance, all high-CFS La³⁺, Y³⁺, Lu³⁺, and Sc³⁺ cations (Table S1) manifest narrow p-distributions in AS glasses.^5,88,89 Indeed, although the La³⁺ ion is markedly larger than Mg²⁺—as is reflected in MD-derived average coordination numbers of 6.0–6.6 in La₂O₃–Al₂O₃–SiO₂ glasses⁸⁸— CFS_La = 0.52 Å^–2 is slightly higher than CFS_Mg = 0.46 Å^–2, yielding a coordination spread of . Incidentally, the Sc³⁺ cation with a substantial CFS of 0.68 Å^–2 reveals very similar {Sc^[p]} populations in Sc₂O₃–Al₂O₃–SiO₂ glasses compared to the {Mg^[p]} populations of the present MgNaK–R glasses (Figure 2), with and spanning 5.1–5.4 and 0.68–0.71, respectively (see Figure 3 of Okhotnikov et al.⁸⁹).

A frequently suggested but hitherto unsettled issue concerns the possibility that a subensemble of Mg²⁺ does not act as a network modifier but merely assumes a network-forming role in the guise of MgO₄ tetrahedra.^22,90−92 Yet, discriminating between those two distinct structural scenarios is not straightforward even from glass models because precise and reliable criteria are difficult to formulate given that all electropositive M^z+ cations coordinate a significant number of BO sites at the SiO₄ and BO_p moieties (section 3.4). Table 3 reveals that Mg^[4] accounts for 12–22% of the Mg speciations of the present BS glasses. Even if {Mg^[4]} would assume a partial network-forming role, it is likely to be only minor. Indeed, eq 9 rests on the assumption that all Na⁺/Mg²⁺ cations act as modifiers, where the excellent agreement between the MD-derived NBO populations and those obtained experimentally via eq 9 (Table 1) suggests that a vast majority of the Mg²⁺ cations (if not all) assume the expected network-modifying role. To widen the perspective, non-negligible Na^[4] populations are also predicted in MD-derived oxide-glass models (see Table 3 and refs (56) and (87)), but a potential network-forming capacity of the archetypal Na⁺modifier has to our knowledge not yet been suggested.

3.4. CFS-Dependent Preferences for M–BO/NBO Bonding

We now examine the MD-derived BO/NBO partitioning in the first coordination shells of the Na⁺ and Mg²⁺ cations in each BS glass structure, i.e., the distribution of the O^[2]/O^[1] coordinations of the respective {NaO_p} and {MgO_p} ensembles presented in Table 4. Because CFS_Ca is intermediate between CFS_Na and CFS_Mg (Table S1) and the relative preferences for M^[p]–BO/NBO bonding are strongly CFS-dependent, Table 4 also includes modeled data from R[0.5CaO–0.5Na₂O]–B₂O₃–KSiO₂ glasses, denoted “CaNaK–R” and discussed in refs (22), (50), and (51). Each NBO and BO fraction in the first coordination shell of an M^z+ cation is denoted by x(M–NBO) and x(M–BO), respectively. It was determined from the MD-derived glass model, whereupon the corresponding preferences for M–NBO and M–BO bonding were calculated by P(M–NBO) = x(M–NBO)/x_NBO, and P(M–BO) = x(M–BO)/x_BO, with x(M–NBO) + x(M–BO) = 1.^56,93 For nonpreferential M–BO/NBO bond formation, P(M–NBO) = P(M–BO) = 1 and the fractions of M–NBO and M–BO contacts match the respective x_NBO and x_BO values of the glass. The cases P(M–O^[q]) > 1 and P(M–O^[q]) < 1 mark the preference and reluctance of M–O^[q] formation, respectively, with the deviation from unity of P(M–O^[q]) conveying the degree of preference/reluctance.

Table 4. MD-Derived Fractional Populations and Preferences for {Na, Ca, Mg}–{BO, NBO} Bonding^a.

		x(M–O[1])			P(M–O[1])			x(M–O[2])			P(M–O[2])
glass	xNBO	Na	Ca	Mg	Na	Ca	Mg	Na	Ca	Mg	Na	Ca	Mg
R = 0.75
Na2.0–0.75	0.054	0.128			2.39			0.872			0.92
CaNa2.0–0.75	0.083	0.114	0.408		1.37	4.91		0.886	0.592		0.97	0.65
MgNa2.0–0.75	0.104	0.155		0.565	1.50		5.44	0.845		0.435	0.94		0.49
Na4.0–0.75	0.039	0.124			3.20			0.876			0.91
CaNa4.0–0.75	0.064	0.131	0.446		2.05	6.95		0.869	0.554		0.93	0.59
MgNa4.0–0.75	0.077	0.165		0.623	2.14		8.07	0.835		0.377	0.90		0.41
R = 2.1
Na2.0–2.1	0.364	0.537			1.48			0.463			0.73
CaNa2.0–2.1	0.363	0.441	0.717		1.22	1.98		0.559	0.283		0.88	0.44
MgNa2.0–2.1	0.382	0.477		0.808	1.25		2.11	0.523		0.192	0.85		0.31
Na4.0–2.1	0.228	0.419			1.84			0.581			0.75
CaNa4.0–2.1	0.238	0.339	0.659		1.42	2.76		0.661	0.341		0.87	0.45
MgNa4.0–2.1	0.253	0.368		0.784	1.45		3.10	0.632		0.216	0.85		0.29
σb	0.001	0.003	0.010	0.007	0.02	0.09	0.07	0.003	0.010	0.007	0.01	0.01	0.01

^aFractional x(M–O^[1]) and x(M–O^[2]) populations out of the entire O speciation for M = {Na, Ca, Mg}, along with the corresponding preferences for M–O^[p] bonding defined by P(M–O^[p])=x(M–O^[p])/, where and . The data from the CaNaK–R glasses were obtained from MD-generated models presented in ref.⁵⁰

^bThe data uncertainties are ±1σ with σ given for each entity.

The data of Table 4 confirm the well-documented propensities of Na⁺ and Ca²⁺ cations to coordinate NBO rather than BO species^{56,60,64,83,87,94−97} and that P(M–NBO) increases concurrently with the M^z+ CFS as follows:^56,64,86,87P(Na–NBO) < P(Ca–NBO) < P(Mg–NBO). The precise P(M–NBO) value depends not only on the M^z+identity but also on the {R, K} parameters of the BS glass: all Na⁺, Ca²⁺, and Mg²⁺ cations strongly prefer NBO coordination, which accentuates for (i) increasing n_Si/n_B ratio (i.e., increasing K) and, in particular, (ii) decreasing x_NBO (i.e., decreasing R). Consequently, all three network-modifier species manifest the overall strongest preference for M–NBO bond formation in the {K, R} = {4.0, 0.75} structures (Table 4).

Trend (ii) is unsurprising, i.e., that the strongest preference for M–NBO bonding occurs in NBO-poor BS glasses. It conforms to the frequently encountered feature of oxide glasses in which their propensity for forming a given structural moiety deviates the most from that predicted by an unrestricted random/statistical distribution whenever its abundance is low. Some examples reported by us and others encompass the following: (I) the clustering of rare-earth cations in RE₂O₃–Al₂O₃–SiO₂ (refs (93) and (98)) and Na₂O–SiO₂ (refs (99) and (100)) glasses (whereas if RE₂O₃ oxides are added to molten SiO₂, strong RE³⁺ clustering occurs at any concentration^100,101); (II) the deviations from an otherwise essentially random spatial distribution of anions in Ca–Na–P–Si–O glasses occur for low P contents (x_P ≲ 0.015), which manifest a minor P–P aggregation.^102,103 (III) The preference for each of the two prevalent P–O–B^[4] and P–O–Si linkages in borophosphosilicate glasses is strongest in Si-rich and B-rich glass structures, respectively, i.e., when the accompanying fractions of P–O–Si and P–O–B^[4] linkages dominate.⁵⁵ Exceptions to this crude “rule of thumb” do indeed exist, such as that the spatial distribution of Na⁺ cations in Na₂O–(CaO)–B₂O₃–SiO₂ glasses, which is most uniform in Na-poor compositions but randomizes in modifier-richer glasses.⁵⁷

The stronger preference for NBO coordination of the higher-CFS Ca²⁺ and Mg²⁺ ions compared to that of Na⁺ (refs (56), (64), (83), (86), (87), and (96)) is also mirrored in the reduced P(Na–NBO) value in each MgNaK–R glass relative to that its ternary NaK–R analog, notwithstanding that some of the mixed-cation glasses manifest slightly larger x(Na–NBO) fractions than their NaK–R counterparts due to their higher NBO contents accompanying Mg²⁺ introduction (Table 4). Yet, except for the overall NBO-richest {K, R}={2.0, 2.1} glasses (x_NBO ≈ 0.36), the low-CFS Na⁺ ion consistently features more BO than NBO contacts throughout all R = 0.75 glasses, for which NBO species accounts only for 11–17% of all Na–O bonds in each {NaO_p} ensemble. That contrasts sharply with the {MgO_p} speciation of the MgNa4.0–0.75 glass, for which NBO anions constitute 62% of all Mg–O^[q] bonds despite the low NBO abundance (x_NBO = 0.077), which is readily rationalized by the 2–3 times stronger Mg–NBO than Na–NBO affinity (Table 4).

The P(M–BO) < 1 values observed in Table 4 throughout all BS glass models and all three network modifier species mirror the as-expected reluctance of M–BO bond formation, which is accentuated for increasing M^z+ CFS values as follows: P(Na–BO) > P(Ca–BO) > P(Mg–BO). Although the much stronger propensity for M–NBO over M–BO contacts in oxide glasses is both intuitive and well documented by computational modeling,^{56,60,64,83,86,87,94,95} it is challenging to quantify the x(M–NBO) and x(M–BO) fractions using experiments. Yet, this was recently accomplished by exploiting double-resonance ¹⁷O{²³Na} and ¹⁷O{²⁷Al} NMR applied to an AS glass of composition 10.8Na₂O–32.3CaO–13.0Al₂O₃–44.0SiO₂.⁹⁶ That yielded estimated x(M–BO)/x(M–NBO) ratios of 1.4 and 0.37 for Na⁺ and Ca²⁺, respectively,⁹⁶ incidentally very close to those of 1.3 (Na⁺) and 0.40 (Ca²⁺) predicted herein for the M–BO/NBO partitioning of the NaCa2.0–2.1 glass (Table 4). However, given the by definition very local structural information encoded by the Na–O^[q] contacts of the NaO_p polyhedra—and M–O^[q̅] bonds in general—coupled with the direct scaling of the x(Na–NBO) and x(Na–BO) fractions with the NBO content of the glass (e.g., see Table 4 and ref (56)), we discourage attempts to draw even qualitative conclusions about medium-range glass-structure features from {x(M–O^[p])} data alone, encompassing inferences about the spatial Na⁺ distribution and its possible implications for the (sub)nanometer-scale glass organization.^96,97

3.5. B^[p]/Si Intermixing Probed by ¹¹B{²⁹Si} REDOR NMR and MD Simulations

3.5.1. Relative Degrees of B^[3]–O–Si and B^[4]–O–Si Bonding

The relative extents of B^[3]–O–Si and B^[4]–O–Si bonding in the Na4.0–0.75, MgNa4.0–0.75, and MgNa4.0–2.1 structures were assessed by ¹¹B{²⁹Si} REDOR NMR experiments. Figure 3 displays the “dephasing” responses observed from the ¹¹BO₃ and ¹¹BO₄ resonances for increasing “dipolar recoupling” periods (τ_rec) of ¹¹B^[p]–²⁹Si through-space interactions that are responsible for the NMR-signal dephasing.^65,104 While the ¹¹B^[p]-resonance dephasing is unaffected by any B^[p]–O–B^[q] or Si–O–Si linkage of the structure, its rate increases concurrently with the number of ¹¹B^[p]–O–²⁹Si linkages, thereby accelerating the progress toward the ΔS/S₀ = 1 limit of “complete dephasing”^24,25,41,42 (Figure 3a–c). As expected from the higher-coordination B^[4] sites and previous ¹¹B{²⁹Si} REDOR NMR reports from other BS glasses,^44,46 their resonances reach the complete limit of complete dephasing well before their ¹¹B^[3] counterparts (Figure 3a–c), except for the MgNa4.0–0.75 glass (vide infra).

Figure 3.

¹¹B{²⁹Si} REDOR NMR dephasing data (ΔS/S₀) plotted against the recoupling/dephasing interval (τ_rec) and recorded at 14.1 T and 9.00 kHz MAS from the (a, d) Na4.0–0.75, (b, e) MgNa4.0–0.75, and (c, f) MgNa4.0–2.1 glasses. The top panels (a–c) displays the entire ¹¹B^[3] and ¹¹B^[3] dephasing curves, whereas the bottom panels (d–f) show zoomed-in views of the initial dephasing regimes. Note that the lines in (a–c) only serve to guide the eye, while those of (d–f) are best-fit results to eq 7 for the data ΔS/S₀ ⩽ 0.20. All data uncertainties are within the symbol sizes.

Numerical fitting of the initial ¹¹B^[p] NMR-signal dephasing regime with “short” τ_rec values (Figure 3d–f) to eq 7 yields an estimate of the dipolar second moment M₂(B^[p]–Si), which reflects the “aggregate” ¹¹B^[p]–²⁹Si contact in the structure^24,41,42 and grows concomitantly with the net number of direct B^[p]–O–Si bridges, N(B^[p]–O–Si), i.e., the average number of Si atoms in the second coordination shell of the {B^[p]} sites: M₂(B^[p]–Si) ≈ N(B^[p]–O–Si). As highlighted by previous reports utilizing M₂/dipolar-interaction-based NMR analyses to make inferences about bonding statistics/preferences,^34,42,64 however, this relationship is only approximate because M₂(B^[p]–Si) involves a sum over all r(B^[p]–O–Si) distances in the structure (eq 6). Although the analysis of section S2 reveals that M₂(B^[p]–Si) approximates well the targeted information about the (average) number of direct B^[p]–O–Si linkages, other factors degrade quantitative assessments of B^[4]–O–Si bonding relative to B^[3]–O–Si, which becomes underestimated by ≈25%.

Table 5 compiles the experimental M₂(B^[p]–Si) data together with those extracted from the glass models via eq 6. The consistently lower experimental M₂(B^[p]–Si) values compared to their MD-derived counterparts (by 40–55%) stem primarily from experimental imperfections, in particular rf inhomogeneity.^66,67 Consequently, we focus our comparisons on the relative M₂(B^[p]–Si) trends among the B^[3]/B^[4] coordinations encoded by each NMR- and MD-derived dipolar second-moment ratio (section S2):

10

Table 5 reveals a very good agreement between the experimental and modeled (B) values, whose deviations only amount to a few percent. Even the largest discrepancy observed for the MgNa4.0–0.75 structure (16%) must be considered decent in view of the large number of potential error sources that could degrade the agreement, notably those of the MD-generated glass models.

Table 5. ¹¹B^[p]–²⁹Si Dipolar Second Moments and F–O–F′ Bonding Preferences^a.

	M2(B[3]–Si)	M2(B[4]–Si)		M2(Si–B[3])	M2(Si–B[4])b		P(Si–O–F)c			P(B[3] −O–F)c		P(B[4]–O–F)c
glass	(104 Hz2)	(104 Hz2)	Mrel2(B)	(104 Hz2)	(104 Hz2)	Mrel2(Si)	Si	B[3]	B[4]	B[3]	B[4]	B[4]
R = 0.75
Na2.0–0.75	5.13	7.58	1.48	9.32	16.61	1.78	0.95	0.90	1.16	0.66	1.38	0.50
MgNa2.0–0.75	5.41	7.16	1.32	14.02	10.14	0.72	0.99	0.99	1.03	0.81	1.27	0.54
Na4.0–0.75	7.08	9.51	1.34	6.70	10.04	1.50	0.99	0.94	1.09	0.76	1.38	0.40
	3.92 ± 0.05	5.65 ± 0.05	1.44 ± 0.05	3.00 ± 0.04	6.99 ± 0.06	2.33 ± 0.08
MgNa4.0–0.75	7.23	8.85	1.22	10.12	5.33	0.53	1.00	1.01	0.99	0.74	1.33	0.49
	3.84 ± 0.03	4.02 ± 0.11	1.05 ± 0.06	5.30 ± 0.04	2.50 ± 0.07	0.47 ± 0.03
R = 2.1
Na2.0–2.1	3.47	6.72	1.94	7.68	12.04	1.57	0.92	0.99	1.19	0.79	1.15	0.52
MgNa2.0–2.1	3.78	6.51	1.72	9.63	9.49	0.99	0.97	0.99	1.11	0.81	1.20	0.54
Na4.0–2.1	5.14	8.84	1.72	4.04	10.75	2.66	0.96	0.95	1.14	0.70	1.28	0.46
MgNa4.0–2.1	5.55	8.45	1.53	6.18	7.51	1.22	0.99	0.98	1.07	0.72	1.29	0.49
	3.37 ± 0.04	5.11 ± 0.04	1.51 ± 0.04	3.75 ± 0.04	4.55 ± 0.04	1.21 ± 0.03
σd	0.04	0.05	0.02	0.10	0.12	0.04	0.01	0.01	0.01	0.05	0.02	0.03

^aHeteronuclear ¹¹B^[p]–²⁹Si dipolar second moments, M₂(B^[p]–Si), calculated from the MD-derived glass model (eq 6) or obtained by fitting ¹¹B{²⁹Si} REDOR NMR data to eq 7 (data on line below the glass entry). (B) = M₂(B^[4]–Si)/M₂(B^[3]–Si).

^bCalculated from eq 11 by using the {M₂(B^[p]–Si)} data. (Si) =M₂(Si–B^[4])/M₂(Si–B^[3]).

^cPreference factor P(F–O–F′) =P(F′–O–F) for F–O–F′ linkage formation of {F, F′} = {B^[3], B^[4], Si}. P(F–O–F′) is defined as the ratio between the as-observed number of F–O–F′ linkages in the glass model relative and that predicted from nonpreferential (statistical) {F, F′} intermixing.

^dThe uncertainties of the MD-derived data are ±1σ, with σ given for each entity.

For a statistical/nonpreferential F–O–F′ linkage-formation among {F, F′}={Si, B^[3], B^[4]} in a BS structure devoid of NBO species, (B) is given by (stat) ≈ 1.18 (section S2). The MD derived (B) data, and notably the experimental counterparts, are markedly larger than (stat) for all glasses but MgNa4.0–0.75, in particular for the NBO-rich R = 2.1 members (Table 5). Two structural factors account for these observations:

• (i)The NBO partitioning among Si, B^[3], and B^[4] species in BS glasses was discussed previously,^34,50,56 suggesting a substantially stronger propensity for B^[3]–NBO bonding relative to B^[4]–NBO. Except for very NBO-rich glasses,¹⁰⁵ the latter is even considered “forbidden” by most scientists in the field^{27,69,106−108} but remains frequently observed to minor extents in numerous MD-derived glass models.^{34,55,56,75,77,82} Consequently, a progressively growing NBO accommodation at the BO₃ groups for increasing x_NBO reduces the possibility of any B^[3]–O–F linkage type, thereby boosting the (B) values of all NBO-rich glasses (Table 5), while the relative dipolar second moment (eq 10) is independent of the degree of Si–NBO bonding.
• (ii)The second effect underlying the increased NMR/MD-derived (B) values in any BS glass is less influential but stems from a slightly higher preference for B^[4]–O–Si bridges than B^[3]–O–Si linkages. Owing to the difficulties in quantifying the preference factor P(F–O–F′) for F–O–F′ bond formation by experiments, however, current quantitative insights stem dominantly from computational modeling;^{55−57,64,109} see ref (50) for experimental attempts to estimate the {P(B^[p]–O–B^[q])} subset. Table 5 lists the MD-derived {P(F–O–F′)} factors of the present BS glasses. Here, P(F–O–F′) = 1 denotes a strictly nonpreferential F/F′ intermixing, whereas P(F–O–F′) > 1 and P(F–O–F′) < 1 imply a preference and reluctance for F–O–F′ linkage formation, respectively. Although the bonding preferences depend slightly on the glass composition, the trends of Table 5 conform well to those discussed previously for Na/Ca-bearing boro(phospho)silicate glasses,^34,50,55,56 revealing the strongest preference (reluctance) for BO₃–BO₄ (BO₄–BO₄) pairs. While B^[3]–O–B^[3] linkages are also disfavored, all remaining Si–O–{Si, B^[3], B^[4]} linkages form nearly statistically, i.e., each fractional population is given roughly by the product of the respective {x_Si, , } molar fractions in the glass structure. Nonetheless, along previous findings from boro(phospho)silicate glasses,^34,50,55,56Table 5 conveys the following subtle trend: P(Si–O–B^[4]) > P(Si–O–B^[3]) ≈ P(Si–O–Si) ≈ 1 [note that P(F–O–F′) = P(F′–O–F)]. Hence, the slightly stronger propensity for forming B^[4]–O–Si linkages relative to B^[3]–O–Si underlies the observed (B)(stat) trend, notably so for the ternary NaK–R glasses (sections 3.6 and S2).

3.5.2. B^[4] Environments with Variable Numbers of Si and B Neighbors

Figure 4a, c, and e shows the REDOR “reference” ¹¹B NMR spectrum, S₀(τ_rec), of each Na4.0–0.75, MgNa4.0–0.75, and MgNa4.0–2.1 glass recorded at the shortest τ_rec = 0.22 ms value, along with two REDOR spectra [S(τ_rec)] observed for long dephasing periods of τ_rec={1.78, 2.67} ms. As expected from the ΔS/S₀ dephasing data of Figure 3, the latter spectra manifest progressively diminished ¹¹B^[3] and (particularly) ¹¹B^[4] resonance intensities for increasing τ_rec, except for the MgNa4.0–0.75 specimen, which reveals comparable NMR-signal decays.

Figure 4.

Selected ¹¹B{²⁹Si} REDOR NMR spectra of the (a, b) Na4.0–0.75, (c, d) MgNa4.0–0.75, and (e, f) MgNa4.0–2.1 glasses for the as-indicated dipolar recoupling periods (τ_rec). (a, c, e) REDOR reference spectra [S₀(0.22 ms)] shown together with the REDOR NMR spectra [S(τ_rec)] obtained for long dephasing intervals. (b, d, f) Zoomed-in view of the ¹¹B^[4] NMR spectral region (normalized to unity maximum intensity throughout), where two peaks attributed to ¹¹ B^[4](4Si) and ¹¹B^[4](3Si) moieties are marked as m = 4 and m = 3, respectively.

We now focus on the ¹¹B^[4] NMR-signal dephasing, which is more pronounced for the low-δ_B spectral region <−1 ppm, as is most transparent from the normalized NMR spectra presented in Figure 4b, d, and f. Along the ¹¹B NMR spectral deconvolution results of the Na4.0–0.75 and NaMg4.0–0.75 glasses presented by Lv et al.,⁵¹ the two peak components at ≈ −1.8 ppm and ≈ −0.3 ppm are attributed to B^[4](OSi)₄ and B^[4](OSi)₃(OB) moieties, respectively,^{35−37,40,72} which are abbreviated as B^[4](4Si) and B^[4](3Si). Owing to its larger number of Si neighbors, the resonance decay of ¹¹B^[4](4Si) is stronger than that of its ¹¹ B^[4](3Si) counterpart. To adequately deconvolute the ¹¹B^[4] NMR signal region of the two Na4.0–0.75 and MgNa4.0–0.75 glasses, however, it was necessary to also include a minor peak at ≈1.2 ppm from ¹¹B^[4](2Si) environments (accounting for 9% and 17% out of all B^[4](mSi) groups, respectively).⁵¹ These NMR signals are not clearly discernible in the REDOR spectra, but they are expected to decay even slower than the ¹¹B^[4](3Si) resonance, as is also hinted in Figure 4.

3.6. Effects from Mg²⁺ on the B^[p]/Si Intermixing

Although all herein discussed BS glasses but MgNa4.0–0.75 manifest a markedly larger number of B^[4]–O–Si linkages than B^[3]–O–Si bridges, the partial replacement of Na⁺ by Mg²⁺ leads to a non-negligible reduction of (B) throughout: contrasting the MD-derived M₂(B^[4]–Si) and M₂(B^[3]–Si) values of the NaK–R glass and its MgNaK–R counterpart in Table 5 reveals that the decrease of (B) stems from an increase of M₂(B^[3]–Si) at the expense of M₂(B^[4]–Si), altogether implying that Mg²⁺-for-Na⁺ substitution is accompanied by an increase (decrease) in the number of B^[3]–O–Si (B^[4]–O–Si) linkages. The predictions of the Na4.0–0.75 and MgNa4.0–0.75 glass models are corroborated by the REDOR NMR experiments, which reveal a significant reduction in M₂(B^[4]–Si) for the MgNa4.0–0.75 glass ((B) = 1.05) relative to (B) = 1.44 for Na4.0–0.75 (yet the experimental M₂(B^[3]–Si) values of both glasses are nearly equal; Table 5). These effects are evident from the nearly coincident ¹¹B^[p]{Si} REDOR NMR dephasing curve observed for MgNa4.0–0.75 (Figure 3b,e), in sharp contrast to those of the other two glasses for which the ¹¹B^[4] NMR-signal dephasing rate consistently exceeds that of ¹¹B^[3].

We stress that although a lower total number of B^[4]–O–Si linkages upon Mg²⁺ incorporation is indeed anticipated from the drastic decrease of the BO₄ population alone (Table 5), that effect is inconsequential for M₂(B^[4]–Si) because its value is independent on {, } (eq 6), in contrast to M₂(Si–B^[4]); see section 3.7. Notably, the MD-generated preference factors of Table 5 rationalize these quantitative trends of dipolar second moments and the number of B^[p]–O–Si bonds as originating from the more fundamental feature of a decrease in P(B^[4]–O–Si) upon the introduction of the high-CFS Mg²⁺ cation, which for all NBO-poor (Mg)NaK–0.75 glasses is moreover emphasized by a concomitant increase of P(B^[3]–O–Si) (see section S2).

The weakened B^[4]/Si contacts upon Mg²⁺-for-Na⁺ substitution reflect a general trend of a linearly decreasing fraction of B^[4]–O–Si linkages (out of all B^[4]–O–Si/B bridges) for increasing CFS_M, as deduced from ¹¹B^[4] MAS NMR spectra deconvolutions.⁵¹ These CFS effects are also mirrored in the ²⁹Si MAS NMR spectra of the Na4.0–0.75, MgNa4.0–0.75, and MgNa4.0–2.1 glasses shown in Figure 5: as expected from amorphous (boro)silicates, the net ²⁹Si resonance is broad but consistent with solely ²⁹SiO₄ groups.^{24,82,110−113} Hence, the ²⁹Si MAS NMR information content is very limited, at best offering qualitative inferences.^113,114 Even for the (almost) NBO-free Na4.0–0.75 and MgNa4.0–0.75 glasses, the net ²⁹Si NMR peak stems from a plethora of unresolved ²⁹Si(OB^[3])_p(OB^[4])_q(OSi)_4–p−q resonances.

Figure 5.

²⁹Si MAS NMR spectra recorded at 9.4 T and 14.00 kHz MAS from the Na4.0–0.75, MgNa4.0–0.75, and MgNa4.0–2.1 glasses. The higher structural complexity of the NBO-rich MgNa4.0–2.1 structure is mirrored in its ≈5 ppm wider resonance relative to that of its MgNa4.0–0.75 counterpart with a low NBO content, along with a more deshielded chemical shift at the peak maximum (i.e., a higher value of δ_Si).

The ²⁹Si chemical-shift dispersion and the precise value of the most probable shift located at the maximum NMR-peak intensity depend predominantly on the {Si, B^[3], B^[4]} distribution in the second coordination sphere of ²⁹Si, where a 3–5 ppm increase of δ_Si accompanies each ²⁹Si–O–Si → ²⁹Si–O–B^[4] bond replacement, whereas ²⁹Si–O–Si → ²⁹Si–O–B^[3] substitutions leave the ²⁹Si chemical shift essentially invariant.^82,110−113 Hence, the markedly reduced number of Si–O–B^[4] bonds in the MgNa4.0–0.75 structure and the concurrently increased number of Si–O–B^[3] and Si–O–Si linkages rationalize its net ²⁹Si resonance-displacement toward lower shifts relative to Na4.0–0.75, while the evident high-ppm “tail” of the NMR stems from the few(er) remaining ²⁹Si–O–B^[4] sites (Figure 5). The structural complexity is accentuated further in the NBO-rich(er) MgNa4.0–2.1 glass, which additionally exhibits variable numbers of Si–NBO bonds among the SiO₄ groups. Besides rationalizing the broader ²⁹Si NMR peak, it accounts for the 5.8 ppm higher value relative to that observed for MgNa4.0–0.75 (Figure 5), stemming from a typically 7–12 ppm ²⁹Si chemical-shift increase for each BO → NBO bond replacement.^82,110−113

It is known that Si-rich but M^z+-poor M_z/2O–B₂O₃–SiO₂ glasses exhibit compositional regions of liquid immiscibility, which widen for high-CFS M^z+ cations.^37,115−119 A glass-in-glass separation occurs upon cooling, typically identified as (or often merely assumed to involve) one Si-dominated phase coexisting with a B-rich borate/BS counterpart^37,115−118 and manifested by ²⁹Si MAS NMR peak displacement toward more negative chemical shifts¹¹⁷ near ≈ –110 ppm observed for vitreous SiO₂.¹¹³ Incidentally, that is also observed for the MgNa4.0–0.75 glass (Figure 5) but is readily attributed to the significantly fewer Si–O–B^[4] bonds in the MgNa4.0–0.75 structure relative to Na4.0–0.75 (note that both²⁹Si–O–Si/B^[3] environments resonate at near-equal shifts¹¹³). Notably, previous heteronuclear ¹¹B/²⁹Si NMR studies on the bearings from thermal annealing of BS glasses, which may induce structural inhomogeneities and/or phase separation, also reveal that the number of Si–O–B^[3] bonds increases relative to Si–O–B^[4],⁴⁵⁻⁴⁸ thereby mirroring the herein observed ²⁹Si shielding accompanying Mg²⁺ incorporation into a Na₂O–B₂O₃–SiO₂ glass. However, backscatter scanning electron microscopy (SEM) images (not shown) did not indicate phase separation. Yet nanometer-scale inhomogeneities cannot be excluded, as they would remain undetected both over the ≳1 μm and <0.5 nm length scales probed by our SEM and NMR experimentation, respectively.

Remarkably, despite numerous reports on phase-separated BS glasses, the precise chemical compositions of the two (assumed) coexisting phases remain surprisingly poorly defined. Interestingly, all of the few studies using techniques that directly inform on the Si/B^[p] intermixing (i.e., heteronuclear NMR) of annealed/phase-separated BS glasses do not point toward a categorical separation into Si-rich and B-rich phases, but merely to a significant reduction of the number of Si–O–B^[4] linkages, while the number of Si–O–B^[3] bonds remains invariant, or even increases,⁴⁵⁻⁴⁸ within one, or several, borosilicate phase(s). A recently reported elemental analysis of a phase-separated Na BS glass did reveal two such phases with distinct Si and B contents; however, both contents were substantial in each phase.¹¹⁹ This issue should be investigated further to better define what phases coexist in BS glasses attributed to exhibit nanometer-range structural inhomogeneities.

3.7. Relative Degrees of Si–O–B^[3]/B^[4] Bonding

This section discusses the dipolar second moments {M₂(Si–B^[p])} listed in Table 5. They are accessible from the {M₂(B^[p]–Si)} set by the general expression M₂(S–I) for S–I spin-pairs,⁶³ which for S = 1/2 (²⁹Si) and I = 3/2 (¹¹B) evaluates to

11

where for ¹¹B, whereas ²⁹Si constitutes ≈100% of all Si sites in the isotopically enriched glasses . Although the M₂(Si–B^[p]) and M₂(B^[p]–Si) values are directly related for a given glass specimen, they nonetheless convey complementary information: while M₂(B^[p]–Si) is proportional to the number of B^[p]–O–Si linkages at the BO_p ensemble, M₂(Si–B^[p]) relates to the number of Si–O–B^[p] bridges at {SiO₄}.

The crucial utility of eq 11 is its straightforward route to derive both M₂(Si–B^[3]) and M₂(Si–B^[4]) values, whose separate estimation is very difficult to accomplish accurately with current state-of-the-art heteronuclear NMR methodology. For instance, the less informative M₂(Si–B) entity, i.e., the aggregate dipolar second moment across the entire {B^[3], B^[4]} ensemble, may be obtained from a ²⁹Si{¹¹B} REAPDOR NMR experiment,¹⁰⁴ with the caveat that the experimental protocol along with its subsequent data analysis yield less accurate dipolar second-moment estimates than the M₂(B^[3]–Si) and M₂(B^[4]–Si) outcomes of ¹¹B{²⁹Si} REDOR NMR, from which moreover both M₂(Si–B^[3]) and M₂(Si–B^[4]) results are readily extracted via eq 11.^57,63,64 Hence, sole¹¹B{²⁹Si} REDOR NMR application to a BS glass specimen unveils all four independent M₂(B^[3]–Si), M₂(B^[4]–Si), M₂(Si–B^[3]), and M₂(Si–B^[4]) results.

From its definition (eq 6), it follows that M₂(B^[p]–Si) is independent on , while M₂(Si–B^[p]) scales linearly with the B^[p] population of the glass but is independent of the Si content. That feature rationalizes the excellent agreement observed between the NMR/MD-derived (Si) ≡ M₂(Si–B^[4])/M₂(Si–B^[3]) data for the Mg-bearing glasses of Table 5 (whose , sets accord very well), whereas the significantly underestimated MD-derived BO₄ population of Na4.0–0.75 (section 3.1) accounts for the lower modeled (Si) result. The direct M₂(Si–B^[p]) dependence on renders the (Si) ratio a very sensitive probe of the reduced Si–O–B^[4] bonding in the glass structure upon Mg²⁺ incorporation, as mirrored in the markedly lower {(Si)} values relative to their {(B)} counterparts (Table 5) and underscoring the time/effort-saving benefits of having both {M₂(B^[p]–Si)} and {M₂(Si–B^[p])} data available from one sole NMR experiment and eq 11 (see refs (57), (63), and (64) for further examples.

Both (B) and (Si) scale approximately as (eq 6 and section S2), while (Si) is additionally proportional to (eq 11). These parameter ratios are listed in Table S4, along with those of P^rel(Si) ≡ P(Si–O–B^[4])/P(Si–O–B^[3]). Disregarding the MgNa4.0–0.75 glass with much weaker Si/B^[4] contacts, P^rel(Si) > 1 holds throughout. When taken together with N^rel(stat) = 1.44 reflecting a nonpreferential/statistical Si–O–B^[3]/B^[4] bond formation (section S2), the product of the , P^rel(Si), and components is consistently slightly larger than (Si) but yields overall good predictions for all BS glasses (Table S4), despite that the crude approximations made are only expected to capture the R = 0.75 glasses with low NBO contents (section S2). The good predictions also observed for the experimental(Si) data are particularly gratifying because they assumed the MD-derived and P^rel(Si) parameters.

Notably, although illustrated in the context of the M₂(Si–B^[p]) and (Si) entities, the various glass composition/structure parameters discussed above are readily replaced by their analogs underpinning any M₂(F–F′) entity reflecting the number of F–O–F′ linkages at FO_p polyhedra that may interlink with two (or several) FO_p, F′O_p′, and F″O_p″ polyhedral types.

3.8. Inferred Si/B^[p] Intermixing Versus Current BS Glass-Structure Descriptions

The findings herein of an overall larger extent of B^[4]–O–Si than B^[3]–O–Si bonding—yet with both linkage-types being abundant throughout all glasses (Table 5)—suggest BS glass networks with substantial {Si, B^[3], B^[4]} intermixing. When combined with previous inferences of the coexistence of all three B^[3]–O–B^[3], B^[3]–O–B^[4] and B^[4]–O–B^[4], linkages,^34,50 the results portrays a network with all six F–O–F′ linkage-types among {Si, B^[3], B^[4]} encountered in significant populations, each scaling with the molar fractions of its constituents but with higher-than-statistical numbers of the most preferred B^[3]–O–B^[4] and Si–O–B^[4] bonds, while B^[4]–O–B^[4] linkages are present but disfavored (Table 5).

Bray and co-workers introduced a structural description,^52,53 herein referred to as the Yun–Dell–Bray–Xiao (YDBX) model, which accurately reproduces the experimental {, } fractions across the entire glass-formation region of the Na₂O–B₂O₃–SiO₂ system and has found widespread recognition, particularly within the NMR/glass community. Notwithstanding its success in predicting the borate speciation, the YDBX model attempts—from routine ¹¹B NMR experimental information alone—to furnish a very detailed (but oversimplified) structural description, involving a (very) confined set of larger BO_p/SiO₄ molecular aggregates (“superstructural units”^27,70,120). Superstructural units are indeed known to build many crystalline borate/BS phases,²⁷ and numerous Raman studies support their existence for glasses as well.^{28,110,120,121} While BS-based glass structures most likely do comprise some larger BO_p/SiO₄ molecular aggregates, it is difficult to reconcile the main body of experimental reports with any dominating role thereof (except for limiting cases, such as vitreous B₂O₃^27,122,123).

Indeed, ¹¹B, ²⁹Si, and ¹⁷O (3Q)MAS NMR reports suggest markedly more disordered BS networks than the (for a glass) exceptionally high medium-range order postulated by the YDBX model, notably its Si/B^[3]/B^[4]-intermixing predictions.^{36,37,39,40,47,108,110,111} We guide the reader to the thoughtful but critical remarks made by Möncke et al.⁴⁷ Notably, the YDBX model predicts that only Si–O–Si/B^[4] and B^[3]–O–B^[3]/B^[4] bonds occur in the present NaK–0.75 glasses.^52,53 Hence, Si–O–B^[3] are absent, in sharp qualitative disagreement with the results for any glass of Table 5, encompassing direct experimental proof of significant Si–O–B^[3] bonding in the Na4.0–0.75 glass, which is accentuated in the Mg-bearing glass structures. Furthermore, earlier findings from similar NMR experimentation compared to that employed herein suggested that SiO₄ interlinks extensively with both BO₃/BO₄ moieties in BS glasses.^44,46−49 The distinctly different Si/B^[p] intermixing predicted by the YDBX model and experimental/modeling findings herein and in refs (36), (37), (39), (40), and (46−48) stems from the YDBX-postulated but grossly underestimated degree of Si–O–B^[3] bonding (i.e., P(Si–O–B^[3]) ≈ 0), whereas in fact P(Si–O–B^[3]) ≲ P(Si–O–B^[4]), yielding P(Si–O–B^[4])/P(Si–O–B^[3]) ≈ 1.2 (Table 5). Analogous contradictions with many experimental findings also plague the alternative branch of “random-network model” (RNM) descriptions^{69,106,107,124} originating from Zachariasen and Warren,^125,126 which overestimate the structural disorder by largely ignoring F–O–F′ bonding preferences (see section 3.5 and comments in ref (50)).

To reconcile the orthogonal implications for the structural order from the too categorical RNM^{69,106,107,124} and “superstructural-unit”^27,70,120 borate/BS glass descriptions, we recently highlighted a “hybrid” model thereof.⁵⁰ While we are unaware of previous explicit outlines or discussions of a BS glass network being built from some superstructural units along with near-randomly intermixed BO₃/BO₄/SiO₄ groups across a <1 nm scale, even the well established and noncontroversial structure of vitreous B₂O₃ conforms to such a hybrid structural picture. Here, superstructural B₃O₆ units (boroxol rings, which comprise ≈70% of all B^[3] sites^27,123) coexist with ring-interlinking BO₃ groups.^27,122,123 The introduction of network modifiers along with another network former (Si) naturally increases the structural disorder. This is, for instance, manifested by the strongly altered Si intermixing with both B^[3] and B^[4] accompanying the replacement of a low-CFS Na⁺ cation by Mg²⁺ (section 3.5) and also mirrored in B^[3]–O–B^[3]/B^[4] and B^[4]–O–B^[4] populations somewhat closer to a statistical {B^[p]–O–B^[q]} intermixing in Mg-bearing BS glasses.⁵⁰ Hence, all experimental/modeled results herein and in ref (50) as well as previous work^{36,37,39,40,47−49,110,111} are consistent with a hybrid random/superstructural-unit BS glass-network description, which nonetheless remains to be concretized from a quantitative standpoint.

4. Conclusions

We investigated the structural alterations occurring across a subnanometer scale when Mg²⁺ replaces Na⁺ in four RNa₂O–B₂O₃–KSiO₂ base glass compositions with K = n(SiO₂)/n(B₂O₃) = {2.0, 4.0} and R=[n(MgO) + n(Na₂O)]/n(B₂O₃) = {0.75, 2.1}. Na⁺ and Mg²⁺ exhibit average coordination numbers that vary across and among the glasses, where Na^[6] and Mg^[5] coordinations are most abundant. A non-negligible fraction (12–22%) of Mg^[4] species is observed but without evidence supporting a network-forming role of the MgO₄ groups. The higher CFS of Mg²⁺ is manifested in narrower distributions of coordination numbers {Mg^[p]} compared with {Na^[p]}, as well as a 2–3 times stronger preference for NBO coordination relative to Na⁺. Although all M^z+ species prefer M–NBO bonding over M–BO, the preference increases concurrently with the CFS along the series P(Na–NBO) < P(Ca–NBO) < P(Mg–NBO), where the precise preference factors also depend on the BS glass composition and, in particular, on the NBO content. Hence, whereas Na⁺, Ca²⁺, and Mg²⁺ all exhibit the strongest affinity for M–NBO bonding in the R = 0.75 glasses with low NBO contents (x_NBO < 0.1), the fractions of M–BO bonds are significant for all glasses/cations, amounting to roughly 86%, 57%, and 40% out of all BO/NBO species in the first coordination shells of Na⁺, Ca²⁺, and Mg²⁺, respectively.

The partial replacement of Na⁺ by Mg²⁺ induces primarily the following structural alterations:

• (i)Significant BO₃ → BO₄ and BO → NBO conversions occur. Whereas BO₄ accounts for 50–60% of the borate speciations of the Na₂O–B₂O₃–SiO₂ glasses, they only amount to 30–45% in the MgO–Na₂O–B₂O₃–SiO₂ structures.
• (ii)The average Na–NBO bond length is 5–8 pm longer in the MgNaK–R glasses relative to that of ≈240 pm in the absence of Mg²⁺. That effect likely stems from the sharing of many NBO sites between Na⁺ and Mg²⁺, where the stronger capacity of the high-CFS cation to maintain its short Mg–NBO distance (≈203 pm) perturbs the local Na coordination environment.
• (iii)All RNa₂O–B₂O₃–KSiO₂ glass structures reveal a clear preference for B^[4]–O–Si linkages over B^[3]–O–Si. Upon Mg²⁺-for-Na⁺ substitution, however, the preference for B^[4]–O–Si is reduced (i.e., P(B^[4]–O–Si) is decreased), whereas the propensity for B^[3]–O–Si bridges either remain unaffected (for NBO-rich glasses; R = 2.1) or even increases (for NBO-poor glasses; R = 0.75). This results in P(B^[4]–O–Si) ≈ P(B^[3]–O–Si) for both MgNa2.0–0.75 and MgNa4.0–0.75 glasses, whereas B^[4]–O–Si linkage formation remains (slightly) preferred for the R = 2.1 analogs. We stress that these bonding preferences are independent on the borate speciation. However, effects (i) and (iii) together account for effect (iv) below, namely
• (iv)a significant reduction in the number of B^[4]–O–Si bonds in the quaternary MgNaK–R glasses. The former bridges dominate throughout all Na₂O–B₂O₃–SiO₂ glasses, notably so for the NBO-rich members, where the stronger propensity for B^[3]–NBO over B^[4]–NBO bonding leads to glass networks with 1.7–2 times more B^[4]–O–Si linkages compared with B^[3]–O–Si, as estimated from dipolar second moments obtained from either ¹¹B{²⁹Si} REDOR NMR experiments or MD simulations.

The diminished B^[4]–O–Si bonding upon Mg²⁺ incorporation is most evident for the MgNa4.0–0.75 glass network, which exhibits similar numbers of B^[4]–O–Si and B^[3]–O–Si linkages, whereas the Na4.0–0.75 analog features ≈1.4 times more B^[4]–O–Si bridges than B^[3]–O–Si. The lower number of B^[4]–O–Si linkages in the MgNa4.0–0.75 structure is also mirrored in its ²⁹Si MAS NMR spectrum, whose broad resonance is centered around a chemical shift close to that of vitreous SiO₂ but stemming from ²⁹Si–O–Si as well as²⁹Si–O–B^[3] environments, both of which resonate at very similar chemical shifts.^82,110−113 We found no indications of phase separation of the MgNa4.0–0.75 glass specimen, which most likely constitutes a single amorphous borosilicate phase with emphasized B^[3]/Si contacts as compared to its NBO-richer MgNa4.0–2.1 counterpart or any NaK–R glass structure. The findings herein of reduced B^[4]/Si contacts in the Mg-bearing glasses echo those deduced from heteronuclear ¹¹B/²⁹Si NMR experimentation on heat-treated BS glasses⁴⁵⁻⁴⁸ that are often taken to imply separation into Si and B dominated phases from face-value interpretations of routine infrared, Raman, or ²⁹Si NMR spectra.

More detailed information about the borate environments and the Si/B^[p] intermixing require reliable ¹¹B MAS NMR spectral deconvolutions, which are far from straightforward concerning the ¹¹B^[3] resonances^22,113 but could give more quantitative information about the M₂(B^[4]–Si) dipolar second moments for the B^[4](2Si), B^[4](3Si), and B^[4](4Si) environments that coexist in the glass and together yielding each average M₂(B^[4]–Si) value. This topic, along with the prospects for realistic ¹¹B NMR spectral deconvolutions, will be discussed in upcoming publications.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpcb.4c01840.

• Discussion on the glass-composition dependence of Z̅_Na and the relationship between the dipolar second moment M₂(F–F′) and the number of F–O–F′ linkages in the glass; tables with cation field strengths, MD simulation parameters, the dependence on M₂ on structural parameters, and MD-derived F–O–F′ bond lengths (PDF)

Author Present Address

⁴ Science Division, New York University Abu Dhabi, P.O. Box 129188, Abu Dhabi, United Arab Emirates

The authors declare no competing financial interest.