# Nonlinear composition dependence of structure-related Young's modulus in tin boro-phosphate glasses

K. Izumi, K. Mitsui, R. Miyoshi, Gregory Tricot, A. Saitoh

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K. Izumi, K. Mitsui, R. Miyoshi, Gregory Tricot, A. Saitoh. Nonlinear composition dependence of structure-related Young's modulus in tin boro-phosphate glasses. Journal of the Ceramic Society of Japan, 2025, J. Ceram. Soc. Jpn., 133, 10.2109/jcersj2.24115. hal-04977171

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# Nonlinear composition dependence of structure-related Young's modulus in tin boro-phosphate glasses

Kosuke Izumi<fnref n="1" />, Kazuki Mitsui<fnref n="1" />, Ryuta Miyoshi<fnref n="1" />, Grégory Tricot<fnref n="2" /> and Akira Saitoh<fnref n="1" />,†

We have studied the effect of B₂O₃ substitution by P₂O₅ on the elastic property of four composition series of SnO–P₂O₅–B₂O₃ glasses. The nonlinear evolution of Young's modulus with increasing B₂O₃/P₂O₅ substitution has substantial implications for understanding the elastic properties of the oxide glasses. The observed Young's modulus can be simulated using structural information based on the Makishima-Mackenzie model. The proportions of tetrahedral and trigonal units in the network were elucidated using high-field ¹¹B and ³¹P MAS-NMR spectroscopy, but this does not entirely match the observed Young's modulus. When the simulation is performed, satisfactory results are obtained so that some of the PO₄–BO₄-related bonds are not counted twice. However, the specific type of local structure, mainly consisting of tetrahedral units, is discussed.

Key-words : Boro-phosphate glass, Young's modulus, Structure

[Received November 15, 2024; Accepted December 6, 2024; Published online January 10, 2025]

## 1. Introduction

We have recently reported that the properties of boro-phosphate glasses are linked to their short-range structure.<fnref n="1" />) The structure is widely defined by the connectivity of the phosphate and/or borate units that form the glass network and the oxygen coordination number of the center cations, though strictly defined as the bond length, bond angle, and dihedral angle between the cation and the adjacent oxygen. In the case of invert glasses, where the characteristics of the modifier ions are reflected, the latter is strongly related to the properties.

The elastic properties of oxide glasses, especially Young's modulus, are empirically expressed as the product of the ion filling factor and the dissociation energy of the unit according to Makishima and Mackenzie (MM)<fnref n="2" />) developed a model to predict Young's modulus, *E*, by considering the electrostatic attraction between oxide ions and metal cations and the atomic packing fraction of those ions, $V_i$, according to

$$E = 2 \sum_{i} V_{i} \cdot D_{i} X_{i}, \qquad (1)$$

where $D_i$ is the dissociation energy per unit volume of the oxide component “i” and $X_i$ is the oxide molar fraction, calculated from tabulated ion sizes. Inaba et al.³) noted that the use of oxide dissociation energies did not properly account for the different types of bonds associated with

glass-forming oxides like B₂O₃ and P₂O₅. They have modified the MM model using dissociation energies calculated from the number of oxygen bonds linking a metal cation to a neighboring cation. The former is related to the density defined by the short-range structure, while the latter is the summation of the binding energies of the constituent units.

In a recent study of the mechanical properties of BaO–SnO–P₂O₅ glasses by Shimizu et al.,⁴) the Inaba model was modified by using metal-oxygen bond distances and metal-ion coordination numbers determined from X-ray and neutron diffraction studies,⁵⁻⁷) to calculate ionic packing fractions ($V_p$). In addition, Okamoto et al.⁸) modified the dissociation energies used by Inaba et al. to account for different coordination environments, particularly for the pyramidal and bi-pyramidal Sn<fnref n="2" />⁺-polyhedra, and to account for the increased rigidity of PO₄³⁻ (Q⁰) tetrahedra, compared to tetrahedra that link to neighboring P-tetrahedra through one (Q<fnref n="1" />) or two (Q<fnref n="2" />) bridging oxygen bonds. Here, $n$ (= 0–2) in Qⁿ is the number of bridging oxygens on a P-tetrahedron.

To date, we have reported the relationship between the density, glass transition temperature, refractive index, and short-range structure in the tin boro-phosphate glasses with varying SnO compositions from 50 mol % of SnO (abbreviated as 50SnO hereafter) to 72SnO in phosphate concentrations.<fnref n="1" />) The B₂O₃ substitution effect in pyrophosphate glasses realized cross-linking connectivity like –PO₄–BO₄–PO₄– with shorter phosphate chains with three domains in the evolution of properties.⁹,¹⁰)

This study aims to identify a relationship between the elastic property of Young's modulus and anionic chain

<footnote n="1">¹Graduate School of Science and Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama 790-8577, Japan</footnote>

<footnote n="2">²University of Lille, CNRS, UMR 8516, Laboratoire de Spectrochimie Infrarouge et Raman, F-59655 Lille, France</footnote>

<footnote>† Corresponding author: A. Saitoh; E-mail: asaito@ehime-u.ac.jp</footnote>
structures in different tin boron-phosphate glass systems. We assert that the nonlinear change in Young's modulus is correlated with the nonlinear change in the local structure of the glass via the MM model. Specifically, in the model, the tetrahedral units of phosphate and borate, and triangle unit of borate play a decisive role in determining $V_i$ and $D_i$. For this purpose, Young's moduli for different SnO concentration systems were calculated using the structural model obtained from the ³¹P and ¹¹B MAS-NMR spectra analyses.

## 2. Experimental procedure

The preparation conditions of the SnO–P₂O₅–B₂O₃ glasses are reported elsewhere.¹,⁹,¹¹,¹²) All glass samples were made by a conventional melt quenching method in a silica tube furnace under flowing argon to avoid oxidation of Sn²⁺.⁹) Four series of glass compositions were prepared from reagent grade SnO (99.5 %), P₂O₅ (99.99 %), and B₂O₃ (99.9 %). The 50SnO–(50 – x)P₂O₅–xB₂O₃, 60SnO–(40 – x)P₂O₅–xB₂O₃, 67SnO–(33 – x)P₂O₅–xB₂O₃, and 72SnO–(28 – x)P₂O₅–xB₂O₃ systems present constant SnO content (50, 60, 67, and 72 mol %) and B₂O₃ content from x = 0–50, x = 0–40, x = 0–33, and x = 0–15 in molar %, respectively. The last system can contain B₂O₃ content from x = 0–15 mol % caused by the limit of devitrified composition.

Young's modulus measurements were made using an ultrasonic system with a 10 MHz transducer. These values were reproducible to ±2 GPa. The density (ρ) of the bulk sample was determined by the Archimedes method with kerosene as an immersion liquid for the glasses at room temperature. Depending on composition, the estimated uncertainties due to sample fluctuation were ±0.01–0.03 g/cm³.

## 3. Results and discussion

**Figure 1** shows observed Young's modulus for the 50SnO–(50 – x)P₂O₅–xB₂O₃, 60SnO–(40 – x)P₂O₅–xB₂O₃, 67SnO–(33 – x)P₂O₅–xB₂O₃, and 72SnO–(28 – x)P₂O₅–xB₂O₃ glasses. Young's modulus in the left axis at the same B₂O₃ substitution decreases with the SnO concentration increase. The evolutions similarly behave in three trends against B₂O₃ substitution: steep, negligible, and moderate evolution. This trend is also seen in glass transition temperature and refractive index.¹,¹³) In each system, Young's modulus generally increases with the increase of B₂O₃ concentration, as well as the density depicted in the right axis. Because the density (ρ) and molecular weight (MW), based on the batched compositions, according to

$$V_i = \frac{N_A \sum x_i \cdot \left( \frac{4}{3} \pi r_i^3 \right)}{MW/\rho}, \quad (2)$$

where $N_A$ is Avogadro number, $r_i$ is the ionic radius of constituting ions.¹⁴)

**Table 1** lists the proportions of Qⁿ units ($n$ = 0–2), Q⁰/¹B, and BO₃ (B³) and BO₄ (B⁴) units for the 50SnO–(50 – x)P₂O₅–xB₂O₃, 60SnO–(40 – x)P₂O₅–xB₂O₃, 67SnO–(33 – x)P₂O₅–xB₂O₃, and 72SnO–(28 – x)P₂O₅–xB₂O₃ glass systems, which were determined from ³¹P and ¹¹B MAS-NMR spectroscopies.¹)

**Table 2** presents the unit models, atomic packing fraction ($V_i$), and dissociation energy ($D_i$) for the tin boron-phosphate glasses used in the calculation depicted in Fig. 3. The calculations, based on the values by the MM equation and referencing the values from the papers,²,⁷) are of significant importance in understanding the properties of these glasses. For the B ion, the ionic radius of 0.16 Å

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Observed Young's modulus on B₂O₃ concentration for (a) 50SnO–(50 – x)P₂O₅–xB₂O₃, (b) 60SnO–(40 – x)P₂O₅–xB₂O₃, (c) 67SnO–(33 – x)P₂O₅–xB₂O₃, and (d) 72SnO–(28 – x)P₂O₅–xB₂O₃ glass series. The regions (I)–(III) are divided according to B₂O₃ concentration. The density against B₂O₃ concentration is also shown on the right axis. The lines are guides for the eyes.</figcaption></figure>
<table><caption>**Table 1.** Proportions of Q'' (n = 0, 1, 2) (n: the number of bridging oxygens on a P-tetrahedron), Q<sup>1/0</sup><sub>B</sub>, and borate units [trigonal (B<sup>3</sup>) and tetrahedral (B<sup>4</sup>) units] determined by <sup>31</sup>P and <sup>11</sup>B MAS-NMR<sup>1)</sup> for (a) 50SnO–(50–x)P<sub>2</sub>O<sub>5</sub>–xB<sub>2</sub>O<sub>3</sub>, (b) 60SnO–(40–x)P<sub>2</sub>O<sub>5</sub>–xB<sub>2</sub>O<sub>3</sub>, (c) 67SnO–(33–x)P<sub>2</sub>O<sub>5</sub>–xB<sub>2</sub>O<sub>3</sub>, and (d) 72SnO–(28–x)P<sub>2</sub>O<sub>5</sub>–xB<sub>2</sub>O<sub>3</sub> glasses. The B<sup>4</sup> units are further classified into four B<sup>4</sup><sub>(1)</sub>–B<sup>4</sup><sub>(4)</sub>.<sup>1,9,10)</sup> The uncertainties of the fractions are ±0.2. For each of the phosphate and borate units, the total proportions are 100%</caption><thead><tr><th rowspan="2">x (mol %)</th><th colspan="5">Proportions of phosphate units (%)</th><th colspan="6">Proportions of borate units (%)</th></tr><tr><th>Q<sup>2</sup></th><th>Q<sup>1</sup></th><th>Q<sup>0</sup></th><th>Q<sup>1</sup><sub>B</sub></th><th>Q<sup>0</sup><sub>B</sub></th><th>B<sup>3</sup><sub>(1)</sub></th><th>B<sup>3</sup><sub>(2)</sub></th><th>B<sup>4</sup><sub>(1)</sub></th><th>B<sup>4</sup><sub>(2)</sub></th><th>B<sup>4</sup><sub>(3)</sub></th><th>B<sup>4</sup><sub>(4)</sub></th></tr></thead><tbody><tr><td colspan="12" style="text-align: left;">(a) 50SnO–(50–x)P<sub>2</sub>O<sub>5</sub>–xB<sub>2</sub>O<sub>3</sub></td></tr><tr><td>0</td><td>95.8</td><td>4.2</td><td>0</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>10</td><td>42.8</td><td>21.1</td><td>0</td><td>36.1</td><td>0</td><td>0</td><td>0</td><td>93.4</td><td>6.6</td><td>0</td><td>0</td></tr><tr><td>15</td><td>25.0</td><td>42.5</td><td>2.3</td><td>25.2</td><td>5.1</td><td>3.8</td><td>0</td><td>81.7</td><td>13.1</td><td>1.4</td><td>0</td></tr><tr><td>25</td><td>4.7</td><td>43.1</td><td>5.9</td><td>11.7</td><td>34.6</td><td>23.7</td><td>17.9</td><td>13.3</td><td>21.1</td><td>21.1</td><td>3.0</td></tr><tr><td>40</td><td>0</td><td>13.2</td><td>10.1</td><td>0</td><td>76.7</td><td>38.9</td><td>26.4</td><td>0.8</td><td>8.8</td><td>18.8</td><td>6.4</td></tr><tr><td>45</td><td>0</td><td>13.5</td><td>11.7</td><td>0</td><td>74.9</td><td>39.2</td><td>26.6</td><td>0.4</td><td>8.2</td><td>19.5</td><td>6.1</td></tr><tr><td>50</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>39.3</td><td>26.5</td><td>0</td><td>6.0</td><td>20.3</td><td>8.0</td></tr><tr><td colspan="12" style="text-align: left;">(b) 60SnO–(40–x)P<sub>2</sub>O<sub>5</sub>–xB<sub>2</sub>O<sub>3</sub></td></tr><tr><td>0</td><td>54.9</td><td>45.1</td><td>0</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>5</td><td>9.0</td><td>65.7</td><td>2.2</td><td>11.9</td><td>10.7</td><td>0</td><td>0</td><td>96.3</td><td>2.6</td><td>1.0</td><td>0</td></tr><tr><td>10</td><td>4.0</td><td>56.9</td><td>6.2</td><td>10.1</td><td>22.5</td><td>9.0</td><td>0</td><td>53.1</td><td>29.7</td><td>7.0</td><td>1.2</td></tr><tr><td>15</td><td>0</td><td>39.8</td><td>6.0</td><td>3.9</td><td>50.3</td><td>29.3</td><td>10.2</td><td>11.8</td><td>28.2</td><td>13.2</td><td>7.3</td></tr><tr><td>20</td><td>0</td><td>21.4</td><td>10.5</td><td>0</td><td>68.2</td><td>37.7</td><td>16.8</td><td>3.12</td><td>23.2</td><td>14.4</td><td>4.8</td></tr><tr><td>25</td><td>0</td><td>12.4</td><td>8.2</td><td>0</td><td>79.4</td><td>43.3</td><td>19.0</td><td>1.6</td><td>18.3</td><td>14.0</td><td>3.1</td></tr><tr><td>30</td><td>0</td><td>10.6</td><td>6.4</td><td>0</td><td>83.0</td><td>43.8</td><td>20.5</td><td>0.7</td><td>15.8</td><td>15.0</td><td>4.2</td></tr><tr><td>35</td><td>0</td><td>3.5</td><td>5.4</td><td>0</td><td>91.1</td><td>40.7</td><td>24.2</td><td>1.0</td><td>13.6</td><td>15.9</td><td>4.6</td></tr><tr><td colspan="12" style="text-align: left;">(c) 67SnO–(33–x)P<sub>2</sub>O<sub>5</sub>–xB<sub>2</sub>O<sub>3</sub></td></tr><tr><td>0</td><td>0</td><td>94.0</td><td>6.0</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>1</td><td>0</td><td>64.1</td><td>22.9</td><td>0</td><td>13.0</td><td>0</td><td>0</td><td>97.1</td><td>2.9</td><td>0</td><td>0</td></tr><tr><td>3</td><td>0</td><td>37.0</td><td>27.9</td><td>0</td><td>35.1</td><td>0</td><td>0</td><td>86.3</td><td>10.1</td><td>1.7</td><td>0</td></tr><tr><td>5</td><td>0</td><td>15.9</td><td>28.1</td><td>0</td><td>50.9</td><td>0</td><td>0</td><td>60.6</td><td>28.8</td><td>4.7</td><td>0</td></tr><tr><td>7</td><td>0</td><td>15.9</td><td>28.1</td><td>0</td><td>56.0</td><td>0</td><td>0</td><td>21.8</td><td>49.4</td><td>5.2</td><td>0</td></tr><tr><td>20</td><td>0</td><td>5.0</td><td>2.0</td><td>0</td><td>93.0</td><td>0</td><td>0</td><td>0.5</td><td>10.5</td><td>24.4</td><td>2.4</td></tr><tr><td>25</td><td>0</td><td>3.0</td><td>1.5</td><td>0</td><td>95.5</td><td>0</td><td>0</td><td>0.6</td><td>8.0</td><td>23.3</td><td>4.9</td></tr><tr><td>30</td><td>0</td><td>2.0</td><td>2.0</td><td>0</td><td>96.0</td><td>0</td><td>0</td><td>0.2</td><td>6.8</td><td>23.6</td><td>6.6</td></tr><tr><td colspan="12" style="text-align: left;">(d) 72SnO–(28–x)P<sub>2</sub>O<sub>5</sub>–xB<sub>2</sub>O<sub>3</sub></td></tr><tr><td>0</td><td>0</td><td>26.1</td><td>73.9</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>1</td><td>0</td><td>16.7</td><td>63.4</td><td>0</td><td>19.9</td><td>0</td><td>0</td><td>82.5</td><td>17.5</td><td>0</td><td>0</td></tr><tr><td>2</td><td>0</td><td>8.7</td><td>58.3</td><td>0</td><td>33.0</td><td>0</td><td>8.6</td><td>47.7</td><td>37.0</td><td>6.7</td><td>0</td></tr><tr><td>5</td><td>0</td><td>2.7</td><td>32.2</td><td>0</td><td>65.1</td><td>0</td><td>29.3</td><td>13.6</td><td>48.0</td><td>7.6</td><td>1.6</td></tr><tr><td>7</td><td>0</td><td>3.7</td><td>23.8</td><td>0</td><td>72.6</td><td>35.0</td><td>19.7</td><td>2.1</td><td>25.9</td><td>14.4</td><td>3.0</td></tr><tr><td>10</td><td>0</td><td>3.4</td><td>18.2</td><td>0</td><td>78.4</td><td>45.1</td><td>19.7</td><td>0.9</td><td>10.6</td><td>19.4</td><td>4.4</td></tr><tr><td>12</td><td>0</td><td>3.8</td><td>11.3</td><td>0</td><td>84.9</td><td>47.4</td><td>20.0</td><td>0.4</td><td>8.4</td><td>17.9</td><td>5.4</td></tr><tr><td>15</td><td>0</td><td>2.3</td><td>5.4</td><td>0</td><td>92.3</td><td>46.5</td><td>21.6</td><td>0</td><td>9.1</td><td>15.9</td><td>7.0</td></tr></tbody></table>

was used when oxygen was in a three-coordinate state, and the ionic radius of 0.26 Å was used when oxygen was in a four-coordinate state. Indeed, it is meaningless to count the ionic radii of P<sup>5+</sup> and B<sup>3+</sup> ions, and the ionic radius of oxygen (1.40 Å) influences to determine V<sub>i</sub>.

According to the <sup>31</sup>P MAS-NMR spectra,<sup>1,9,10)</sup> a three-component model based on two phosphate sites is not connected to boron (Q<sup>1</sup> and Q<sup>0</sup> species, a P connected to Sn and one P atoms and a P connected to Sn and two P atoms) and one site is involved in mixed P–O–B bond (Q<sup>1/0</sup><sub>B</sub> species, a P connected to Sn atoms and engaged in the P–O–B linkages).<sup>1)</sup> This model could be directly applied to the 72SnO system, while the 50SnO and 60SnO decomposition require the addition of a Q<sup>1</sup><sub>B</sub> species<sup>1,15)</sup> in the model.

Specifically, the same value of V<sub>i</sub> is used even though the oxygen coordination number of SnO can be three or four,<sup>3)</sup> so the values used are 6.0 cm<sup>3</sup>/mol for three-coordinated oxygen and 8.0 cm<sup>3</sup>/mol for four-coordinated oxygen.<sup>8)</sup> It is known experimentally that the number of

oxygen atoms coordinating with Sn<sup>2+</sup> is three or four depending on the amount of SnO, so the value of V<sub>i</sub> also changes depending on the local structure model.

Also, according to the <sup>11</sup>B MAS-NMR spectra,<sup>1,9,10)</sup> a triangle B and four tetrahedral B signals can be observed, indicating the presence of four species in the structure. The first species, B<sup>4</sup><sub>(1)</sub>, is assigned to a borate unit only connected to tetrahedral BO<sub>4</sub> and/or PO<sub>4</sub> species, and the second species B<sup>4</sup><sub>(2)</sub>, is assigned to borate connected to at least one B<sup>3</sup> unit. The third species B<sup>4</sup><sub>(3)</sub> species can be assigned to borate connected to Sn<sup>2+</sup> ions. The fourth species, B<sup>4</sup><sub>(4)</sub>, only present at low intensity in the glasses, has not been elucidated. Therefore, in Table 2, a unit model for B<sup>4</sup><sub>(4)</sub> is absent and does not account for the B<sup>4</sup><sub>(4)</sub> proportion in Young’s modulus calculation. Here, we emphasize that the V<sub>i</sub> and D<sub>i</sub> values for B<sup>4</sup><sub>(1)</sub> and B<sup>4</sup><sub>(2)(3)</sub> are significantly different. The reason for this is that, as will be shown in the inset of Fig. 5, B<sup>4</sup><sub>(1)</sub> is assumed to have a local structure in which a PO<sub>4</sub> unit (Q<sup>0</sup>) is surrounded by four BO<sub>4</sub> units.
<table><caption>**Table 2.** The reported and modified values for $V_i$ and $D_i$ in this study. The values are taken from papers by Inaba-Fujino³) and Okamoto et al.⁸) for the calculations. The simple unit models¹) used to calculate $V_i$ and $D_i$, as well as the oxygen coordination number (C.N.), are presented. The ionic radii were used when the Sn ion was +2, the P ion was 5+, the B ion was +3, and the oxygen ion was -2¹⁴)</caption><thead><tr><th rowspan="2">Oxide</th><th rowspan="2">C.N.</th><th rowspan="2">Unit model</th><th colspan="2">V<sub>i</sub> (cm<sup>3</sup>/mol)</th><th colspan="2">D<sub>i</sub> (kJ/cm<sup>3</sup>)</th></tr><tr><th>Previous studies<sup>3)</sup></th><th>This study</th><th>Previous studies<sup>3)</sup></th><th>This study</th></tr></thead><tbody><tr><td rowspan="2">SnO</td><td>3</td><td></td><td>8.9</td><td>6.0</td><td>39.9</td><td>39.9</td></tr><tr><td>4</td><td></td><td>8.9</td><td>8.0</td><td>39.9</td><td>39.9</td></tr><tr><td>P<sub>2</sub>O<sub>5</sub></td><td>4</td><td></td><td>34.6</td><td>—</td><td>28.2</td><td>—</td></tr><tr><td>Q<sup>2</sup></td><td>4</td><td></td><td>34.6</td><td>34.6</td><td>28.2</td><td>28.2</td></tr><tr><td>Q<sup>1</sup></td><td>4</td><td></td><td>34.6</td><td>34.6</td><td>28.2</td><td>28.2</td></tr><tr><td>Q<sup>0</sup></td><td>4</td><td></td><td>34.6</td><td>34.6</td><td>56.4</td><td>56.4</td></tr><tr><td>Q<sup>1</sup><sub>B</sub></td><td>4</td><td></td><td>—</td><td>90.0</td><td>—</td><td>139.2</td></tr><tr><td>Q<sup>0</sup><sub>B</sub></td><td>4</td><td></td><td>—</td><td>55.4</td><td>—</td><td>139.2</td></tr><tr><td rowspan="2">B<sub>2</sub>O<sub>3</sub></td><td>3</td><td></td><td>15.2</td><td>15.2</td><td>15.6</td><td>15.6</td></tr><tr><td>4</td><td></td><td>20.8</td><td>20.8</td><td>82.8</td><td>82.8</td></tr><tr><td>B<sup>4</sup><sub>(1)</sub></td><td>4</td><td></td><td>—</td><td>159.2</td><td>—</td><td>195.6</td></tr><tr><td>B<sup>4</sup><sub>(2)</sub></td><td>4</td><td></td><td>—</td><td>36.0</td><td>—</td><td>98.4</td></tr><tr><td>B<sup>4</sup><sub>(3)</sub></td><td>4</td><td></td><td>—</td><td>20.8</td><td>—</td><td>82.8</td></tr><tr><td>B<sup>4</sup><sub>(4)</sub></td><td>4</td><td>Not elucidated.</td><td>—</td><td>—</td><td>—</td><td>—</td></tr></tbody></table>

Because an aggregated structure consisting of five tetrahedral units is assumed, the $V_i$ and $D_i$ values for $B^4_{(1)}$ are particularly large rather than those for $B^4_{(2),(3)}$.

**Figures 2(a)–2(d)** compares the observed and calculated Young's moduli (E) for the 50SnO-(50–x)P₂O₅–xB₂O₃, 60SnO-(40–x)P₂O₅–xB₂O₃, 67SnO-(33–x)P₂O₅–xB₂O₃, and 72SnO-(28–x)P₂O₅–xB₂O₃ glass systems. The calculated values for $V_i$ and $D_i$ in previous studies³,⁸) are used as shown in Table 2. Here, we assume that the values of $V_i$ and $D_i$ are the same for three- and four-coordinated $Sn^{2+}$. However, the coordination number of oxygen of $Sn^{2+}$ varies from four to three with increasing SnO concentration in binary phosphate glasses.⁶) In binary SnO–B₂O₃ glasses, on the other hand, the oxygen coordination number is supposed to be constant at four, irrespective of the SnO concentration, from the corresponding crystal.¹⁶⁻¹⁸) Therefore, even with compositions in which P₂O₅ is substituted by B₂O₃, the influence of the $V_i$ and $D_i$ values regarding the coordination number of $Sn^{2+}$ on the elastic property will be small. This is because, although there is no data exhibiting that the oxygen coordination number of $Sn^{2+}$ in the tin boro-phosphate glasses depends on the concentration of SnO or the B₂O₃/P₂O₅ ratio, the values for SnO in the 50SnO-(50–x)P₂O₅–xB₂O₃ system

were 8.0 cm³/mol for four-coordinated oxygen for $V_i$ and 39.9 kJ/cm³ for $D_i$. For the 72SnO-(28–x)P₂O₅–xB₂O₃ system, 6.0 cm³/mol for three-coordinated oxygen for $V_i$ and 39.9 kJ/cm³ for $D_i$ (see Table 2). In the other composition systems of the 60SnO-(40–x)P₂O₅–xB₂O₃ and 67SnO-(33–x)P₂O₅–xB₂O₃, we assumed that the oxygen coordination numbers of $Sn^{2+}$ are 3.6 and 3.3, respectively, the values of $V_i$ were set to 7.2 cm³/mol and 6.6 cm³/mol, which are proportional to the oxygen coordination numbers.

It's worth noting that the measured Young's moduli often do not align with the existing $V_i$ and $D_i$ in every series. This discrepancy is particularly evident when B₂O₃ replaces P₂O₅, a composition dependence not reflected in the calculation. Therefore, accurately predicting the observed Young's modulus is not just a goal but a necessity, requiring a reflection of the phosphate and borate structures already obtained by MAS-NMR.

**Figure 3** shows the calculated Young's moduli for the four series of tin boro-phosphate glass systems. The parameters ($V_i$, $D_i$) described in the columns of the present study in Table 2 have been input without any simulation. The following parameters are prepared: the constituent models for tetrahedral PO₄ and BO₄ units and triangle BO₃
<figure><img src="image_3.png" /><figcaption>Fig. 2. Comparisons of Young's moduli (E) for the observed and calculated values for (a) 50SnO–(50–x)P₂O₅–xB₂O₃, (b) 60SnO–(40–x)P₂O₅–xB₂O₃, (c) 67SnO–(33–x)P₂O₅–xB₂O₃, and (d) 72SnO–(28–x)P₂O₅–xB₂O₃ glass systems. The calculated values are used to combine data derived from the previous papers.³,⁸</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Calculated Young's moduli using the $V_i$ and $D_i$ in this study, based on the structural information, for (a) 50SnO–(50–x)P₂O₅–xB₂O₃, (b) 60SnO–(40–x)P₂O₅–xB₂O₃, (c) 67SnO–(33–x)P₂O₅–xB₂O₃, and (d) 72SnO–(28–x)P₂O₅–xB₂O₃ glass systems, together with the observed ones. The definition of regions (I)–(III) follows the Refs. 1, 9, 10, 13. The solid and dashed lines are guides for the eyes.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 4. Best-fitted values (red triangles) of Young's moduli for (a) 50SnO–(50–x)P₂O₅–xB₂O₃, (b) 60SnO–(40–x)P₂O₅–xB₂O₃, (c) 67SnO–(33–x)P₂O₅–xB₂O₃, and (d) 72SnO–(28–x)P₂O₅–xB₂O₃ glass systems by a simulation, together with the observed ones. Insets are comparisons of Young's moduli (E) for the observed and simulated values.</figcaption></figure>

units are first established. Next, the minimal connectivity via a corner-sharing manner to satisfy the structural models are supposed. Lastly, the $V_i$ and $D_i$ are automatically calculated.

There is a significant discrepancy between the measured values in all regions with small and large B₂O₃ substituted compositions for all systems. Depending on how we estimate the structural models, perhaps the calculated $V_i$ is within ±10%, so this is not the reason for the discrepancy seen in Fig. 3. But for $D_i$, for example, the BO₄ unit is calculated as 82.8 kJ/cm³ while the BO₃ unit is 15.6 kJ/cm³.³) The difference is more than five times. The proportion of BO₄ units is minor in high B₂O₃ composition [region (III)], and conversely, the proportion of BO₄ units is present in low B₂O₃ composition [region (I)].¹⁰) This underscores the need for accurate structural models in our research.

In addition, the unfitted reasons with the observed values are the over- or under-estimation due to double counting of the parameters. Namely, in terms of ³¹P MAS-NMR spectra, Q⁰B is a structure in which the BO₄ tetrahedron is connected to the PO₄ tetrahedron by a corner-sharing of oxygen, whereas B⁴(1) is the same structure in terms of ¹¹B MAS-NMR spectra, and both structures are essentially the same in the region (I). Thus, not only the Q⁰B unit and B⁴(1) but also, for example, the BO₃ or BO₄ units may overlap with B⁴(2) and the BO₄ units with B⁴(3). Alternatively, B⁴(1) has been elucidated from the analysis of ³¹P MAS-NMR spectra to be a giant cluster containing four corner-shared oxygens with four PO₄ units centered on a BO₄ unit, but the Q⁰B above considers only the connected units of PO₄ and BO₄ tetrahedra. In this case, lower

values against the measured Young's moduli will return after the calculation.

On the other hand, in the region (II), the primary units of Q¹/₀B and B⁴(2) bring about radical decreases in the calculated Young's moduli, even though the measured values show moderate increases. Further, the compositional evolution of the computed values in the region (III) is opposite to the observed ones.

The best-fitted values in Young's modulus are shown in Fig. 4. In this simulation, we assume that the $V_i$ and $D_i$ values for each structure unit are fixed, but the proportions of those units are variable. The variable parameters here are the ratios of each unit (Qⁿ, Q¹/₀B, B⁴(1)–(3)), which have been clarified by analyses of the ³¹P and ¹¹B MAS-NMR spectra. By appropriately increasing or decreasing the proportions of these units when the above variable parameters are changed independently, the observed Young's modulus can be fitted. Using the units $V_i$ and $D_i$ shown in Table 2, we show the possibility of predicting Young's modulus observed from the local structure.

The MM model has succeeded in predicting Young's moduli by multiplying the atomic packing fraction and the dissociation energy of the bonds that consist of the simple unit. In contrast, the information obtained by analyzing NMR spectra¹) is not only the form of the unit but also the form of their connection manner. Here, since there is an overlap between the local structure obtained from the MM model and the structure obtained from the NMR spectra, it was clarified which units were superimposed in the calculation.

However, it is not possible to explain whether the values of $V_i$ and $D_i$ are thermodynamically reasonable. We need
<figure><img src="image_6.png" /><figcaption>Fig. 5. Simulated proportions for the best-fitted values of Young's moduli for the 50SnO-(50-x)P₂O₅-xB₂O₃, 60SnO-(40-x)P₂O₅-xB₂O₃, 67SnO-(33-x)P₂O₅-xB₂O₃, and 72SnO-(28-x)P₂O₅-xB₂O₃ glasses. Colored areas are the local structures that overlap between phosphate- (left) and borate-related (right) units in the simulation. Insets are the corresponding structures that are perhaps calculated in a doubly way.</figcaption></figure>

to discuss whether the local structure that becomes the B⁴(1) is a metastable state. It might be right that the structural fluctuations in the bond length, bond angle, and dihedral angle of local structures such as B⁴(1) are not reflected in Vᵢ and Dᵢ. A detailed discussion of the Madelung constant described in the MM model also seems impossible, accordingly.

**Figure 5** shows the change in the proportions input to the simulation. Here, the units identified from the NMR analyses are fitted using the least-squares method with the Vᵢ and Dᵢ values (Table 2) unchanged, and proportions

of the Q²–Q⁰, Q¹/₀B, B⁴(1)–(3) units are as a free parameter. Note that the B⁴(4) unit, to which no structure was assigned,¹,⁹,¹⁰) was not used as a variable parameter during the simulation. In Figs. 5(a)–5(d), the proportions obtained from the MAS-NMR analyses are supposed to be 1 if there is no correction for the duplicate units. In the best-fitted simulation, the proportions of Q⁰B and B⁴(1) units that relate to the shorter –PO₄–BO₄–PO₄– connectivity are smaller, which means those Vᵢs and Dᵢs are overestimated due to similar structures. Of the structures revealed by the analyses of the NMR spectra,¹) the primary units counted
in duplicate are illustrated and extracted. The most overlapping units are Q⁰B and Q¹B, which contain a four-coordinate borate in the phosphate chain, and B⁴(1) and B⁴(2) of the borate units with large Vᵢ. Secondly, the Q₁ or Q⁰ unit in the phosphate chain possibly overlaps with the B⁴(1) unit. Thus, the main structure units of Q¹ and B⁴(1) in the boro-phosphate glasses overlap significantly in the calculation based on the MM model, and by making corrections that take these into account, it is possible to account for the composition dependence of Young's moduli using the elucidated structures. In order to discuss the observed Young's moduli with the structures, it is necessary to address the excess and deficiency of each unit quantitatively, but it actually seems complicated to perfectly explain the ion filling ratio of the boro-phosphate glasses using individual units.

The effects of B₂O₃ substitution on Young's modulus are almost identical regardless of SnO concentration. Namely, when the amount of B₂O₃ substitution is small, Young's modulus increases rapidly with the formation of tetrahedral bonds between minor BO₄ and major PO₄ units. When the amount of B₂O₃ becomes the same as that of P₂O₅, the effect of forming –PO₄–BO₄– linkages proceeds. The compositional evolutions of the nonlinear Young's modulus reflect the characteristics of the structures as segmented in the region (I)–(III).

Finally, concerns regarding the validity of the Vᵢ and Dᵢ values for borates exist, but we humbly address that it is meaningful to apply the MM model that includes borates as a network component in oxide glasses as a new insight, albeit being applied to phosphate glasses.⁴,⁸) The only variable parameter is the proportions of units that exhibit different local structures. The double counting cannot immediately predict the observed Young's modulus, which would be a limit of the present method using MM models, which infers information on the local structure.

## 4. Conclusions

The effects of B₂O₃ substitution on P₂O₅ are systematically investigated Young's modulus. The composition dependence of Young's modulus and density is similar, and both can be divided into steep increases [region (I)], moderate increases [region (II)], and substantial increases [region (III)]. The anion mixing effects in tin boro-phosphate glass systems have been unveiled by analyzing ³¹P and ¹¹B MAS-NMR spectra. The main results of this study are summarized as follows:

(1) Young's modulus was calculated using the nominal glass composition and analyzed proportions of tetrahedral phosphate and borate units, triangle borate unit, and pyra-

midal and bi-pyramidal divalent tin units based on the Makishima-Mackenzie equation.

(2) The calculated Young's moduli, based on the ion packing fraction (Vᵢ) and dissociation energy (Dᵢ) of the structural units constructed, show deviation from the measured values in all the regions.

(3) The reason the measured Young's modulus cannot be predicted simply by using the elucidated structural unit models with analyzed proportions may be the double-counting of phosphate and borate tetrahedral units, which can be understood by simulations.

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