(4)
where x(j) and M(j) are the mole fraction and molar mass of oxide j, respectively, and ρ the density of the glass.
As for Poisson's ratio μ, it is calculated as
(5)
The manner in which Young's modulus varies when a glass composition of (74 − x)SiO2, 10 CO, 16 Na2O (by wt) is modified by an addition of x wt % of another oxide is illustrated in Figure 8. Young's modulus can be raised by the addition of P, Li, B, Zr, Mg, and Al oxides, and in contrast lowered by oxides of heavy mono‐ and divalent ion oxides. True, such small additions do not yield major overall effects but the tendency is clearly shown.
4.3 Thermal Expansion Coefficient
Incremental oxide factors for the calculation of the thermal expansion coefficient are compiled in Table 4. Again, the factors are taken from a widely accepted earlier publication [24], see also [15]. When inspecting the entries in Table 4, the reader will notice a number of conditions that have to be obeyed for specific compositions. These conditions reflect intrinsic structural changes, which can be understood only if the chapters in Part II are consulted. As an example, the sophisticated condition for the increment of boron oxide by the factor φ reflects the expected coordination change between [BO3] and [BO4] units in the glass (Chapter 7.6). Predictions based on these factors yield estimates of the thermal expansion coefficient within an error margin of ±0.3 ppm/K. The effect of addition of x wt % of another oxide is shown in Figure 9 for the base glass composition 74 SiO2·10 CaO·(16 − x) Na2O. Clearly, alkali oxides raise the thermal expansion coefficient whereas especially oxides of highly charged ions decrease it. Iso lines of the expansion coefficient within the ternary sodium borosilicate system are plotted in Figure 10. Optional compositions for so‐called “hard” glasses with α = 4 ppm/K are marked. The base composition 6 Na2O·10 B2O3·84 SiO2 may be considered as a starting point for further adjustment. When taking 6 Na2O → 4 Na2O, 10 B2O3 → 13 B2O3, 84 SiO2 → 81 SiO2 + 2 Al2O3, then the composition obtained is indeed very similar to those of the commercial products Duran or Pyrex having a thermal coefficient of expansion α20–300 = 3.3 ppm/K between 20 and 300 °C.
Table 4 Empirical factors for the calculation of the thermal expansion coefficient α20–300 in ppm/K; it is calculated from the molar fractions of oxides j like α20–300 = ∑x(j)·α(j).
| Oxide j | Increment α(j) | Condition |
|---|---|---|
| SiO2 | 10.5–10·x(SiO2) | x(SiO2) > 0.67 |
| 3.8 | Otherwise | |
| TiO2 | 10.5–15·x(SiO2) | 0.5 < x(SiO2) < 0.8 |
| ZrO2 | −6.0 | |
| Al2O3 | −3.0 | |
| B2O3 | −1.26·φ | φ < 4 |
| −5.0 | Otherwise | |
| MgO | 6.0 | |
| CaO | 13.0 | |
| BaO | 20.0 | |
| MnO | 10.5 | |
| ZnO | 5.0 | |
| PbO | 13.0 | ∑ x(R2O) < 0.03 |
| 13.0 | (1/ψ)·∑ x(R2O) < 3 | |
| 11.5 + 0.5·∑ x(R2O) | Otherwise | |
| Li2O | 27.0 | |
| Na2O | 41.0 | in Na2O–SiO2 |
| 39.5 | Otherwise | |
| K2O | 46.5 | in K2O–SiO2 |
| 42.0 | Otherwise |
φ = (Na2O + K2O + BaO + 0.7·CaO + 0.7·PbO + 0.3·Li2O + 0.3·MgO + 0.3·ZnO − Al2O3)/B2O3;
ψ = ∑ (RO + R2O3 + R2O5);
Oxide amounts to be inserted as molar fractions.
Figure 9 Change of the thermal expansion coefficient α20–300 in the base glass composition 74 SiO2 10 CaO 16 Na2O upon the replacement of x wt % of calcia by another oxide.