The potential cannot be generated for certain B2O3-containing compositions. This is because of the way the composition-dependent B-O parameters are determined.
In the original paper, only simpler glass compositions have been investigated and the Dell–Bray–Xiao (DBX) model [1] is used. This only considers the R=Na2O/B2O3 and K=SiO2/B2O3 ratios and from R and K compute the N4 value. This is motivated by NMR measurements that investigate the ratio of 3- and 4-fold coordinated B.
Unfortunately, there exist no unified approach for this (to the best of my knowledge).
Generally, on can motivate this approach by looking at charge compensation. Na (and most likely all other alkali ions) lead to charge compensation and help converting BO₃ → BO₄⁻. For earth-alkali ions the situation is more complicated, because their double negative charge will not purely lead to the same type of charge compensation. Instead, they also lead to non‑bridging oxygens (NBO). From Tuheen & Du, Int. J. Applied Glass Science (2022):
Modifier cations with higher field strength such as Mg²⁺ are less effective for charge compensation of four‑fold coordinated boron and favor NBO generation.
For now, I leave this open for a discussion. I'll have a deeper look into literature soon. Our current approach is as follows for treating the modifiers:
modifier_sum = cLi2O + cNa2O + cK2O + cBeO + cMgO + cCaO + cSrO + cBaO - cAl2O3
K = cSiO2 / cB2O3
R = max(modifier_sum / cB2O3, 0)
Because the mol% of earth-alkali ions is used in the equation above without further modification, this implicitely assumes a factor of 0.5 of how they lead to charge compensation.
By the way, the original error I faced was the following:
"N4 could not be calculated: R=5.999979899093449, K=2.0999819091841037, R_MAX=0.6312488693240065"
[1] Dell WJ, Bray PJ, Xiao SZ. 11B NMR studies and structural modeling of Na2O-B2O3-SiO2 glasses of high soda content. J Non Cryst Solids. 1983, 58 (1), 1-16.
The potential cannot be generated for certain B2O3-containing compositions. This is because of the way the composition-dependent B-O parameters are determined.
In the original paper, only simpler glass compositions have been investigated and the Dell–Bray–Xiao (DBX) model [1] is used. This only considers the R=Na2O/B2O3 and K=SiO2/B2O3 ratios and from R and K compute the N4 value. This is motivated by NMR measurements that investigate the ratio of 3- and 4-fold coordinated B.
Unfortunately, there exist no unified approach for this (to the best of my knowledge).
Generally, on can motivate this approach by looking at charge compensation. Na (and most likely all other alkali ions) lead to charge compensation and help converting BO₃ → BO₄⁻. For earth-alkali ions the situation is more complicated, because their double negative charge will not purely lead to the same type of charge compensation. Instead, they also lead to non‑bridging oxygens (NBO). From Tuheen & Du, Int. J. Applied Glass Science (2022):
For now, I leave this open for a discussion. I'll have a deeper look into literature soon. Our current approach is as follows for treating the modifiers:
Because the mol% of earth-alkali ions is used in the equation above without further modification, this implicitely assumes a factor of 0.5 of how they lead to charge compensation.
By the way, the original error I faced was the following:
[1] Dell WJ, Bray PJ, Xiao SZ. 11B NMR studies and structural modeling of Na2O-B2O3-SiO2 glasses of high soda content. J Non Cryst Solids. 1983, 58 (1), 1-16.