Structural Evaluation of Silicate Melts by Performing Impedance Measurements and Quasichemical Model Calculations


Structures of molten oxides (including silicates) strongly affect their physical properties. Although structural analysis is typically performed experimentally by nuclear magnetic resonance and Raman spectroscopy or theoretically by employing thermodynamic quasichemical and cell models, these techniques are time-consuming, and their results are not sufficiently accurate. In this study, high-precision structural evaluations were performed within a relatively short timeframe by combining impedance measurements with the quasichemical model. For this purpose, correlation equations, including the thermodynamic parameters of the quasichemical model and equivalent circuit components obtained by impedance measurements, were derived. Using the proposed method, the evaluation accuracy was improved by approximately 3.6 times as compared with the data generated by FactSage software.

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  1. 1.

    K.C. Mills, A.B. Fox, ISIJ Int. 43, 1479–1486 (2003)

    CAS  Article  Google Scholar 

  2. 2.

    K. Raj, K.K. Prasad, N.K. Bansal, Nucl. Eng. Des. 236, 914–930 (2006)

    CAS  Article  Google Scholar 

  3. 3.

    K.C. Mills, ISIJ Int. 33, 148–155 (1993)

    CAS  Article  Google Scholar 

  4. 4.

    T. Iida, H. Sakai, Y. Kita, K. Shigeno, ISIJ Int. 40, S110–S114 (2000)

    CAS  Article  Google Scholar 

  5. 5.

    M. Chen, D. Zhang, M. Kou, B. Zhao, ISIJ Int. 54, 2025–2030 (2014)

    CAS  Article  Google Scholar 

  6. 6.

    L. Shartsis, S. Spinner, W. Capps, J. Am. Ceram. Soc. 35, 155–160 (1952)

    CAS  Article  Google Scholar 

  7. 7.

    E.F. Riebling, J. Chem. Phys. 39, 3022–3030 (1963)

    CAS  Article  Google Scholar 

  8. 8.

    Y.E. Lee, D.R. Gaskell, Metall. Trans. 5, 853–860 (1974)

    CAS  Article  Google Scholar 

  9. 9.

    W.D. Kingery, J. Am. Ceram. Soc. 42, 6–10 (1959)

    CAS  Article  Google Scholar 

  10. 10.

    M. Nakamoto, A. Kiyose, T. Tanaka, L. Holappa, M. Hamalainen, ISIJ Int. 47, 38–43 (2007)

    CAS  Article  Google Scholar 

  11. 11.

    H. Hasegawa, T. Kowatari, Y. Shiroki, H. Shibata, H. Ohta, Y. Waseda, Metall. Mater. Trans. B 43B, 1413–1419 (2012)

    Article  CAS  Google Scholar 

  12. 12.

    Y. Kim, K. Morita, ISIJ Int. 54, 2077–2083 (2014)

    CAS  Article  Google Scholar 

  13. 13.

    H. Maekawa, T. Maekawa, K. Kawamura, T. Yokokawa, J. Phys. Chem. 95, 6822–6827 (1991)

    CAS  Article  Google Scholar 

  14. 14.

    S. Sukenaga, K. Kanehashi, H. Shibata, N. Saito, K. Nakashima, Metall. Mater. Trans. B 47B, 2177–2181 (2016)

    Article  CAS  Google Scholar 

  15. 15.

    D.R. Neuville, Chem. Geol. 229, 28–41 (2006)

    CAS  Article  Google Scholar 

  16. 16.

    B. Hehlen, D.R. Neuville, J. Phys. Chem. B 119, 4093–4098 (2015)

    CAS  Article  Google Scholar 

  17. 17.

    A.D. Pelton, M. Blander, Metall. Trans. B 17, 805–815 (1986)

    Article  Google Scholar 

  18. 18.

    Y. Sasaki, K. Ishii, Tetsu-to-Hagane 88, 419–429 (2002)

    CAS  Article  Google Scholar 

  19. 19.

    S. Sukenaga, T. Nagahisa, K. Kanehashi, N. Saito, K. Nakashima, ISIJ Int. 51, 333–335 (2011).

    CAS  Article  Google Scholar 

  20. 20.

    A.A. Ariskin, V.B. Polyakov, Geochem. Int. 46, 467–486 (2008)

    Article  Google Scholar 

  21. 21.

    N. Saito, K. Kusada, S. Sukenaga, Y. Ohta, K. Nakashima, ISIJ Int. 52, 2123–2129 (2012)

    CAS  Article  Google Scholar 

  22. 22.

    Y. Harada, K. Kusada, S. Sukenaga, H. Yamamura, Y. Ueshima, T. Mizoguchi, N. Saito, K. Nakashima, ISIJ Int. 54, 2071–2076 (2014)

    CAS  Article  Google Scholar 

  23. 23.

    Y. Harada, S. Sakaguchi, T. Mizoguchi, N. Saito, K. Nakashima, ISIJ Int. 57, 1313–1318 (2017)

    CAS  Article  Google Scholar 

  24. 24.

    Y. Harada, N. Saito, K. Nakashima, ISIJ Int. 57, 23–30 (2017)

    CAS  Article  Google Scholar 

  25. 25.

    Y. Harada, N. Saito, K. Nakashima, ISIJ Int. 59, 421–426 (2019)

    CAS  Article  Google Scholar 

  26. 26.

    A. Romero-Serrano, A.D. Pelton, Metall. Mater. Trans. B 26B, 305–315 (1995)

    CAS  Article  Google Scholar 

  27. 27.

    A.D. Pelton, S.A. Degterov, G. Eriksson, C. Robelin, Y. Dessureault, Metall. Mater. Trans. B 31B, 651–659 (2000)

    CAS  Article  Google Scholar 

  28. 28.

    A.D. Pelton, P. Chartrand, Metall. Mater. Trans. A 32A, 1355–1360 (2001)

    CAS  Article  Google Scholar 

  29. 29.

    C.W. Bale, E. Belisle, P. Chartrand, S.A. Decterov, G. Eriksson, A.E. Gheribi, K. Hack, I.H. Jung, Y.B. Kang, J. Melancon, A.D. Pelton, S. Petersen, C. Robelin, J. Sangster, P. Spencer, M.A. Van Ende, CALPHAD 54, 35–53 (2016)

    CAS  Article  Google Scholar 

  30. 30.

    P. Wu, G. Eriksson, A.D. Pelton, J. Am. Ceram. Soc. 76, 2059–2064 (1993)

    CAS  Article  Google Scholar 

  31. 31.

    M. Blander, A.D. Pelton, Geochim. Cosmochim. Acta 51, 85–95 (1987)

    CAS  Article  Google Scholar 

  32. 32.

    G. Eriksson, P. Wu, A.D. Pelton, CALPHAD 17, 189–205 (1993)

    CAS  Article  Google Scholar 

  33. 33.

    G. Eriksson, A.D. Pelton, Metall. Trans. B 24, 807–816 (1993)

    Article  Google Scholar 

  34. 34.

    B.C. Riggs, G.E. Plopper, J.L. Paluh, T.B. Phamduy, D.T. Corr, D.B. Chrisey, Proc. SPIE 8371, 83711F1-10 (2012)

    Google Scholar 

  35. 35.

    Y. Harada, S. Sukenaga, N. Saito, K. Nakashima, ISIJ Int. 59, 1956–1965 (2019)

    CAS  Article  Google Scholar 

  36. 36.

    Y. Harada, N. Nishioka, N. Saito, K. Nakashima, ISIJ Int. 60, 42–50 (2020)

    CAS  Article  Google Scholar 

  37. 37.

    A.P. Sandoval, J.M. Feliu, R.M. Torresi, M.F. Suarez-Herrera, RSC Adv. 4, 3383–3391 (2014)

    CAS  Article  Google Scholar 

  38. 38.

    R.H. Fowler, E.A. Guggenheim, Statistical Thermodynamics (Cambridge University Press, Cambridge, 1939), pp. 350–366

    Google Scholar 

  39. 39.

    H. Gaye, J. Welfringer, in Proceedings of the 2nd International Symposium on Metallurgical Slags and Fluxes. ed. by H.A. Fine, D.R. Gaskel (TMS-AIME, Warrendale, 1984), pp. 357–75

    Google Scholar 

  40. 40.

    C. Le Losq, D.R. Neuville, J. Non-cryst. Solids 463, 175–188 (2017)

    Article  CAS  Google Scholar 

  41. 41.

    C. Spearman, Am. J. Psychol. 15, 72–101 (1904)

    Article  Google Scholar 

  42. 42.

    C.I. Merzbacher, B.L. Sherriff, J.S. Hartman, W.B. White, J. Non-cryst. Solids 124, 194–206 (1990)

    CAS  Article  Google Scholar 

  43. 43.

    H. Maekawa, T. Maekawa, K. Kawamura, T. Yokokawa, J. Non-cryst. Solids 127, 53–64 (1991)

    CAS  Article  Google Scholar 

  44. 44.

    T. Matsumiya, K. Shimoda, K. Saito, K. Kanehashi, W. Yamada, ISIJ Int. 47, 802–804 (2007)

    CAS  Article  Google Scholar 

  45. 45.

    T. Iida, Y. Kita, M. Ueda, K. Mori, K. Nakashima, Viscosity of Molten Slag and Glass (Agne Gijutsu Center, Tokyo, 2003), p. 77

    Google Scholar 

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Correspondence to Noritaka Saito.

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Manuscript submitted July 24, 2020; accepted January 3, 2021.



(A) Calculation of Thermodynamic Parameters in Quasichemical Model

In this study, ternary molten oxide systems were assumed as quasi binary system consisting of NWF and NWM oxides. The calculations of the thermodynamic parameters of the quasichemical model in quasi binary systems, ω (the change in the enthalpy of the pair formation reaction) and η (the change in the non-configurational entropy of the pair formation reaction), are described below for the SiO2-Al2O3-CaO system as an example. In this case, the mole fraction of each oxide is represented by \(X_{{{\text{SiO}}_{2} }} ,\;X_{{{\text{Al}}_{2} {\text{O}}_{3} }} ,\) and XCaO.

First, the method for calculating NBO/T by the pair fraction of the system by “Equilib module” of FactSage is explained. The Equilib module of FactSage can calculate the bond information for bonds such as Si-Si-O-O. In the system of SiO2-Al2O3-CaO, the mole fractions of Si-Si-O-O, Si-Al-O-O, Si-Ca-O-O, Al-Al-O-O, Al-Ca-O-O, and Ca-Ca-O-O were obtained. Then, NBO/T was calculated considering that only the Si-Si-O-O bond was the bridging oxygen bond.

Second, the procedure for calculating the thermodynamic parameters ω and η is described below, and the 0.5SiO2-0.2Al2O3-0.3CaO system was separated by each binary system. In this case, Al2O3 was considered as AlO1.5: that is, 0.56SiO2-0.44Al2O3, 0.63SiO2-0.37CaO, and 0.57Al2O3-0.43CaO systems were considered. The thermodynamic parameters ω and η of each separated system were calculated with the help of values from the literature. In the case of the 0.56SiO2-0.44Al2O3 system, for example, when \(X^{\prime}_{{{\text{SiO}}_{2} }}\) was defined as the mole fraction of SiO2 (i.e., \(X^{\prime}_{{{\text{SiO}}_{2} }} = 0.56\)), thermodynamic parameters were given as below based on the literature.[32]

$$ \omega_{{\text{Si - Al}}} = 4800 + 100,784\left( {X^{\prime}_{{{\text{SiO}}_{2} }} } \right)^{3} - 142,068\left( {X^{\prime}_{{{\text{SiO}}_{2} }} } \right)^{5} + 78,571\left( {X^{\prime}_{{{\text{SiO}}_{2} }} } \right)^{7} \;({\text{J}}/{\text{mol}}), $$
$$ \eta_{{\text{Si - Al}}} = 0. $$

Similar calculations were performed to calculate the values of ωSi-Al, ωSi-Ca, ωAl-Ca, ηSi-Al, ηSi-Ca, and ηAl-Ca. From these parameters, the mixing enthalpy ΔH and the non-configurational entropy ΔSnc were calculated using the following equation:

$$ \begin{aligned} \Delta H & = (b_{{{\text{SiO}}_{2} }} X_{{{\text{SiO}}_{2} }} + b_{{{\text{Al}}_{2} {\text{O}}_{3} }} X_{{{\text{Al}}_{2} {\text{O}}_{3} }} + b_{{{\text{CaO}}}} X_{{{\text{CaO}}}} ) \\ & \quad \cdot (R_{{\text{Si - Al}}} \omega_{{\text{Si - Al}}} + R_{{\text{Si - Ca}}} \omega_{{\text{Si - Ca}}} + R_{{\text{Al - Ca}}} \omega_{{\text{Al - Ca}}} ) \cdot \frac{1}{2}, \\ \end{aligned} $$
$$ \begin{aligned} \Delta S & = (b_{{{\text{SiO}}_{2} }} X_{{{\text{SiO}}_{2} }} + b_{{{\text{Al}}_{2} {\text{O}}_{3} }} X_{{{\text{Al}}_{2} {\text{O}}_{3} }} + b_{{{\text{CaO}}}} X_{{{\text{CaO}}}} ) \\ & \quad \cdot (R_{{\text{Si - Al}}} \eta_{{\text{Si - Al}}} + R_{{\text{Si - Ca}}} \eta_{{\text{Si - Ca}}} + R_{{\text{Al - Ca}}} \eta_{{\text{Al - Ca}}} ) \cdot \frac{1}{2}, \\ \end{aligned} $$

where Rij (i, j = Si, Al, or Ca) is the pair fraction of [MiMj] calculated by FactSage and bA (A= SiO2, Al2O3, or CaO) is called the equivalent fraction. The calculation was based on parameters such as \(b_{{{\text{SiO}}_{2} }} = 1.0331,\) where \(\left( {n/4} \right)b_{{{\text{SiO}}_{2} }}\) is given in the case of an n-valent cation, i.e., \(b_{{{\text{Al}}_{2} {\text{O}}_{3} }} = 1.0331\) and bCaO = 0.6887. The thermodynamic parameters of the quasi binary system were calculated using the values of ΔH and ΔSnc.

Then, the 0.5SiO2-0.2Al2O3-0.3CaO (mol pct) system was represented by quasi binary system composed of NWF and NWM oxides. In the case of the 0.5SiO2-0.2Al2O3-0.3CaO (mol pct) system, the amount of Al2O3 is larger than that of CaO. Therefore, it is assumed that Al2O3 was tetracoordinate. NWF and NWM were further defined as follows:

$$ {\text{NWF}}\;{\text{as}}\;0.5{\text{SiO}}_{2} + 0.2{\text{Al}}_{2} {\text{O}}_{3} + 0.2{\text{CaO}} = 0.9{\text{NWF}}\;\left( {X_{{{\text{NWF}}}} = 0.9} \right),\;{\text{and}} $$
$$ {\text{NWM}}\;{\text{as}}\;0.1{\text{CaO}} = 0.1{\text{NWM}}\;\left( {X_{{{\text{NWM}}}} = 0.1} \right). $$

Furthermore, the pair fraction of RNWF–NWF, RNWF–NWM, and RNWM–NWM can be calculated by the pair fraction of the 0.5SiO2-0.2Al2O3-0.3CaO system estimated by FactSage. In particular, both Si and Al worked as NWF; therefore, the [Si-Al] pair, for example, was assumed as RNWF–NWF. Then, the following equations provided the thermodynamic parameters of a quasi binary system, ω and η, considering the values of pair fraction of the quasi binary system (RNWF–NWM), mole fraction of NWF and NWM (XNWF and XNWM), and ΔH and ΔSnc calculated using Eqs. [A1] and [A2].

$$\omega =2\cdot \Delta H\cdot \frac{1}{{R}_{\mathrm{NWF}-\mathrm{NWM}}\cdot ({X}_{\mathrm{NWF}}+{X}_{\mathrm{NWM}})},$$
$$\eta =2\cdot \Delta {S}^{\mathrm{nc}}\cdot \frac{1}{{R}_{\mathrm{NWF}-\mathrm{NWM}}\cdot ({X}_{\mathrm{NWF}}+{X}_{\mathrm{NWM}})}.$$

(B) Calculation of the Pair Fraction of the Quasichemical Model

The calculation of the pair fraction of the quasi binary system was performed by using the thermodynamic parameters ω and η. For the composition mentioned in the above example, XNWF and XNWM were 0.9 and 0.1, respectively. In the quasichemical model, the pair fraction is expressed by the following equation:

$${R}_{\mathrm{NWF}-\mathrm{NWM}}=\frac{4{X}_{\mathrm{NWF}}{X}_{\mathrm{NWM}}}{1+\xi },$$
$$\xi ={\left[1+4{X}_{\mathrm{NWF}}{X}_{\mathrm{NWM}}\cdot \left\{\mathrm{exp}\left(\frac{\omega -\eta T}{RT}\right)-1\right\}\right] }^{1/2}, $$

where R and T are gas constant and absolute temperature, respectively. From the Eqs. [A5] and [A6], when (ωηT) is given, RNWF–NWM was calculated, the RNWF–NWF and RNWM–NWM were calculated by substituting RNWF–NWM for the following equations of mass conservation law:


The fraction of NBO and NBO/T were calculated by the estimated pair numbers.

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Harada, Y., Saito, N. & Nakashima, K. Structural Evaluation of Silicate Melts by Performing Impedance Measurements and Quasichemical Model Calculations. Metall Mater Trans B (2021).

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