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

Abstract

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.

Appendices

Appendix

(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 X_CaO.

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 AlO_1.5: 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 ω_Si-Al, ω_Si-Ca, ω_Al-Ca, η_Si-Al, η_Si-Ca, and η_Al-Ca. From these parameters, the mixing enthalpy ΔH and the non-configurational entropy ΔS^nc were calculated using the following equation:

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

(A1)

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

(A2)

where R_ij (i, j = Si, Al, or Ca) is the pair fraction of [M_i − M_j] calculated by FactSage and b_A (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 b_CaO = 0.6887. The thermodynamic parameters of the quasi binary system were calculated using the values of ΔH and ΔS^nc.

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 R_NWF–NWF, R_NWF–NWM, and R_NWM–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 R_NWF–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 (R_NWF–NWM), mole fraction of NWF and NWM (X_NWF and X_NWM), and ΔH and ΔS^nc calculated using Eqs. [A1] and [A2].

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

(A3)

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

(A4)

(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, X_NWF and X_NWM were 0.9 and 0.1, respectively. In the quasichemical model, the pair fraction is expressed by the following equation:

$${R}_{\mathrm{NWF}-\mathrm{NWM}}=\frac{4{X}_{\mathrm{NWF}}{X}_{\mathrm{NWM}}}{1+\xi },$$

(A5)

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

(A6)

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

$$2{X}_{\mathrm{NWF}}=2{R}_{\mathrm{NWF}-\mathrm{NWF}}+{R}_{\mathrm{NWF}-\mathrm{NWM}},$$

(A7)

$$2{X}_{\mathrm{NWM}}=2{R}_{\mathrm{NWM}-\mathrm{NWM}}+{R}_{\mathrm{NWF}-\mathrm{NWM}}.$$

(A8)

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