Prediction Model for SiO₂ Activity in the CaO-Al₂O₃-SiO₂-MgO Quaternary Slag System

Abstract

Activity models based on the ion and molecule coexistence theory have been widely used in the refining of metallurgical slags, with the SiO₂ content of slag playing a crucial role in improving the mechanical properties of refining slag-based cementitious materials. In order to improve the reactivity of SiO₂ in slag, this study established a SiO₂ activity prediction model for the CaO-Al₂O₃-SiO₂-MgO quaternary slag system based on the ion and molecule coexistence theory, validating the prediction results using reference values from the literature. Following this, the effects of w(SiO₂), w(CaO), w(CaO)/w(Al₂O₃), and R(w(CaO)/w(SiO₂)) on SiO₂ activity were explored (where w and R represent content and alkalinity, respectively). The results show that the model could accurately predict the SiO₂ activity of refining slag at 1873k. When the SiO₂ content was increased from 10% to 30%, with 60% w(CaO) and a w(MgO)/w(Al₂O₃) ratio of 0.25, the SiO₂ activity exhibited a trend of initially increasing and then decreasing, with a maximum activity value of 0.1359 reached at 17.5% w(SiO₂). When slag contained 15% w(SiO₂) and a w(MgO)/w(Al₂O₃) ratio of 0.25, the SiO₂ activity decreased with increasing CaO content, reaching a maximum activity value of 0.1268 when 55% w(CaO) was present. Therefore, by controlling the ratio of w(CaO)/w(Al₂O₃) and w(CaO)/w(SiO₂) in the slag to maintain a ratio of 3, the activity of SiO₂ can be effectively increased.

1. Introduction

Geopolymers were developed by Davidovits in 1978 and have been widely utilized since then, as they exhibit properties similar to those of silicate cements [1]. The basic raw materials used in the formation of geopolymers are a class of substances containing silica-aluminate active components, with the specific nature of different raw materials affecting the final geopolymers’ performance. Industrial development has resulted in increased levels of production of industrial waste materials, such as steel slag, blast furnace slag, red mud, fly ash, and carbide slag [2,3,4,5], which typically contain silicate or silicoaluminate as a main mineral component. Therefore, these waste materials can be used as a raw material for the production of geopolymers, which can not only provide a sustainable and efficient industrial waste management solution, but also solve the environmental problems arising from the storage of these waste materials.

Refining slag is a typical type of steel slag, which is the waste slag produced during the process of steel refining. SiO₂ is the main chemical component of refining slag and the further addition of pure SiO₂ to slag has been shown to eliminate free calcium oxide, make the refining slag more stable and significantly improving the grindability of the refining slag [6]. When using refining slag to prepare cementitious materials, the pozzolanic activity of SiO₂ consumes a large amount of Ca(OH)₂. In addition, due to the crystallization nucleation and filling effect of SiO₂, the hydration capacity of refining slag is enhanced, improving the mechanical properties of the final refining slag-based cementitious material [7]. Therefore, improving the reactivity of SiO₂ in slag is of great importance for the preparation of gelling materials. The CaO-Al₂O₃-SiO₂-MgO system is a typical refining slag system that is used in the process of preparing geopolymers from refining slag [8,9]. CaO, SiO₂, Al₂O₃, and MgO are the main components of refining slag, accounting for about 90% of refining slag mass, with their relative proportions and subsequent influence, being much larger than other components (such as Fe₂O₃ and SO₂). However, at present, there are relatively few studies on the activity of refining slag components. Therefore, establishing a calculation model for the activity of CaO-Al₂O₃-SiO₂-MgO quaternary slag components and predicting the activity of related components are of great significance for further research on the mechanical properties of refining slag-based geopolymers.

So far, many slag thermodynamic models have been proposed, such as the Regular Solution Model (RSM) [10], ion model [11], Wilson equation [12], the new generation of solution geometric model [13], KTH model [14], molecular interaction volume model (MIVM) [15], Monte Carlo model [16], Aggregation model [17], molecular ion coexistence theoretical model [18], etc. However, most prediction models involve manually defined parameters and have a limited scope of application. All models have their advantages and disadvantages. Up to now, no model can be applied to all slag systems, so thermodynamic models need to be further developed and improved. Many scholars’ research and analysis show that [19] the theoretical model of molecular ion coexistence has certain applicability, and only the standard Gibbs free energy of relevant reactions is used in the calculation process (ΔG^θ). The calculation process is simple and has a good prospect for application. Therefore, the molecular ion coexistence theory is selected in this paper to establish a prediction model for SiO₂ activity of CaO-Al₂O₃-SiO₂-MgO quaternary slag system. It is verified by the literature value in order to reduce the calculation error caused by the theoretical basic assumption. On this basis, the influence of component SiO₂, CaO content, CaO/Al₂O₃, and alkalinity on SiO₂ activity in the CaO-Al₂O₃-SiO₂-MgO slag system is analyzed. After the rules are obtained through calculation, the conditions for preparing high-strength geopolymers can be achieved by changing the experimental parameters (alkalinity, content of acid-base oxides), which provides a prospect for preparing building materials, paving roads, and reducing environmental impact in the future.

2. Thermodynamic Activity Model of the CaO-SiO₂-Al₂O₃-MgO System

2.1. Structural Units and Mass Action Concentrations in the CaO-SiO₂-Al₂O₃-MgO System

The molecular-ion coexistence theory (IMCT) is a model used to study the structural units of slag and their interactions, to determine the relationship between slag components and properties [20]. The calculations used by the IMCT model are simple, as it only uses the thermodynamic value ΔG^θ and the IMCT model can calculate the active concentration of all structural units present in slag material, unlike other comparable models such as the Van Laar equation, which cannot be used for multivariate systems. The IMCT model has been successfully applied to the analysis of slag composed of CaO-Al₂O₃ [21], CaO-Al₂O₃-Ce2O3 [22], CaO-Al₂O₃-SiO₂ [21], CaO-Al₂O₃-MgO [23], CaO-SiO₂-MgO-Al₂O₃-Na₂O [24], CaO-SiO₂-MgO-Al₂O₃-FeO [25], and CaO-SiO₂-MgO-Al₂O₃-Cr2O3 [26]. Furthermore, the use of the IMCT model has been well established for prediction of the sulfur and phosphorus capacity of metallurgical slag, as well as for determining the sulfur partition ratio in metallurgical processes [27,28,29].

The IMCT model assumes that complex molecules are formed via chemical reactions that exhibit chemical kinetic equilibrium between simple molecules and ion pairs produced by simple ions, allowing activity values to be obtained when all components of the system are in equilibrium for all reactions. This assumption replaces activity in the traditional sense, with the mass action concentration of structural units or ion pairs in the slag system. Based on the IMCT and the phase diagram of CaO-SiO₂, MgO-SiO₂, CaO-Al₂O₃, MgO-Al₂O₃, CaO-SiO₂-MgO, CaO-Al₂O₃-SiO₂, and Al₂O₃-SiO₂-MgO [30], the structural units present in the CaO-Al₂O₃-SiO₂-MgO quaternary slag system are shown in

Table 1. Structural units present in the CaO-Al₂O₃-SiO₂-MgO system.

Slag System	Structural Units
	Ca2+, O2−, Mg2+, Al2O3, SiO2
CaO-SiO2	CaO·SiO2, 2CaO·SiO2, 3CaO·SiO2
MgO-SiO2	MgO·SiO2, 2MgO·SiO2
CaO-Al2O3	3CaO·Al2O3, 12CaO·7Al2O3, CaO·Al2O3, CaO·2Al2O3, CaO·6Al2O3
MgO-Al2O3	MgO·Al2O3
Al2O3-SiO2	3Al2O3·2SiO2
CaO-MgO-SiO2	CaO·MgO·2SiO2, CaO·MgO·SiO2, 2CaO·MgO·2SiO2, 3CaO·MgO·2SiO2
CaO-Al2O3-SiO2	2CaO·Al2O3·SiO2, CaO·Al2O3·2SiO2

.

The active concentration of each structural unit in the CaO-Al₂O₃-SiO₂-MgO slag system can be determined using Equations (1)–(4) as follow:

$$\mathrm{b}=\sum {{\mathrm{X}}_{\mathrm{C}}}_{\mathrm{aO}}$$

(1)

$${\mathrm{a}}_1=\sum {\mathrm{X}}_{{\mathrm{Al}}_2{\mathrm{O}}_3}$$

(2)

$${\mathrm{a}}_2=\sum {\mathrm{X}}_{{\mathrm{SiO}}_2}$$

(3)

$${\mathrm{a}}_3=\sum {\mathrm{X}}_{\mathrm{MgO}}$$

(4)

where b, a₁, a₂, and a₃ are the total moles of the components CaO, Al₂O₃, SiO₂, and MgO, respectively, in the slag at equilibrium; $\sum \mathrm{X}$ indicates the total number of moles at equilibrium; and ${\mathrm{N}}_{\mathrm{i}}$(i = 1, 2, 3,…, n) is the active concentration of each component in the refining slag, which obeys the law of mass action. The structural unit is defined as: N₁ = ${\mathrm{N}}_{\mathrm{CaO}}$, N₂ = ${\mathrm{N}}_{{\mathrm{Al}}_2{\mathrm{O}}_3}$, N₃ = ${\mathrm{N}}_{{\mathrm{SiO}}_2}$, N₄ = ${\mathrm{N}}_{\mathrm{MgO}}$, N₅ = ${\mathrm{N}}_{{\mathrm{CaSiO}}_3}$, N₆ = ${\mathrm{N}}_{{\mathrm{Ca}}_2{\mathrm{SiO}}_4}$, N₇ = ${\mathrm{N}}_{{\mathrm{Ca}}_3{\mathrm{SiO}}_5}$, N₈ = ${\mathrm{N}}_{{\mathrm{MgSiO}}_3}$, N₉ = ${\mathrm{N}}_{{\mathrm{Mg}}_2{\mathrm{SiO}}_4}$, N₁₀ = ${\mathrm{N}}_{\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3}$, N₁₁ = ${\mathrm{N}}_{12\mathrm{CaO}·7{\mathrm{Al}}_2{\mathrm{O}}_3}$, N₁₂ = ${\mathrm{N}}_{3\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3}$, N₁₃ = ${\mathrm{N}}_{\mathrm{CaO}·2{\mathrm{Al}}_2{\mathrm{O}}_3}$, N₁₄ = ${\mathrm{N}}_{3\mathrm{CaO}·6{\mathrm{Al}}_2{\mathrm{O}}_3}$, N₁₅ = ${\mathrm{N}}_{2\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3·{\mathrm{SiO}}_2}$, N₁₆ = ${\mathrm{N}}_{\mathrm{CaO}·{\mathrm{Al}}_2{\mathrm{O}}_3·2{\mathrm{SiO}}_2}$, N₁₇ = ${\mathrm{N}}_{3{\mathrm{Al}}_2{\mathrm{O}}_3·2{\mathrm{SiO}}_2}$, N₁₈ = ${\mathrm{N}}_{\mathrm{CaO}·\mathrm{MgO}·{\mathrm{SiO}}_2}$, N₁₉ = ${\mathrm{N}}_{3\mathrm{CaO}·\mathrm{MgO}·2{\mathrm{SiO}}_2}$, N₂₀ = ${\mathrm{N}}_{\mathrm{CaO}·\mathrm{MgO}·2{\mathrm{SiO}}_2}$, N₂₁ = ${\mathrm{N}}_{2\mathrm{CaO}·\mathrm{MgO}·2{\mathrm{SiO}}_2}$, and N₂₂ = ${\mathrm{N}}_{\mathrm{MgO}·{\mathrm{Al}}_2{\mathrm{O}}_3}$.

2.2. Thermodynamic Data and Chemical Equilibrium Equation

Thermodynamic data for the formation reactions of each complex compound in slag and the respective chemical equilibrium equations were established according to the IMCT model and are listed in

Table 2. Thermodynamic data for complex slag component chemical reactions and the corresponding chemical equilibrium equations.

Number	Formula or Reaction Equation	ΔGθ/(J·mol−1)	${\mathbf{N}}_{\mathbf{i}}$	${\mathbf{X}}_{\mathbf{i}}$
1	Ca2+ + O2− = CaO		N1 = NCa2+ + NO2− = 2X1/$\sum \mathrm{X}$	2X1 = N1$\sum \mathrm{X}$
2	Mg2+ + O2− = MgO		N4 = NMg2+ + NO2− = 2X4/$\sum \mathrm{X}$	2X4 = N4$\sum \mathrm{X}$
3	SiO2		N3 = X3/$\sum \mathrm{X}$	X3 = N3$\sum \mathrm{X}$
4	Al2O3		N2 = X2/$\sum \mathrm{X}$	X2 = N2$\sum \mathrm{X}$
5	(Ca2+ + O2−) + SiO2 = CaSiO3	−92,500 − 2.5 T	N5 = K1N1N3	X5 = N5$\sum \mathrm{X}$
6	2(Ca2+ + O2−) + SiO2 = Ca2SiO4	−118,800 − 11.3 T	N6 = K2${\mathrm{N}}_1^2$N3	X6 = N6$\sum \mathrm{X}$
7	3(Ca2+ + O2−) + SiO2 = Ca3SiO5	−118,800 − 6.7 T	N7 = K3${\mathrm{N}}_1^3$N3	X7 = N7$\sum \mathrm{X}$
8	(Mg2+ + O2) + SiO2 = MgSiO3	−41,100 + 6.1 T	N8 = K4N3N4	X8 = N8$\sum \mathrm{X}$
9	2(Mg2+ + O2) + SiO2 = Mg2SiO4	−67,200 + 4.31 T	N9 = K5N3${\mathrm{N}}_4^2$	X9 = N9$\sum \mathrm{X}$
10	(Ca2+ + O2−) + Al2O3 = CaO·Al2O3	−18,000 − 18.83 T	N10 = K6N1N2	X10 = N10$\sum \mathrm{X}$
11	12(Ca2+ + O2−) + 7Al2O3 = 12CaO·7Al2O3	−86,100 − 205.1 T	N11 = K7${\mathrm{N}}_1^{12}{\mathrm{N}}_2^7$	X11 = N11$\sum \mathrm{X}$
12	3(Ca2+ + O2−) + Al2O3 = 3CaO·Al2O3	−12,600 − 24.69 T	N12 = K8${\mathrm{N}}_1^3$N2	X12 = N12$\sum \mathrm{X}$
13	(Ca2+ + O2−) + 2Al2O3 = CaO·2Al2O3	−16,700 − 25.52 T	N13 = K9N1${\mathrm{N}}_2^2$	X13 = N13$\sum \mathrm{X}$
14	3(Ca2+ + O2−) + 6Al2O3 = 3CaO·6Al2O3	−16,380 − 37.58 T	N14 = K10${\mathrm{N}}_1^3{\mathrm{N}}_2^6$	X14 = N14$\sum \mathrm{X}$
15	2(Ca2++ O2−) + Al2O3 + SiO2 = 2CaO·Al2O3·SiO2	−61,964.64 − 60.29 T	N15 = K11${\mathrm{N}}_1^2$N2N3	X15 = N15$\sum \mathrm{X}$
16	(Ca2++ O2−) + Al2O3 + 2SiO2 = CaO·Al2O3·2SiO2	−13,816.44 − 55.266 T	N16 = K12N1N2${\mathrm{N}}_3^2$	X16 = N16$\sum \mathrm{X}$
17	3Al2O3 + 2SiO2 = 3Al2O3·2SiO2	−8600 − 17.41 T	N17 = K13${\mathrm{N}}_2^3{\mathrm{N}}_3^2$	X17 = N17$\sum \mathrm{X}$
18	(Ca2+ + O2−) + (Mg2+ + O2) + SiO2 = CaO·MgO·SiO2	−124,766.6 + 3.768 T	N18 = K14N1N3N4	X18 = N18$\sum \mathrm{X}$
19	3(Ca2+ + O2−) + (Mg2+ + O2) + 2SiO2 = 3CaO·MgO·2SiO2	−315,469 + 24.786 T	N19 = K15${\mathrm{N}}_1^3{\mathrm{N}}_3^2$N4	X19 = N19$\sum \mathrm{X}$
20	(Ca2+ + O2−) + (Mg2+ + O2) + 2SiO2 = CaO·MgO·2SiO2	−80,387 − 51.916 T	N20 = K16N1${\mathrm{N}}_3^2$N4	X20 = N20$\sum \mathrm{X}$
21	2(Ca2+ + O2−) + (Mg2+ + O2) + 2SiO2 = 2CaO·MgO·2SiO2	−73,688 − 63.639 T	N21 = K17${\mathrm{N}}_1^2{\mathrm{N}}_3^2$N4	X21 = N21$\sum \mathrm{X}$
22	(Mg2+ + O2) + Al2O3 = MgO·Al2O3	−35,600 − 2.09 T	N22 = K18N2N4	X22 = N22$\sum \mathrm{X}$

Note: According to the theory of molecular-ion coexistence, CaO and MgO exist in slag in a solid state, forming a face-centered cubic ionic lattice structure. In addition, they exist in the form of Ca²⁺, Mg²⁺, and O²⁻, all of which can remain independent without forming a molecular state; ${\mathrm{N}}_{\mathrm{i}}$ is the mass action concentration of the component i; ${\mathrm{X}}_{\mathrm{i}}$ is the mole fraction of component i; ${\mathrm{K}}_{\mathrm{i}}$ is the equilibrium constant of the chemical reaction; and $\Delta {\mathrm{G}}^{\mathsf{\theta }}$ is the standard Gibbs free energy change of the chemical reaction to form the complex molecule i.

[30,31,32,33,34].

2.3. Mass Balance Equation Construction

As the IMCT assumes that a dynamic chemical equilibrium exists between atoms and molecules in the slag system, a mass balance equation was established based on the conservation of mass before and after chemical reactions, as shown in Equation (5):

$${{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{i}=22}{\mathrm{N}}_{\mathrm{i}}=1$$

(5)

where the total number of moles of CaO, Al₂O₃, SiO₂, and MgO before and after the reactions remained constant, as shown in Equations (6)–(9):

$$\begin{gathered} \mathrm{b}=\sum \mathrm{X}(0.5{\mathrm{N}}_1+{\mathrm{N}}_5+2{\mathrm{N}}_6+3{\mathrm{N}}_7+{\mathrm{N}}_{10}+12{\mathrm{N}}_{11}+3{\mathrm{N}}_{12}+{\mathrm{N}}_{13}+3{\mathrm{N}}_{14}+2{\mathrm{N}}_{15}+{\mathrm{N}}_{16}+3{\mathrm{N}}_{17}+{\mathrm{N}}_{18}+3{\mathrm{N}}_{19}+{\mathrm{N}}_{20}+ \\ 2{\mathrm{N}}_{21}) \end{gathered}$$

(6)

$${\mathrm{a}}_1=\sum \mathrm{X}({\mathrm{N}}_2+{\mathrm{N}}_{10}+7{\mathrm{N}}_{11}+{\mathrm{N}}_{12}+2{\mathrm{N}}_{13}+6{\mathrm{N}}_{14}+{\mathrm{N}}_{15}+{\mathrm{N}}_{16}+{\mathrm{N}}_{17}+{\mathrm{N}}_{22})$$

(7)

$$\begin{gathered} {\mathrm{a}}_2=\sum \mathrm{X}({\mathrm{N}}_3+{\mathrm{N}}_5+{\mathrm{N}}_6+{\mathrm{N}}_7+{\mathrm{N}}_8+{\mathrm{N}}_9+{\mathrm{N}}_{15}+2{\mathrm{N}}_{16}+2{\mathrm{N}}_{17}+{\mathrm{N}}_{18}+2{\mathrm{N}}_{19}+2{\mathrm{N}}_{20}+ \\ 2{\mathrm{N}}_{21}) \end{gathered}$$

(8)

$${\mathrm{a}}_3=\sum \mathrm{X}(0.5{\mathrm{N}}_4+{\mathrm{N}}_8+2{\mathrm{N}}_9+{\mathrm{N}}_{18}+{\mathrm{N}}_{19}+{\mathrm{N}}_{20}+{\mathrm{N}}_{21}+{\mathrm{N}}_{22})$$

(9)

By combining Equations (6) and (7), it is possible to eliminate $\sum \mathrm{X}$ for b and a₁ as described by Equation (10) as follows:

b(0.5N₄ + N₈ + 2N₉ + N₁₈ + N₁₉ + N₂₀ + N₂₁ + N₂₂) − a₃(0.5N₁ + N₅ + 2N₆ + 3N₇ + N₁₀ + 12N₁₁ + 3N₁₂ + N₁₃ + 3N₁₄
+ 2N₁₅ + N₁₆ + 3N₁₇ + N₁₈ + 3N₁₉ + N₂₀ + 2N₂₁) = 0

(10)

By combining Equations (7) and (9), it is possible to eliminate $\sum \mathrm{X}$ for a₁ and a₃ as described by Equation (11) as follows:

a₁(0.5N₄ + N₈ + 2N₉ + N₁₈ + N₁₉ + N₂₀ + N₂₁ + N₂₂) − a₃(N₂ + N₁₀ + 7N₁₁ + N₁₂ + 2N₁₃ + 6N₁₄ + N₁₅ + N₁₆ + N₁₇ +
N₂₂) = 0

(11)

By combining Equations (8) and (9), it is possible to eliminate $\sum \mathrm{X}$ for a₂ and a₃ as described by Equation (12) as follows:

a₂(0.5N₄ + N₈ + 2N₉ + N₁₈ + N₁₉ + N₂₀ + N₂₁ + N₂₂) − a₃(N₃ + N₅ + N₆ + N₇ + N₈ + N₉ + N₁₅ + 2N₁₆ + 2N₁₇ + N₁₈ +
2N₁₉ + 2N₂₀ + 2N₂₁) = 0

(12)

Using this approach, Equations (10)–(12) can be simplified, replacing ${\mathrm{N}}_{\mathrm{i}}$ (i = 5, 6,..., 22) with ${\mathrm{N}}_{\mathrm{i}}$ (i = 1, 2, 3, 4), allowing the system of nonlinear equations for ${\mathrm{N}}_1,{ \mathrm{N}}_2,{ \mathrm{N}}_3,{ \mathrm{and} \mathrm{N}}_4$ to be collated, where Equation (5) was rectified to form Equation (13) as follows:

$$\begin{gathered} {\mathrm{N}}_1+{\mathrm{N}}_2+{\mathrm{N}}_3+{\mathrm{N}}_4+{\mathrm{K}}_1{\mathrm{N}}_1{\mathrm{N}}_2+{\mathrm{K}}_2{\mathrm{N}}_1^2{\mathrm{N}}_2+{\mathrm{K}}_3{\mathrm{N}}_1^3{\mathrm{N}}_2+{\mathrm{K}}_4{\mathrm{N}}_2{\mathrm{N}}_3+{\mathrm{K}}_5{\mathrm{N}}_2{\mathrm{N}}_3^2+{\mathrm{K}}_6{\mathrm{N}}_1{\mathrm{N}}_4+{\mathrm{K}}_7{\mathrm{N}}_1^{12}{\mathrm{N}}_4^7+{\mathrm{K}}_8{\mathrm{N}}_1^3{\mathrm{N}}_4+ \\ {\mathrm{K}}_9{\mathrm{N}}_1{\mathrm{N}}_4^2+{\mathrm{K}}_{10}{\mathrm{N}}_1{\mathrm{N}}_4^6+{\mathrm{K}}_{11}{\mathrm{N}}_1^2{\mathrm{N}}_2{\mathrm{N}}_4+{\mathrm{K}}_{12}{\mathrm{N}}_1{\mathrm{N}}_2^2{\mathrm{N}}_4+{\mathrm{K}}_{13}{\mathrm{N}}_2^2{\mathrm{N}}_4^3+{\mathrm{K}}_{14}{\mathrm{N}}_1{\mathrm{N}}_2{\mathrm{N}}_3+{\mathrm{K}}_{15}{\mathrm{N}}_1^3{\mathrm{N}}_2^2{\mathrm{N}}_3+{\mathrm{K}}_{16}{\mathrm{N}}_1{\mathrm{N}}_2^2{\mathrm{N}}_3+ \\ {\mathrm{K}}_{17}{\mathrm{N}}_1^2{\mathrm{N}}_2^2{\mathrm{N}}_3+{\mathrm{K}}_{18}{\mathrm{N}}_3{\mathrm{N}}_4=1 \end{gathered}$$

(13)

Equation (10) was rectified to form Equation (14) as follows:

$$\begin{gathered} \mathrm{b}(0.5{\mathrm{N}}_4+{\mathrm{K}}_4{\mathrm{N}}_2{\mathrm{N}}_3+2{\mathrm{K}}_5{\mathrm{N}}_2{\mathrm{N}}_3^2+{\mathrm{K}}_{14}{\mathrm{N}}_1{\mathrm{N}}_2{\mathrm{N}}_3+{\mathrm{K}}_{15}{\mathrm{N}}_1^3{\mathrm{N}}_2^2{\mathrm{N}}_3+{\mathrm{K}}_{16}{\mathrm{N}}_1{\mathrm{N}}_2^2{\mathrm{N}}_3+{\mathrm{K}}_{17}{\mathrm{N}}_1^2{\mathrm{N}}_2^2{\mathrm{N}}_3+{\mathrm{K}}_{18}{\mathrm{N}}_3{\mathrm{N}}_4)-{\mathrm{a}}_3(0.5{\mathrm{N}}_1+ \\ {\mathrm{K}}_1{\mathrm{N}}_1{\mathrm{N}}_2+2{\mathrm{K}}_2{\mathrm{N}}_1^2{\mathrm{N}}_2+3{\mathrm{K}}_3{\mathrm{N}}_1^3{\mathrm{N}}_2+{\mathrm{K}}_6{\mathrm{N}}_1{\mathrm{N}}_4+12{\mathrm{K}}_7{\mathrm{N}}_1^{12}{\mathrm{N}}_4^7+3{\mathrm{K}}_8{\mathrm{N}}_1^3{\mathrm{N}}_4+{\mathrm{K}}_9{\mathrm{N}}_1{\mathrm{N}}_4^2+3{\mathrm{K}}_{10}{\mathrm{N}}_1{\mathrm{N}}_4^6+2{\mathrm{K}}_{11}{\mathrm{N}}_1^2{\mathrm{N}}_2{\mathrm{N}}_4+ \\ {\mathrm{K}}_{12}{\mathrm{N}}_1{\mathrm{N}}_2^2{\mathrm{N}}_4+3{\mathrm{K}}_{13}{\mathrm{N}}_2^2{\mathrm{N}}_4^3+{\mathrm{K}}_{14}{\mathrm{N}}_1{\mathrm{N}}_2{\mathrm{N}}_3+3{\mathrm{K}}_{15}{\mathrm{N}}_1^3{\mathrm{N}}_2^2{\mathrm{N}}_3+{\mathrm{K}}_{16}{\mathrm{N}}_1{\mathrm{N}}_2^2{\mathrm{N}}_3+2{\mathrm{K}}_{17}{\mathrm{N}}_1^2{\mathrm{N}}_2^2{\mathrm{N}}_3)=0 \end{gathered}$$

(14)

Equation (11) was rectified to form Equation (15) as follows:

$$\begin{gathered} {\mathrm{a}}_1(0.5{\mathrm{N}}_4+{\mathrm{K}}_4{\mathrm{N}}_2{\mathrm{N}}_3+2{\mathrm{K}}_5{\mathrm{N}}_2{\mathrm{N}}_3^2+{\mathrm{K}}_{14}{\mathrm{N}}_1{\mathrm{N}}_2{\mathrm{N}}_3+{\mathrm{K}}_{15}{\mathrm{N}}_1^3{\mathrm{N}}_2^2{\mathrm{N}}_3+{\mathrm{K}}_{16}{\mathrm{N}}_1{\mathrm{N}}_2^2{\mathrm{N}}_3+{\mathrm{K}}_{17}{\mathrm{N}}_1^2{\mathrm{N}}_2^2{\mathrm{N}}_3+{\mathrm{K}}_{18}{\mathrm{N}}_3{\mathrm{N}}_4)-{\mathrm{a}}_3({\mathrm{N}}_2+ \\ {\mathrm{K}}_6{\mathrm{N}}_1{\mathrm{N}}_4+7{\mathrm{K}}_7{\mathrm{N}}_1^{12}{\mathrm{N}}_4^7+{\mathrm{K}}_8{\mathrm{N}}_1^3{\mathrm{N}}_4+2{\mathrm{K}}_9{\mathrm{N}}_1{\mathrm{N}}_4^2+6{\mathrm{K}}_{10}{\mathrm{N}}_1{\mathrm{N}}_4^6+{\mathrm{K}}_{11}{\mathrm{N}}_1^2{\mathrm{N}}_2{\mathrm{N}}_4+{\mathrm{K}}_{12}{\mathrm{N}}_1{\mathrm{N}}_2^2{\mathrm{N}}_4+{\mathrm{K}}_{13}{\mathrm{N}}_2^2{\mathrm{N}}_4^3+{\mathrm{K}}_{18}{\mathrm{N}}_3{\mathrm{N}}_4)=0 \end{gathered}$$

(15)

Equation (12) was rectified to form Equation (16) as follows:

$$\begin{gathered} {\mathrm{a}}_2(0.5{\mathrm{N}}_4+{\mathrm{K}}_4{\mathrm{N}}_2{\mathrm{N}}_3+2{\mathrm{K}}_5{\mathrm{N}}_2{\mathrm{N}}_3^2+{\mathrm{K}}_{14}{\mathrm{N}}_1{\mathrm{N}}_2{\mathrm{N}}_3+{\mathrm{K}}_{15}{\mathrm{N}}_1^3{\mathrm{N}}_2^2{\mathrm{N}}_3+{\mathrm{K}}_{16}{\mathrm{N}}_1{\mathrm{N}}_2^2{\mathrm{N}}_3+{\mathrm{K}}_{17}{\mathrm{N}}_1^2{\mathrm{N}}_2^2{\mathrm{N}}_3+{\mathrm{K}}_{18}{\mathrm{N}}_3{\mathrm{N}}_4)-{\mathrm{a}}_3({\mathrm{N}}_3+ \\ {\mathrm{K}}_1{\mathrm{N}}_1{\mathrm{N}}_2+{\mathrm{K}}_2{\mathrm{N}}_1^2{\mathrm{N}}_2+{\mathrm{K}}_3{\mathrm{N}}_1^3{\mathrm{N}}_2+{\mathrm{K}}_4{\mathrm{N}}_2{\mathrm{N}}_3+{\mathrm{K}}_5{\mathrm{N}}_2{\mathrm{N}}_3^2+{\mathrm{K}}_{11}{\mathrm{N}}_1^2{\mathrm{N}}_2{\mathrm{N}}_4+2{\mathrm{K}}_{12}{\mathrm{N}}_1{\mathrm{N}}_2^2{\mathrm{N}}_4+2{\mathrm{K}}_{13}{\mathrm{N}}_2^2{\mathrm{N}}_4^3+{\mathrm{K}}_{14}{\mathrm{N}}_1{\mathrm{N}}_2{\mathrm{N}}_3+ \\ 2{\mathrm{K}}_{15}{\mathrm{N}}_1^3{\mathrm{N}}_2^2{\mathrm{N}}_3+2{\mathrm{K}}_{16}{\mathrm{N}}_1{\mathrm{N}}_2^2{\mathrm{N}}_3+2{\mathrm{K}}_{17}{\mathrm{N}}_1^2{\mathrm{N}}_2^2{\mathrm{N}}_3)=0 \end{gathered}$$

(16)

Equations (13)–(16) describe the activity calculation model for the CaO-Al₂O₃-SiO₂-MgO slag system and the above calculation models were solved using Matlab 2021a software and the fsolve function [35], allowing ${\mathrm{N}}_{\mathrm{i}}$(i = 5, 6,..., 22) to be obtained from the equilibrium relationships, as expressed in Table 2.

2.4. CaO-Al₂O₃-SiO₂-MgO Quaternary Slag System Activity Model Verification

In order to verify the reliability of the model, this paper compared the SiO₂ activity values (${\mathrm{N}}_{\left({\mathrm{SiO}}_2\right)}$) obtained from the molecular-ionic coexistence theory prediction model of CaO-Al₂O₃-SiO₂-MgO quaternary refining slag system at 1873 K with the literature-referenced SiO₂ activity values (${\mathrm{a}}_{\left({\mathrm{SiO}}_2\right)}$) [23], and the results are shown in Figure 1. As can be seen from Figure 1, the SiO₂ activity values (${\mathrm{N}}_{\left({\mathrm{SiO}}_2\right)}$) predicted by the molecular-ionic coexistence theory and the literature-referenced trend of SiO₂ activity values (${\mathrm{a}}_{\left({\mathrm{SiO}}_2\right)}$) are consistent with the literature references, and their fit is 0.9086, which has a good linear relationship. Therefore, the molecule-ion coexistence theory model can be applied to calculate the activity of the tetragonal slag system group elements.

2.5. Slag Sample Configuration

The slag sample used for analysis was selected in the model to explore the effect of w(SiO₂), w(CaO), w(CaO)/w(Al₂O₃), and R(w(CaO)/w(SiO₂)) on SiO₂ activity in the CaO-Al₂O₃-SiO₂-MgO slag system. The selected slag CaO-Al₂O₃-SiO₂-MgO quaternary composition and activity values are presented in

Table 3. Slag composition after equilibrium and the activities of components in the CaO-Al₂O₃-SiO₂-MgO system.

	Slag Composition				Activity
Sample No.	${\mathbf{X}}_{\left(\mathbf{C}\mathbf{a}\mathbf{O}\right)}$	${\mathbf{X}}_{(\mathbf{S}\mathbf{i}{\mathbf{O}}_2)}$	${\mathbf{X}}_{\left(\mathbf{M}\mathbf{g}\mathbf{O}\right)}$	${\mathbf{X}}_{(\mathbf{A}{\mathbf{l}}_2{\mathbf{O}}_3)}$	${\mathbf{N}}_{\left(\mathbf{C}\mathbf{a}\mathbf{O}\right)}$	${\mathbf{N}}_{(\mathbf{S}\mathbf{i}{\mathbf{O}}_2)}$	${\mathbf{N}}_{\left(\mathbf{M}\mathbf{g}\mathbf{O}\right)}$	${\mathbf{N}}_{(\mathbf{A}{\mathbf{l}}_2{\mathbf{O}}_3)}$
1	0.660	0.103	0.092	0.145	0.0936	0.1199	0.0432	0.0029
2	0.648	0.126	0.083	0.143	0.1005	0.1256	0.0348	0.0028
3	0.652	0.152	0.076	0.120	0.0950	0.1334	0.0210	0.0034
4	0.648	0.177	0.068	0.107	0.0999	0.1359	0.0203	0.0032
5	0.646	0.199	0.060	0.095	0.1405	0.1218	0.0161	0.0019
6	0.641	0.225	0.052	0.082	0.1090	0.1074	0.0150	0.0031
7	0.636	0.248	0.044	0.070	0.0968	0.1254	0.0143	0.0036
8	0.537	0.158	0.119	0.186	0.0603	0.1051	0.0811	0.0044
9	0.560	0.157	0.110	0.173	0.0669	0.1250	0.0524	0.0046
10	0.584	0.156	0.101	0.159	0.0807	0.1221	0.0509	0.0034
11	0.607	0.155	0.093	0.145	0.0840	0.1268	0.0390	0.0036
12	0.630	0.153	0.084	0.133	0.1013	0.1261	0.0339	0.0027
13	0.652	0.152	0.076	0.120	0.0906	0.1242	0.0227	0.0038
14	0.675	0.151	0.068	0.106	0.1348	0.1191	0.0260	0.0018
15	0.633	0.160	0.032	0.174	0.2006	0.1163	0.0168	0.000867
16	0.666	0.157	0.031	0.146	0.1980	0.1169	0.0117	0.000942
17	0.689	0.155	0.031	0.125	0.1221	0.1248	0.0175	0.0023
18	0.706	0.153	0.031	0.110	0.1116	0.1244	0.0159	0.0028
19	0.719	0.152	0.030	0.099	0.1698	0.1120	0.0081	0.0014
20	0.730	0.151	0.030	0.089	0.1127	0.1012	0.0120	0.0030
21	0.740	0.149	0.030	0.081	0.1053	0.1084	0.0130	0.0033

.

3. Results and Discussion

3.1. Influence of SiO₂ Content on SiO₂ Activity

Figure 2a shows the slag composition of the (CaO + MgO)-SiO₂-Al₂O₃ pseudo-ternary system, in which the proportion of w(CaO) (=60%) and the ratio of w(MgO)/w(Al₂O₃) (=0.25) remained constant, while the influence of w(SiO₂) on the activity of SiO₂ was analyzed. The pattern of variation in the activity of SiO₂ is shown in Figure 2b. When w(SiO₂) was increased from 10%~17.5%, SiO₂ activity exhibited an upward trend, reaching a maximum value of 0.1359 at 17.5% w(SiO₂). Further increases in the content of SiO₂ resulted in SiO₂ activity exhibiting a downward trend.

The reason for the initial increase and subsequent decrease in the modelled activity value was analyzed, showing that as the w(SiO₂) increased from 10% to 17.5%, CaO existed in the form of a highly alkaline oxide. When w(CaO) remained constant and R(w(CaO)/w(SiO₂)) decreased, the generation of dissociated O²⁻ from the alkaline oxide was limited, inhibiting Equations (5)–(7) shown in Table 2 from proceeding and causing the SiO₂ activity in slag to continually increase. When w(SiO₂) exceeded 17.5%, its activity decreased accordingly. Combined with our preliminary experimental analysis in the laboratory, it can be seen that SiO₂ mostly existed in a crystalline structural form, with the chemical components involved in the reaction being few and limited. At this time, the addition of a pure silicon source can improve the reaction rate to a certain extent, promoting the forward progression of Equations (5) to (7) in Table 2 and causing the SiO₂ activity to decrease.

3.2. Influence of CaO Content on SiO₂ Activity

The influence of w(CaO) on the activity of SiO₂ was analyzed with 60% w(SiO₂) and a w(MgO)/w(Al₂O₃) ratio of 0.25 kept constant. As shown in Figure 3, when w(CaO) = 55%, w(MgO) = 6%, and w(Al₂O₃) = 24%, SiO₂ activity reached a maximum value of 0.1268, then exhibited a decreasing trend when the w(CaO) was increased gradually to 72.5%.

The reason for the decrease in theoretical activity was analyzed, with an increasing gradient of w(CaO) from 55% to 72.5%, while w(MgO)/w(Al₂O₃) remained constant at 0.25 and w(Al₂O₃) decreased relatively. CaO is a strongly alkaline oxide that can dissociate to produce O²⁻ in slag (CaO → Ca²⁺ + O²⁻), with the dissociated O²⁻ participating in the generation of silicate ions and aluminate ions. Due to the reduction of w(Al₂O₃), the generated O²⁻ consumes the acidic oxide SiO₂, resulting in a continual decrease in the activity of SiO₂ in slag.

3.3. Influence of w(CaO)/w(Al₂O₃) on SiO₂ Activity

The influence of w(CaO)/w(Al₂O₃) on the activity of SiO₂ in slag was analyzed, with 60% w(SiO₂) and 2% w(MgO) remaining constant. The red triangle area shown in Figure 4 indicates the area used for the reference slag selection value, while Figure 5 shows the variation in SiO₂ activity with varying ratios of w(CaO)/w(Al₂O₃). When the w(CaO)/w(Al₂O₃) ratio was 1, SiO₂ activity reached only 0.0185, while increasing the w(CaO)/w(Al₂O₃) from 1 to 3 resulted in an increase in N(SiO₂), reaching a peak value of 0.1248 when w(CaO)/w(Al₂O₃) = 3. With further increases in w(CaO)/w(Al₂O₃), the activity of SiO₂ exhibited a decreasing trend (Figure 5).

The reason for the observed reduction in SiO₂ activity value when w(CaO)/w(Al₂O₃) was increased to >3, is that the content of CaO gradually increases with higher w(CaO)/w(Al₂O₃) ratios, causing the generated Ca²⁺ + O²⁻ ion pair to react with SiO₂ and Al₂O₃ to form calcium silicate and calcium aluminate complexes, such as CaO-SiO₂, CaO-2SiO₂, CaO-Al₂O₃, and CaO-2Al₂O₃. In addition, the content of Al₂O₃ was found to decrease with the increase in w(CaO)/w(Al₂O₃), with the reaction between the Ca²⁺ + O²⁻ ion pair and Al₂O₃ (as shown in Equations (10) to (14) in Table 2) being weakened, reducing the activity of calcium aluminate complex molecules, such as CaO-Al₂O₃ and CaO-2Al₂O₃. Therefore, as shown in Equations (5)–(7) in Table 2, CaO consumes more SiO₂ to generate calcium silicate composite molecules, which consequently reduces the activity of SiO₂.

3.4. Influence of R(w(CaO)/w(SiO₂)) on SiO₂ Activity

Table 4. Effect of alkalinity on the activity of components in the CaO-Al₂O₃-SiO₂-MgO system.

Sample No.	R (Alkalinity)	Slag Composition				Activity
		$X_{\left(CaO\right)}$	$X_{(Si{O}_2)}$	$X_{\left(MgO\right)}$	$X_{(A{l}_2{O}_3)}$	$N_{\left(CaO\right)}$	$N_{(Si{O}_2)}$	$N_{\left(MgO\right)}$	$N_{(A{l}_2{O}_3)}$
1	1.0	0.478	0.446	0.030	0.046	0.0118	0.0037	0.0100	0.1159
2	1.1	0.499	0.424	0.050	0.045	0.0171	0.0049	0.0124	0.0720
3	1.25	0.529	0.395	0.030	0.046	0.0309	0.0090	0.0213	0.0327
4	1.5	0.570	0.355	0.030	0.045	0.0768	0.0229	0.0167	0.0080
5	2	0.630	0.295	0.029	0.046	0.2840	0.0974	0.0054	0.0005372
6	2.5	0.674	0.251	0.029	0.046	0.1203	0.0892	0.0100	0.0026
7	3	0.706	0.220	0.029	0.045	0.1179	0.1326	0.0105	0.0026
8	3.5	0.731	0.195	0.029	0.045	0.1603	0.1029	0.0069	0.0016
9	4	0.750	0.175	0.029	0.046	0.1437	0.1063	0.0057	0.0020

shows the slag sample configurations used to study the effect of R on SiO₂ activity. The influence of w(CaO)/w(Al₂O₃) on SiO₂ activity was analyzed with 60% w(SiO₂) and 2% w(MgO) kept constant. The law of variation of theoretical SiO₂ activity under varying alkalinity conditions is shown in Figure 6. Results show that when w(CaO)/w(SiO₂) was increased from 1 to 3, N(SiO₂) continually increased, reaching a maximum activity value of 0.1326 when w(CaO)/w(SiO₂) = 3. When w(CaO)/w(SiO₂) was increased to >3, the SiO₂ activity exhibited a downward trend, which is consistent with the results presented in Section 3.1 and Section 3.2.

The reason for the initial increase and subsequent decrease in theoretical activity value was that as R increased from 1 to 3, the relative increase in w(CaO) enhanced the generation of Ca²⁺ + O²⁻ ion pairs, which reacted with SiO₂ and Al₂O₃ to form calcium silicate and calcium aluminate phases, such as CaO-SiO₂, CaO-2SiO₂, CaO-Al₂O₃, and CaO-2Al₂O₃. Simultaneously, SiO₂ decreased with the increase in R, weakening the reaction between the Ca²⁺ + O²⁻ ion pair and SiO₂, as shown in Equations (10) to (12), which reduced the activity of calcium silicate composite molecules, such as CaO-SiO₂ and CaO-2SiO₂, increasing the activity of SiO₂ molecules. When R was increased further to >3 and the content of Al₂O₃ in the reference slag remained constant, the generation of aluminate ions by the reaction between Al₂O₃ and O²⁻ (dissociated from CaO) was weakened and the generated O²⁻ consumed more acidic oxide SiO₂, resulting in a continuous decrease in SiO₂ activity.

4. Conclusions

Based on the molecular ion coexistence theory, an activity calculation model of the CaO-SiO₂-MgO-Al₂O₃ slag system was established and the influence of varying slag composition on SiO₂ activity was calculated, allowing the following conclusions to be obtained:

(1)

According to the degree of coincidence between the theoretical prediction value and the literature value, the activity prediction model based on the molecule-ion coexistence theory is feasible, using simple model calculations to generate reasonable and high-precision predictions.

(2)

When w(CaO) = 60% and w(MgO)/w(Al₂O₃) = 0.25, an increased w(SiO₂) caused the SiO₂ activity to increase initially and then decrease gradually, with SiO₂ activity reaching a maximum value of 0.1359 when w(SiO₂) was 17.5%.

(3)

When w(SiO₂) = 15% and w(MgO)/w(Al₂O₃) = 0.25, the w(CaO) in slag was negatively correlated with SiO₂ activity. With increasing w(CaO), SiO₂ activity gradually decreased, reaching a maximum value of 0.1268 when the w(CaO) was 55%.

(4)

SiO₂ activity can be maintained at a high level by controlling w(CaO)/w(Al₂O₃) and w(CaO)/w(SiO₂). When w(SiO₂) was 15%, it was found that controlling the w(CaO)/w(Al₂O₃) to ≤3 can effectively improve SiO₂ activity. In addition, when w(MgO) was 2%, w(Al₂O₃) was 8%, and w(CaO)/w(SiO₂) was 3, the highest level of SiO₂ activity was obtained, while further increases in w(CaO)/w(SiO₂) from 3 to 4 caused the activity of SiO₂ to decrease from 0.1326 to 0.1063.