Use of a Structural Model to Calculate the Viscosity of Liquid Silicate Systems

Abstract

A viscosity model for binary and ternary silicate melts is proposed in this article. The temperature dependence of viscosity is expressed using the Arrhenius equation and the composition dependence is made through the concentration of oxygen bridges (Si–O–Si) in the silica structure. A previous proposed structural thermodynamic model is used to calculate the content of oxygen bridges. The model requires only three parameters to obtain a good agreement between experimental and calculated data for the SiO₂–CaO, SiO₂–MgO, SiO₂–MnO, and SiO₂–Na₂O binary systems. The viscosity of ternary systems is calculated with the model assuming a linear function of the parameters from binary systems; however, the content of oxygen bridges is calculated using the structural thermodynamic model for ternary systems. Comparison is made between the experimental and model results for the SiO₂–CaO–MgO, SiO₂–CaO–MnO, and SiO₂–Na₂O–MgO systems. The viscosity model can take into account the effect of substituting one metal oxide for another in the ternary systems.

1. Introduction

Slag viscosity is an essential parameter for the metal manufacturing since it affects the appropriate refining process and separation of the slag and the liquid metal, and the rate of tapping from the furnace. The slag viscosity is related to the structure of the oxide melts and is very sensitive to the changes of temperature and composition. Viscosities of liquid slag systems have been extensively studied and many models have been developed. The early models are based on the well-known Arrhenius equation to describe the temperature dependence of silicate viscosities; however, the dependency on the composition is fully empirical. The Urbain approach¹⁾ is one of the most widely used slag viscosity models and it considers that the slag constituents are classified into three categories: glass former (SiO₂), modifiers (CaO), and amphoterics (Al₂O₃). In another viscosity model, Riboud et al.²⁾ classified the oxide components of the slag into five categories, depending on their ability to break or form polymeric chains in the molten slag.

Structural based models have been developed recently considering that the silica network structure breaks down with the addition of basic oxides giving a more depolymerized structure. These models consider that the silicate melts contain three types of oxygen: (1) bridging oxygen bonded to two silicon atoms (O°); (2) nonbridging oxygen bonded only to one silicon atom (O⁻), and (3) free oxygen bonded to no silicon atom (O²⁻). The main concern in these structural models is to estimate the three types of oxygen. Zhang et al.³⁾ proposed a viscosity model considering complete bridge breaking. Zhang and Jahanshahi⁴⁾ used the cell model to estimate the fractions of the bridging oxygens and free oxygen, whereas Kondratiev and Jak⁵⁾ calculated the concentration of the three types of oxygens using the quasi-chemical thermodynamic model.

Another important feature of some structural-based viscosity models is the extrapolation for ternary or high order systems. Zhang and Jahanshahi⁴⁾ assumed that the model parameters for higher-order systems are linear functions of those of binary silicate systems, that is, CaO and MgO play a similar role in the SiO₂–CaO–MgO system. Han et al.⁶⁾ introduced a ‘probability term’ to describe the fraction of cations (Ca²⁺ or Mg²⁺) to break the Si–O–Si network.

The present model uses the structural model formerly proposed by Lin and Pelton⁷⁾ and subsequently extended by Romero and Pelton⁸⁾ to estimate the concentration of the types of oxygens in binary and ternary silicate systems. This structural model has been used to calculate the thermodynamic properties and the phase diagrams for binary and ternary systems as well as to estimate the sulphide capacity of binary silicate melts.

2. The Model

2.1. Thermodynamic Model

A detailed development was given previously.⁸⁾ Only a brief summary will be presented here. In a binary solution MO–SiO₂ the mole fractions of the components are:   

$${\text{X}}_{\text{MO}}=\frac{{\text{n}}_{\text{MO}}}{{\text{n}}_{\text{MO}}+{\text{n}}_{{\text{SiO}}_{\text{2}}}}$$(1)

  

$${\text{X}}_{{\text{SiO}}_{\text{2}}}=1-{\mathrm{X}}_{\mathrm{M}\mathrm{O}}$$(2)

where n_MO and n_SiO2 are the number of moles of MO and SiO₂. Let N_O^2-, N_O^- and N_O°, represent the numbers of moles of the various oxygen species per mole of solution:   

$${\text{N}}_{{\text{O}}^{\text{2}-}}=\frac{{\text{n}}_{{\text{O}}^{\text{2}-}}}{{\text{n}}_{\text{MO}}+{\text{n}}_{{\text{SiO}}_{\text{2}}}}$$(3)

  

$${\text{N}}_{{\text{O}}^{-}}=\frac{{\text{n}}_{{\text{O}}^{-}}}{{\text{n}}_{\text{MO}}+{\text{n}}_{{\text{SiO}}_{\text{2}}}}$$(4)

  

$${\text{N}}_{\text{O}°}=\frac{{\text{n}}_{\text{O}°}}{{\text{n}}_{\text{MO}}+{\text{n}}_{{\text{SiO}}_{\text{2}}}}$$(5)

It is assumed that every silicon atom is bonded to four oxygen atoms. Thus, mass balance considerations require that:   

$${\text{N}}_{\text{O}°}=\text{2}\mathrm{\hspace{0.17em} }{\text{X}}_{{\text{SiO}}_{\text{2}}}-\frac{{\text{N}}_{{\text{O}}^{-}}}{2}$$(6)

  

$${\text{N}}_{{\text{O}}^{\text{2}-}}={\text{X}}_{\text{MO}}-\frac{{\text{N}}_{{\text{O}}^{-}}}{2}$$(7)

The configurational entropy (S^c) is physically related to the number of ways in which particles themselves can be distributed in space. The non-configurational entropy (S^nc), or thermal entropy, results from the motion of particles in the system, such as translation, rotation, vibration, etc., and is proportional to the amount of energy stored as chaotic molecular motion. The configurational entropy of the melt is calculated through the multiplicity for the random distribution of the Si atoms and O²⁻ ions on the sites (Ω₁) and the multiplicity of the random distribution of the N_O° bridging oxygens atoms over the number of moles of neighboring Si–Si pairs positions (Ω₂)   

$${\mathrm{S}}^{\mathrm{c}}=(\mathrm{R}/\mathrm{N}°)\mathrm{\hspace{0.17em} }\mathrm{l}\mathrm{n}\mathrm{\hspace{0.17em} }[({\mathrm{\Omega }}_1{\mathrm{\Omega }}_2]$$(8)

where R is the gas constant and N° is Avogadro’s number. The structural model assumes that the depolymerization reaction is associated with the Gibbs energy change containing an enthalpic (ω) and entropic (η) term:   

$${\mathrm{O}}^{2-}+\mathrm{O}°=2{\mathrm{O}}^{-}$$(9)

  

$$\mathrm{\Delta }\text{H}-{\text{TS}}^{\text{nc}}=\left(\frac{{\text{N}}_{{\text{O}}^{-}}}{\text{2}}\right)\omega -\text{T}\left(\frac{{\text{N}}_{{\text{O}}^{-}}}{\text{2}}\right)\eta$$(10)

Finally, ω and η are expanded as polynomials:   

$$\omega ={\omega }_{\text{0}}+{\omega }_{\text{1}}\mathrm{\hspace{0.17em} }{\text{X}}_{{\text{SiO}}_{\text{2}}}+{\omega }_{\text{2}}\mathrm{\hspace{0.17em} }{\text{X}}_{{\text{SiO}}_{\text{2}}}^{\text{2}}+\mathrm{...}$$(11)

  

$$\eta ={\eta }_{\text{0}}+{\eta }_{\text{1}}\mathrm{\hspace{0.17em} }{\text{X}}_{{\text{SiO}}_{\text{2}}}+{\eta }_{\text{2}}\mathrm{\hspace{0.17em} }{\text{X}}_{{\text{SiO}}_{\text{2}}}^{\text{2}}+\mathrm{...}$$(12)

The coefficients ω_i and η_i are the parameters of the thermodynamic model, which are obtained by optimisation of the data. Given a composition X_SiO2 and values of the parameters ω_i and η_i, the actual value of N_O^- can be calculated by minimising the Gibbs energy at constant X_SiO2, ω and η:   

$$\mathrm{\Delta }\mathrm{G}=\mathrm{\Delta }\mathrm{H}–\mathrm{T}({\mathrm{S}}^{\mathrm{c}}+{\mathrm{S}}^{\mathrm{n}\mathrm{c}})$$(13)

Figure 1 shows concentrations of the O°, O⁻, and O²⁻ species calculated by the model at 1873 K for SiO₂–CaO, SiO₂–MgO, SiO₂–MnO, and SiO₂–FeO systems. It is worth to note that the ‘basicity’ of the oxides increase in the order FeO < MnO < MgO < CaO and the concentration of the broken oxygen bridges (N_O^-) also increases in the same order.

Fig. 1.

Concentration of oxygen species calculated by the structural model at 1873 K.

Flood and Förland⁹⁾ proposed the concept of slag basicity in terms of the activity of the oxygen ion ($a_{O^{2-}}$) and the reported order of the oxides, with increasing basicity, is the same as the order mentioned above. This same order of the basicity was obtained by Duffy and Ingram¹⁰⁾ using the concept of optical basicity.

In the thermodynamic model for SiO₂-MO systems (M = Ca, Mg, Mn, etc.), the configurational entropy was calculated solely from the distribution of the various oxygen species. In SiO₂-rich melts, this may be taken to mean that the M²⁺ cations remain associated with the broken bridges in order to maintain local charge neutrality. In the binary silicate melt SiO₂–Na₂O in the highly acid region, it is similarly assumed that both the Na⁺ cations remain associated with the broken bridges:   

$$\text{Si}-\text{O}-\text{Si}+{\text{Na}}_{\text{2}}\text{O }=\begin{array}{c}\mathrm{N}{\mathrm{a}}^{+}\mathrm{\hspace{0.17em} }\mathrm{N}{\mathrm{a}}^{+}\\ \text{Si}-\text{O}\hspace{1em}\text{O}-\text{Si}\end{array}$$(14)

This assumption has been tested using the limiting slope criteria of the liquidus curve at X_SiO2→1 in the SiO₂–Na₂O phase diagram.⁸⁾ This liquidus curve calculated by the model agrees well with the experimental results.

To expand the model for ternary systems we considered 5 kinds of oxygen.¹¹⁾ Let us consider the general SiO₂-AO-BO ternary system where A and B are divalent cations, such as Ca²⁺ and Mn²⁺:

1. O^°: Oxygen bridge

2. O_A²⁻: ‘Free oxygen ions’ introduced in the system by the AO species

3. O_B²⁻: ‘Free oxygen ions’ introduced in the system by the BO species

4. O_A⁻: Broken bridges for the AO species

5. O_B⁻: Broken bridges for the BO species

There are two depolymerization reactions given by:   

$$\begin{array}{ccc}& \mathrm{A}& \\ \mathrm{S}\mathrm{i}-\mathrm{O}-\mathrm{S}\mathrm{i}+\mathrm{A}\mathrm{O}=& \mathrm{S}\mathrm{i}-\mathrm{O}\mathrm{\hspace{1em}}\mathrm{O}-\mathrm{S}\mathrm{i}& \hspace{1em}\hspace{1em}{(\omega -\eta \mathrm{T})}_{\mathrm{A}\mathrm{O}}\end{array}$$(15)

  

$$\begin{array}{ccc}& \mathrm{B}& \\ \mathrm{S}\mathrm{i}-\mathrm{O}-\mathrm{S}\mathrm{i}+\mathrm{B}\mathrm{O}=& \mathrm{S}\mathrm{i}-\mathrm{O}\hspace{1em}\mathrm{O}-\mathrm{S}\mathrm{i}& \hspace{1em}\hspace{1em}{(\omega -\eta \mathrm{T})}_{\mathrm{B}\mathrm{O}}\end{array}$$(16)

The mass balance considerations now require that:   

$${\text{N}}_{\text{O}°}=\text{2}\mathrm{\hspace{0.17em} }{\text{X}}_{{\text{SiO}}_{\text{2}}}-\frac{{\text{N}}_{{\text{O}}_{\text{A}}^{-}}+{\text{N}}_{{\text{O}}_{\text{B}}^{-}}}{\text{2}}$$(17)

  

$${\text{N}}_{{\text{O}}_{\text{A}}^{\text{2}-}}={\text{X}}_{\text{AO}}-\frac{{\text{N}}_{{\text{O}}_{\text{A}}^{-}}}{\text{2}}$$(18)

  

$${\text{N}}_{{\text{O}}_{\text{B}}^{\text{2}-}}={\text{X}}_{\text{BO}}-\frac{{\text{N}}_{{\text{O}}_{\text{B}}^{-}}}{\text{2}}$$(19)

The expression of the entropy is obtained by making the statistical distributions of the different types of oxygens as well as the neighboring Si–Si pairs. The excess free energy expression for the ternary system is obtained by the addition of the interaction energy terms (ω-ηT) for each bridge breaking reaction, Eqs. (15) and (16), which are known in the two binary systems (SiO₂-AO and SiO₂-BO) from the binary optimisations. This expression also must include the contribution of the excess free energy for the AO-BO binary system, G^E_AO-BO, which is multiplied by the fraction of free oxygen ions in the quasi lattice whose sites are occupied by O²⁻ ions and Si atoms.

The metallic oxides, such as CaO and MnO, must play a different role in the silicate network. Assuming that A represents Ca and B represents Mn and combining the Eqs. (15) and (16) results:   

$$\begin{array}{ll}\hspace{1em}\hspace{1em}\mathrm{M}{\mathrm{n}}^{2+}& \hspace{1em}\hspace{1em}\mathrm{C}{\mathrm{a}}^{2+}\\ \mathrm{S}\mathrm{i}-\mathrm{O}\mathrm{\hspace{1em}}\mathrm{O}-\mathrm{S}\mathrm{i}+\mathrm{C}\mathrm{a}\mathrm{O}=& \mathrm{S}\mathrm{i}-\mathrm{O}\mathrm{\hspace{1em}}\mathrm{O}-\mathrm{S}\mathrm{i}+\mathrm{M}\mathrm{n}\mathrm{O}\mathrm{\hspace{1em}}\mathrm{\Delta }\omega \end{array}$$(20)

where Δω = ${(\omega -\eta \mathrm{T})}_{\mathrm{C}\mathrm{a}\mathrm{O}-\mathrm{S}\mathrm{i}{\mathrm{O}}_2}$ − ${(\omega -\eta \mathrm{T})}_{\mathrm{M}\mathrm{n}\mathrm{O}-\mathrm{S}\mathrm{i}{\mathrm{O}}_2}$. At 1873 K and X_SiO2 = 1/3 the value of Δω for the SiO₂–CaO–MnO system is about –69 kJ/mol. Thus, reaction (20) moves to the right and the cations Ca²⁺ will be found associated to the O⁻ species and the cations Mn²⁺ will be close to the free oxygen ions O²⁻. Figure 2 shows the oxygen bridges (N_O°) in terms of the X_MnO/(X_CaO+X_MnO) for the SiO₂–CaO–MnO system at X_SiO2 = 1/3 and 1873 K. It is clear that substitution of MnO for CaO increases the oxygen bridges and the change of N_O° is not a linear function.

Fig. 2.

Concentration of oxygen bridges in the SiO₂–CaO–MnO system calculated at 1873 K and X_SiO2 = 1/3 by the structural model.

2.2. Viscosity Model

2.2.1. Binary Systems

In the present study the viscosity for binary SiO₂-MO systems (M = Ca, Mg, Mn, Na₂, etc.) is expressed as follows:   

$$\mathrm{I}\mathrm{n}\mathrm{\hspace{0.17em} }\eta =\mathrm{A}+\frac{\mathrm{B}}{\mathrm{T}}+\mathrm{C}\mathrm{\hspace{0.17em} }{\text{N}}_{\text{O}°}$$(21)

The temperature dependence of viscosity is described by the Arrhenius equation, where η is viscosity in Pa·s, A is the natural logarithm of the pre-exponential term, B is the activation energy over the gas constant (E/R), and T is the absolute temperature. The analysis of the available data on the composition dependence of viscosity for several binary silicate systems showed that, at a given temperature, ln(η) is almost a linear function of the concentration of oxygen bridges (N_O°), as can be shown in Fig. 3, which shows the logarithm of the viscosity vs N_O° for the SiO₂–CaO system at 1873 and 1973 K. Parameter C in Eq. (21) gives to the linear relationship between ln(η) experimentally calculated^12,13,14) and the concentration of oxygen bridges (N_O°) calculated by the structural model.

Fig. 3.

Linear function between oxygen bridges and logarithm of the viscosity for the SiO₂–CaO system at 1873, and 1973 K.

These results show that only three parameters are needed to calculate the viscosity in terms of both, composition and temperature, in binary silicate systems SiO₂-MO. The values of these parameters for the SiO₂-MO (M = Ca, Mg, Mn, Na₂) binary systems were obtained by regression of viscosity data and the results are shown in Table 1.

Table 1. Values of model parameters for different binary silicate systems.

System	A	B	C
SiO2–CaO	−13.2523	18429.96	4.0069
SiO2–MgO	−14.5122	21061.97	4.1265
SiO2–MnO	−12.4586	14048.62	6.0765
SiO2–Na2O	−17.0930	22330.14	4.6667

2.2.2. Ternary Systems

The model is expanded for ternary silicate systems using a linear relationship of the model parameters of the binary silicate systems, that is, if Y represents any of the fitting binary parameters A, B, or C of Eq. (21), Z in the SiO₂-AO-BO ternary system is obtained as:   

$$\mathrm{Z}=\left[\frac{{\mathrm{X}}_{\mathrm{A}\mathrm{O}}}{{\mathrm{X}}_{\mathrm{A}\mathrm{O}}+{\mathrm{X}}_{\mathrm{B}\mathrm{O}}}\right]{\mathrm{Y}}_{\mathrm{S}\mathrm{i}{\mathrm{O}}_2-\mathrm{A}\mathrm{O}}+\left[\frac{{\mathrm{X}}_{\mathrm{B}\mathrm{O}}}{{\mathrm{X}}_{\mathrm{A}\mathrm{O}}+{\mathrm{X}}_{\mathrm{B}\mathrm{O}}}\right]{\mathrm{Y}}_{\mathrm{S}\mathrm{i}{\mathrm{O}}_2-\mathrm{B}\mathrm{O}}$$(22)

However, the bridge breaking concentration (N_O°) of Eq. (21) is calculated with the structural model for ternary systems. This value cannot be considered as a linear function of the binary N_O° parameters, as has been shown in Fig. 2.

3. Modeling Results and Discussion

3.1. Binary Systems

Figure 4 shows the experimental and calculated values^12,13,14,15) of viscosity of the SiO₂–CaO system at 1873 and 1973 K, where it can be seen that they are in good agreement. Schumacher et al.¹⁵⁾ reported the viscosities for the system SiO₂–CaO–MgO with low concentrations of MgO, so we included in Fig. 4 their results for the slag systems with less than 1.5 wt.% MgO. It is worth to note that with only three parameters and the value of the oxygen bridges (N_O°) the model can appropriately represent the experimental data as well as estimate the viscosity of this system at temperatures and compositions not obtained experimentally, even at compositions close to pure SiO₂ and compositions X_SiO2<1/3, where silica structure consists of almost no bridging oxygen ions.

Fig. 4.

Calculated (lines) and experimental (points) viscosities at 1873, and 1973 K of SiO₂–CaO melts.

Figures 5, 6, 7 show that the viscosity values calculated with the model agree well with the experimental data for the systems SiO₂–MgO, SiO₂–MnO, and SiO₂–Na₂O, respectively. The viscosity of these systems as well as the SiO₂–CaO system increases monotonically with increasing silica content.

Fig. 5.

Calculated (lines) and experimental (points) viscosities at 1873, 1973, and 2073 K of SiO₂–MgO melts.

Fig. 6.

Calculated (lines) and experimental (points) viscosities at 1773, and 1873 K of SiO₂–MnO melts.

Fig. 7.

Calculated (lines) and experimental (points) viscosities at 1473, 1573, and 1673 K of SiO₂–Na₂O melts.

It must be stressed that the structural model used in this work was carried out under the assumption of tetrahedral coordination where the number of moles of neighboring Si–Si pairs per mole of solution is 4 (as in the cristobalite structure). This model has been used for calculating the thermodynamic properties and phase diagrams for binary and ternary silicate melts and glasses’ systems.^8,11) In addition, sulphide ion has been incorporated into the model, substituting quasi-lattice sites for O²⁻ to calculate the sulphide capacity of the binary silicate melts.

This thermodynamic model also predicts the chain-length distribution of polymeric silicate chains. It is generally accepted that basic silicate melts (X_SiO2 < 1/3) contain orthosilicate anions SiO₄⁴⁻ which, at higher silica concentrations, can polymerize to form dimers Si₂O₇⁶⁻, trimers Si₃O₈¹⁰⁻, etc., and then increase the viscosity of the melt. Another important feature of this structural model is that the concentrations of these silicate anions can be calculated even though they are not explicitly considered in the formulation of the model.⁸⁾

3.2. Ternary Systems

Figures 8, 9, 10 show the viscosity calculated by the model and the experimental data for the SiO₂–CaO–MgO, SiO₂–CaO–MnO and SiO₂–Na₂O–MgO, respectively. These figures show the limiting liquidus curve at the temperatures considered. It can be observed from these figures that the calculated viscosities agree well with the experimental data.

Fig. 8.

Isoviscosity curves (Pa s) of SiO₂–CaO–MgO melts at 1773 K.

Fig. 9.

Isoviscosity curves (Pa s) of SiO₂–CaO–MnO melts at 1823 K.

Fig. 10.

Isoviscosity curves (Pa s) of SiO₂–Na₂O–MgO melts at 1623 K.

Several works have been dealing with the effect on viscosity of substituting one basic metal oxide with another oxide at a given silica content in ternary systems. Urbain¹⁾ states that the effect of the metal oxides on viscosity is similar, i.e., the viscosity of the ternary silicate system is a linear function of the viscosity of the binary silicate systems. This assumption may not be justified in all the systems and all the compositions. Zhang et al.³⁾ calculated viscosities of binary and ternary silicate melts assuming ‘complete bridge breaking’, i.e., all the metal oxides added to the silicate melt will completely break the oxygen bridges in the acid region (X_SiO2 > 1/3). Figure 1 shows that this approach is valid for all the binary silicate systems SiO₂-MO (M = Ca, Mg, Mn, Fe) at about X_SiO2 > 0.5; however, at X_SiO2 < 0.5 the limiting case of complete oxygen bridge breaking is valid only for the more basic metal oxide such as CaO and Na₂O.

The models for calculating the viscosities of ternary systems, which considers complete oxygen bridge breaking or linear function of the viscosity of binary systems to calculate the viscosity of the ternary system show good agreement between calculated and experimental data in the acid region (X_SiO2 > 0.5). This can be explained with the present model through the plot of N_O° in function of the composition. Let’s consider the N_O° and viscosity calculated by the present model for the system SiO₂–CaO–MnO at 1873 K and at two SiO₂ contents, X_SiO2 = 0.35 and 0.5. Figure 11 shows that at X_SiO2 = 0.5, both N_O° and viscosity in terms of X_MnO/(X_CaO+X_MnO) are almost linear functions of the results of the binary systems, SiO₂–CaO and SiO₂–MnO. However, the results at X_SiO2 = 0.35 show that the N_O° and viscosity of the ternary system cannot be linearly extrapolated from the results of the binary system. The calculated viscosity is higher than that of the linear extrapolation. This can be explained because MnO is not as efficient to break the oxygen bridges as CaO; then, when CaO is replaced by MnO the N_O° increases as well as the viscosity, compared with the linear extrapolation results.

Fig. 11.

Concentration of oxygen bridges and viscosity in the SiO₂–CaO–MnO system calculated by the structural model at 1873 K and X_SiO2 = 0.5 and 0.35.

It must be stressed that the model predicts that the viscosities have a maximum value in the ternary systems such as SiO₂–CaO–MnO close to CaO. Figure 12 shows the calculated and experimental²⁰⁾ viscosity results for the SiO₂–CaO–MnO system at 1773 K. The experimental results show a maximum value in this system when X_MnO/(X_CaO+X_MnO) is between 0.2 and 0.4; however, additional experimental results are required to confirm the maximum viscosity values predicted by this model.

Fig. 12.

Calculated and experimental²⁰⁾ viscosities in the SiO₂–CaO–MnO system at 1773 K and X_SiO2 = 0.45.

In this work, we tried to make the viscosity model as simple as possible and included only three parameters for each binary system, two parameters for the temperature function, and one for the composition function. We did not include additional adjusted parameters for ternary systems. There was a good agreement between experimental and calculated data, even though the model does not consider some intrinsic psychochemical properties of the metal oxides explicitly, such as the electronegativity or ionic radii of metal ions.

Several viscosity models are available in the literature with varying degrees of success. They may be classified into two main groups: structural based models and empirical or semi-empirical models. The difference between the present model and other structural models is that the former considers a coordination number of silicon equals four where each silicon atom is bonded to four oxygen atoms, as has been reported experimentally.³⁰⁾ The model proposed by Kondratiev and Jak⁵⁾ calculated the concentration of the types of oxygens using the quasi-chemical model; however, this model considers a coordination number to fit the experimental thermodynamic data and thus, this number differs from the real one. Another difference between this model and the other structural models is that the present model considers that the replacement of one metallic oxide for another in the ternary silicate systems may change the viscosity at constant SiO₂ content, i.e. there is not a linear function of the concentration of the oxygen bridges of the two binary silicate systems. Finally, the present model requires only three parameters to represent the viscosity of each binary silicate system.

4. Conclusions

A viscosity model to calculate binary silicate melts was developed considering the temperature dependence with the Arrhenius expression. Viscosity was related with the composition of the system through the concentration of the oxygen bridges (N_O°), which was calculated using a structural thermodynamic model. The experimental and model results agreed well for the SiO₂-MO (M = Ca, Mg, Mn, Na₂) binary systems. The viscosity for ternary systems, SiO₂–CaO–MgO, SiO₂–CaO–MnO, and SiO₂–Na₂O–MgO, were calculated using a linear function of the model parameters of binary systems. However, the effect of the content of the different metal oxides in the silicate structure was taken into account through the value of the oxygen bridges calculated with the thermodynamic model for ternary systems. The model may explain why the viscosity of ternary systems can be calculated as linear function of the binary systems in the silica rich composition but it cannot be a linear function in the basic composition of the system.

Acknowledgements

The authors wish to thank the Institutions CONACyT, SNI, COFAA and IPN for the support of this research.

References

• 1)   G.  Urbain: Steel Res., 58 (1987), 111.
• 2)   P. V.  Riboud,  Y.  Roux,  D.  Lucas and  H.  Gaye: Fachber. Huttenprax. Metallweiterverarb., 19 (1981), 859.
• 3)   G.-H.  Zhang,  K.-C.  Chou,  Q.-G.  Xue and  K.  Mills: Metall. Mater. Trans. B, 43B (2012), 64.
• 4)   L.  Zhang and  S.  Jahanshahi: Metall. Mater. Trans. B, 29B (1998), 177.
• 5)   A.  Kondratiev and  E.  Jak: Metall. Mater. Trans. B, 36B (2005), 623.
• 6)   C.  Han,  M.  Chen,  W.  Zhang,  Z.  Zhao,  T.  Evans and  B.  Zhao: Metall. Mater. Trans. B, 47B (2016), 2861.
• 7)   P. L.  Lin and  A. D.  Pelton: Metall. Trans. B, 10B (1979), 667.
• 8)   A.  Romero-Serrano and  A. D.  Pelton: Metall. Mater. Trans. B, 26B (1995), 305.
• 9)   H.  Flood and  T.  Förland: Acta Chem. Scand., 1 (1947), 592.
• 10)   J. A.  Duffy and  M. D.  Ingram: J. Am. Chem. Soc., 93 (1971), 6448.
• 11)   J.  Gutiérrez,  A.  Romero-Serrano,  G.  Plascencia,  F.  Chávez and  R.  Vargas: ISIJ Int., 40 (2000), 664.
• 12)   G.  Urbain,  Y.  Bottinga and  P.  Richet: Geochim. Cosmochim. Acta, 46 (1982), 1061.
• 13)   Y.  Shiraishi and  T.  Saito: J. Jpn. Inst. Met., 29 (1965), 614.
• 14)  J. O’M. Bockris and D. C. Lowe: Proc. R. Soc. Lond. A, A226 (1954), 423.
• 15)   K. J.  Schumacher,  J. F.  White and  J. P.  Downey: Metall. Mater. Trans. B, 46B (2015), 119.
• 16)  J. O’M. Bockris, J. D. MacKenzie and J. A. Kitchener: Trans. Faraday Soc., 51 (1955), 1734.
• 17)   E. F.  Riebling: Can. J. Chem., 42 (1964), 2811.
• 18)   S.  Yagi,  K.  Mizoguchi and  Y.  Suginohara: Kyushu Kogyo Daigaku Kenkyu, Hokoku Kogaku, 40 (1980), 33.
• 19)   V. I.  Sokolov,  S. I.  Popel and  O. A.  Esin: Izv. VUZ Chern. Met., 13 (1970), 40.
• 20)   L.  Segers,  A.  Fontana and  R.  Winard: Electrochim. Acta, 24 (1979), 213.
• 21)   G.  Urbain: Rev. Int. Haut. Temp. Refract., 22 (1985), 39.
• 22)   H. R.  Lillie: J. Am. Ceram. Soc., 22 (1939), 367.
• 23)   L.  Sasek: Silikaty, 21 (1977), 291.
• 24)   M.  Kawahara,  K.  Morinaga and  T.  Yanagase: J. Jpn. Inst. Met., 41 (1977), 1047.
• 25)   O.  Takeda,  T.  Ohnishi and  Y.  Sato: ISIJ Int., 54 (2014), 2045.
• 26)   I. I.  Gultyai: Izv. Akad. Nauk SSR Otdel. Tekhn. Nauk, Metall. Toplivo, 5 (1962), 51.
• 27)   J. S.  Machin and  T. B.  Yee: J. Am. Ceram. Soc., 37 (1954), 177.
• 28)   M.  Kawahara,  K.  Mizoguchi and  Y.  Suginohara: Bull. Kyushu Inst. Tecnol., 43 (1981), 53.
• 29)   F.-Z.  Ji: Metall. Mater. Trans. B, 32B (2001), 181.
• 30)   B. O.  Mysen: Structure and Properties of Silicate Melts, Elsevier, Amsterdam, (1988), 4.