Non-Arrhenius Viscosity Models for Molten Silicate Slags with Constant Pre-Exponential Parameter: A Comparison to Arrhenius Model

Introduction

Viscosity is among the most important thermophysical parameters for molten slags. It plays critical roles in nearly all the chemical reactions and physical transformations. Although measurement [1, 2] and modeling [3–5] of viscosity of silicate melts have been performed for many years, a thorough understanding of slag viscosity is still a challenge [6].

One of the most fundamental issues in viscosity modeling is its dependence on temperature. A brief summary of the temperature dependence of various models is listed in Table 1. Shaw [3] and Bottinga and Weill [4] developed some of the earliest viscosity predicting models, which are based on a number of high-temperature viscosity measurement. In their model, they adopted the classical two-variable Arrhenius equation:

where η is viscosity, T is temperature, A is fitting parameter and E represents the activation energy.

The Arrhenius model has been commonly used in the viscosity model of molten slag. Mills and Sridhar [7] investigated an Arrhenius viscosity model based on the optical basicity corrected for the cation required for the charge balance of AlO₄^5–. Shankar et al. [8] proposed an Arrhenius viscosity model also based on a new definition of optical basicity. Iida et al. [9] derived a famous expression for accurate prediction of the viscosities of various industrial slags, especially the blast furnace slags, using an Arrhenius model based on the concept of network structure. Nakamoto et al. [10] proposed an Arrhenius model to evaluate the viscosity of silicate melts on the basis of the bonding states of oxygen in the silicate structure, considering the flow mechanism of the melts with the network structure. Grundy et al. [11], Brosh et al. [12] and Kim et al. [13] developed a series of Arrhenius-based model that links the viscosities of silicate melts to their thermodynamic properties. Shu et al. [14] and Zhang et al. [15] also developed structurally based Arrhenius models by representing the slag structure through the different types of oxygen ions formed in the melt.

The Eyring equation [16], which can be treated as Arrhenius-type model with constant pre-exponential parameter, is also used by several authors. The Eyring equation is

where h is the Planck’s constant, N_A is Avogadro’s numbers, ρ and M are the density and molecular weight of the melt, respectively, B is related to the activation energy of viscous flow and structural rearrangement of the melt.

Seetharaman and Du [17] presented an Eyring-type model for estimating the viscosities of multicomponent ionic and metallic melts at high temperature, by describing a correlation between the Gibbs energy of activation for viscosities and Gibbs energy for mixing. Kondratiev et al. [18] developed a structurally based quasi-chemical viscosity model in Eyring type for fully liquid slags in Al₂O₃–CaO– “FeO”–MgO–SiO₂ systems, by linking the slag viscosities to the internal structures of the melts through the concentrations of various viscous flow structure units. This model was recently modified by Suzuki and Jak [19].

Another widely used two-variable viscosity model for molten slag is the Weymann [20]–Frenkel [21] (WF) relation

where A and E are fitting parameters.

Urbain et al. [22] developed one of the most widely used slag viscosity prediction model in WF expression based on CaO–Al₂O₃–SiO₂ systems, by classifying the slag constituents into three categories: glass former, modifiers and amphoterics. This model was latterly modified by several researchers to include more constituents [23–26]. Riboud et al. [27] also proposed a similar WF-type model by classifying the slag components into five different categories: network formers, network breakers, X_”Al2O3”, X_CaF2 and X_”Na2O”. Zhang and Jahanshahi [28] proposed a structurally related WF-type model for silicate melts by relating viscosity to the degree of polymerization. Ray and Pal [29] and Hu et al. [30] developed some simple viscosity models in WF form based on the concept of optical basicity. Costa e Silva [31] also presented a WF-type model as a potential alternative to estimating slag viscosity directly from computational thermodynamics software, based on the Riboud model combined with thermodynamic cell model. However, in WF equation, the viscosity may not decrease with increasing temperature at extreme high temperature, which is contract with current viscosity theories [32, 33].

It should be noted that the activation energy is set as a constant in the Arrhenius equation. However, in fact, it is temperature dependent, and generally increases significantly on cooling [34]. For a narrow temperature range, i.e. for molten slag, the activation energy can be regarded as constant. However, over a wide temperature range, where the influence of temperature cannot be ignored for most of the time [35], the viscosity–temperature relationship is super-Arrhenius, namely the log η vs 1/T exhibits an upward curve rather than linear behavior in Arrhenius form.

To describe the non-Arrhenius temperature dependence of viscosity, at least three-variable models should be used. The Vogel [38]–Fulcher [39]–Tammann [40] (VFT) equation is historically the most frequently applied non-Arrhenius viscosity model, although not usually used for slag:

where A, B and C are fitting parameters, and the parameter A is the value of log η at infinite temperature, B corresponds to the pseudo-activation energy associated with viscous flow and C is the temperature at which viscosity becomes infinite.

Mauro, Yue, Ellison, Gupta and Allan (MYEGA) [41] derived a physical meaningful model based on energy landscape analysis and the temperature-dependent constraint model for configurational entropy, which is

where A, B and C are fitting parameters, and the parameter A is also the value of log η at infinite temperature, B is effective activation energy and C relates to the energy difference between intact and broken states of network constraints.

The Arrhenius equation is widely used to describe viscosity of molten slag because it is very simple and fully consistent with the available data. However, due to the nature of non-Arrhenius temperature dependence of viscosity over wide temperature range, the high-temperature viscosity equations may not be accurately extrapolated to low temperature. The low-temperature viscosity is directly associated with the glass transition temperature, T_g, which is often defined as the temperature at the viscosity of 10¹² Pa s [42]. T_g is crucial for understanding glass formation and crystallization kinetics of molten slag, and it is a vital parameter in the glass formation ability criterions [43]. The glass formation ability of metallurgical slags has practical meanings. During continuous casting, the mold flux will form crystalline and glassy layers between steel shell and copper mold. The relative depth of the two kinds of lay is determined to a great extent by the glass-forming ability of the molten mold flux. The formation of crystalline layer will cause an increase in the thermal resistance of mold/slag interface [44, 45]. As a result, crystallization behavior of the mold fluxes significantly affected the slab surface quality [46]. For molten blast furnace slag or copper slag, which are always vitrified to obtain glassy slag and recycled in cement and glass–ceramic industry [47–50], the glass-forming ability is also very important, as glass content is a key determining factor of cement hydraulic activity [51].

In order to predict the low-temperature viscosity from high-temperature values, the three-variable non-Arrhenius models, like VFT or MYEGA equations, should be used. However, these models have one more variable than the Arrhenius equation, which significantly complicates the modeling. An effective solution to simplify the non-Arrhenius models is to set the pre-exponential parameter A as a constant. In both VFT and MYEGA equations, the pre-exponential parameter A has the same meaning of high-temperature viscosity limit. Studies reveal that there is no systematic dependence of the high-temperature viscosity limit on chemical composition for silicate melts, and the viscosity converges to a common value at infinite temperature [52, 53]. Thus, setting the pre-exponential parameter as a constant will simplify the model substantially. Moreover, the fitting accuracy is also kept to a high rate. Recently, the authors have established viscosity models for molten blast furnace slags based on VFT model with a constant pre-exponential parameter [36, 37]. However, the accuracy of non-Arrhenius model with constant pre-exponential parameter over wide composition ranges of molten slags has not been investigated. In this work, the high-temperature viscosity fitting accuracy and low-temperature viscosity extrapolation accuracy of Arrhenius and non-Arrhenius model with constant pre-exponential parameters were compared, based on a large molten slag viscosity database that covers a wide composition ranges.

Method

A liquid silicate slag viscosity database was established by collecting viscosity measurements from open published literatures. In this work, the molten slag corresponds to slags with viscosity lower than 10 Pa s, because the liquidus temperature was not given in most of the studies and, more importantly, the viscosity change from molten to glass states is continuous. The viscosity measurements clearly deviating from Arrhenius equation were removed, in order to exclude the measurements with the existence of solid in melt. The database consists of 824 compositions and 5,038 measurements. It covers silicate melts from binary, ternary, quaternary, quinary systems to multicomponent industrial slags, commercial glasses and natural melts. The viscosity–temperature relationship is shown in Figure 1, and the composition ranges are listed in Table 2. The viscosity fitting was done using a constrained Levenberg–Marquardt algorithm to avoid local minima.

Figure 1:

Viscosity–temperature relationship of molten slags in the database.

Results and discussions

The global viscosity fitting errors of all measurements in the database for Arrhenius and WF models, along with VFT and MYEGA models with different pre-exponential parameter, are shown in Figure 2. It is seen that the fitting error of Arrhenius model is slightly higher than WF model. The fitting errors of VFT and MYEGA models depend on the value of pre-exponential parameter. Over wide ranges of pre-exponential parameter, i.e. –5.4 to –1.8 for VFT model and –5.2 to –1.6 for MYEGA model, the fitting errors of VFT and MYEGA models are lower than the Arrhenius model. The errors reach minima at –2.8 for VFT and –2.3 for MYEGA. It indicates that for molten slag, the VFT and MYEGA models with appropriate pre-exponential parameters are more accurate than the Arrhenius model.

Figure 2:

Molten slag viscosity fitting errors of Arrhenius, WF models and VFT and MYEGA models with constant pre-exponential parameter.

Previous studies suggest that the pre-exponential parameter of VFT and MYEGA models is independent of compositions [52, 53]. Current study comes to the same conclusion. As shown in Figure 3, there is no obvious trend of fitting errors with SiO₂ content or effective network modifier content, expressed as RO + R₂O + 2CaF₂–Al₂O₃–B₂O₃–Fe₂O₃–2TiO₂. Therefore, the VFT and MYEGA models with constant pre-exponential parameter could be applied to multicomponent slag systems.

Figure 3:

Dependences of viscosity fitting error on slag compositions: (a) VFT, SiO₂; (b) VFT, RO + R₂O + 2CaF₂–Al₂O₃–B₂O₃–Fe₂O₃–2TiO₂; (c) MYEGA, SiO₂; (d) MYEGA, RO + R₂O + 2CaF₂–Al₂O₃–B₂O₃–Fe₂O₃–2TiO_2.

For Arrhenius and WF models, there are clear relationships between the two fitted variables in the equations, as shown in Figure 4(a) and (b). This relationship can be used to simplify the models to only one variable [15, 25]. For VFT and MYEGA models with constant pre-exponential parameters, obvious trend can also be found between the two fitting variables, as shown in Figure 4(c) and (d). It indicates that the VFT and MYEGA models with constant pre-exponential parameters can also be potentially simplified to only one variable.

Figure 4:

Relationship between fitted variables for two-variable models: (a) Arrhenius, (b) WF, (c) VFT with constant pre-exponential parameter, (d) MYEGA with constant pre-exponential parameter.

In order to compare the viscosity extrapolation accuracy between different models, the viscosity extrapolation tests from high temperature to low temperature were also performed. In the test, each model was fitting by only high-temperature measurement with viscosity low than 10 Pa s. Then the fitted equations were extrapolated to predict the low-temperature viscosities, and compared with the measured values, as shown in Figure 5. In this work, the Arrhenius, WF, VFT with the pre-exponential parameter of –2.8 and MYEGA with the pre-exponential parameter of –2.3 were used to extrapolate viscosity from lower than 10 Pa s to as high as 10¹¹ Pa s, using the viscosity data of calcium aluminosilicate melts measured by Solvang et al. [54]. It is obvious in Figure 5 that the Arrhenius and WF models significantly underestimate the viscosity at low temperature, because the temperature dependence of viscosity over a wide temperature range is in fact super-Arrhenius. As expected, the VFT and MYEGA models with constant pre-exponential parameter predicted the low-temperature viscosity much more accurate than the Arrhenius model. For VFT and MYEGA models, the extrapolation accuracy also depends on pre-exponential parameter. Figure 6 presents the average prediction error in 10¹⁰ Pa s isokom temperature for the same 10 calcium aluminosilicate melts. It is seen that the prediction errors reach minima at –3.1 for VFT model and –1.8 for MYEGA model, which are close to the values with minimal fitting error. It is also found over the entire range of pre-exponential parameter investigated, the prediction errors are lower than Arrhenius and WF models.

Figure 5:

Viscosity extrapolation from high temperature to low temperature for 10 calcium aluminosilicate melts.

Figure 6:

Prediction error in 10¹⁰ Pa s isokom temperature for ten calcium aluminosilicate melts with different pre-exponential parameters.

The key point to apply VFT and MYEGA model with pre-exponential parameter is to define the optimized pre-exponential parameters. From a general view of point, the values of –2.8 for VFT and –2.3 for MYEGA are recommended, because they correspond to the smallest fitting errors for high-temperature viscosities and satisfactory low-temperature viscosity extrapolation accuracies. It should be noted that the recommended values are best to be used for multisystem models. For a specific system with few constituents, technically speaking, a more desired value of pre-exponential parameter may exist. Nevertheless, selecting an individual value for each system will significantly complicate the modeling. The recommended values were derived from a very large molten slag viscosity database, and the fitting errors are independent of compositions. Therefore, the recommended values can also be applied reliably to subsystems within the viscosity database. Although the recommended values cannot lead to “best fitting” for any molten slag systems, it is still expected that the viscosity fitting and extrapolation accuracies are higher than the Arrhenius model.

Conclusions

In summary, the molten slag viscosity fitting accuracy and viscosity extrapolation accuracy from high to low temperature were compared between Arrhenius model, WF model and VFT and MYEGA model with constant pre-exponential parameter in current work. The studies are based on a molten slag viscosity database consisting of over 800 compositions and 5,000 measurements. It is found that over wide ranges of pre-exponential parameter, the VFT and MYEGA models have lower viscosity fitting errors and much higher low-temperature viscosity extrapolation accuracies than the Arrhenius and WF models.

For a general multisystem model, the pre-exponential parameter values of –2.8 for VFT and –2.3 for MYEGA are recommended, whereas more accurate predictions can be reached if the pre-exponential parameter is selected for the system in question. Nevertheless, the recommended values can also be applied to subsystems within the viscosity database, and higher viscosity fitting and extrapolation accuracies than the Arrhenius model are expected.