Parsimonious viscosity–composition relationships for high-temperature multicomponent glass melts

ABSTRACT The activation energy of glass melt viscosity, η, is nearly constant at temperatures at which η < 100 Pa s. Provided that the preexponential factor is a composition-independent constant, only the activation energy is a function of composition, and viscosity–composition relationships of utmost simplicity can be formulated to provide a welcome advantage in computational fluid dynamics modeling of glass melting furnaces processing multicomponent glasses. Using a dataset with over 3000 viscosity values acquired experimentally for a temperature and composition region of low-activity nuclear waste glasses, we have generated three linear models for viscosity as a function of temperature and composition. Model A quantifies the effects of 20 viscosity-influencing components. Model B achieves a similar prediction accuracy after setting aside volatile components, whose concentrations may vary during glass processing. A parsimonious Model C reduces the number of viscosity-influencing components to a mere seven: Al2O3, B2O3, CaO, Li2O, Na2O, SiO2, and Others. In each model, the “Others” component summarizes the fractions of the remaining components. For all three models, the component coefficients are determined with a high confidence (low standard error) and a high coefficient of determination: 0.972 for Model A, 0.970 for Model B, and 0.949 for Model C.


Introduction
Glass melting furnaces, whether heated by burning fuel or by dissipating electric power, operate at temperatures at which glass melt viscosity is low, typically 1-10 Pa s, to enhance melt flow for faster melting and to allow efficient homogenization and fining. Below 100 Pa s, glass melt is nearly fully depolymerized, the activation energy of viscous flow is independent of temperature, and the Arrhenius equation adequately represents the viscosity-temperature relationship [1][2][3][4][5][6][7]. Because of the constant preexponential factor, the activation energy is then the only compositiondependent parameter, allowing us to express viscosity as a function of temperature and composition with a minimum number of fitting parameters. This is advantageous for both commercial glasses [1,2] and highly multicomponent waste glasses [6].
In this study, we develop Arrhenius models for lowactivity waste (LAW) glasses [8] that contain high fractions of alkali oxides, up to 27 mass% Na 2 O and 5 mass% Li 2 O, and more than 20 components. These glasses were designed for the Hanford Waste Treatment and Immobilization Plant (WTP) that will vitrify nuclear wastes of a large variety of compositions in Joule-heated melters constraining viscosity to 2-8 Pa s at 1150°C [9].
To guarantee acceptable product quality (chemical durability), to enable easy processing (viscosity in 2-8 Pa s range), and to maximize the fraction of waste components in glass (the waste loading), glass formulation is facilitated by constructing mathematical models that represent property-composition relationship in terms of mathematical functions fitted to a set of experimental data [10]. The models allow formulating glasses that minimize the WTP product volume and shorten the duration of waste cleanup [11]. The viscosity model is also used for simulating the melt flow and temperature fields to facilitate a flawless melter operation and evaluate the effects of melter design features on glass production efficiency [12][13][14].
Section 2 presents the viscosity approximation function for the Arrhenius model, Section 3 describes the LAW viscosity database, and Section 4 presents the results in terms of component coefficients and component effects, defines linear-nonlinear crossover, and mentions melt structural causes of component effects. Section 5 compares the Arrhenius model with previous approaches to viscosity-composition relationships. Section 6 summarizes major conclusions and outlines an approach to nonlinear models.

Theory
Glass melt viscosity, η, is a function of glass composition and melt temperature, η(T, x), where T is the temperature and x is the composition vector (glass composition as a point in the composition space). It is conveniently displayed as the logarithm of viscosity versus inverse absolute temperature and expressed in the form of the Arrhenius relationship where η 1 is the preexponential factor (an asymptotic viscosity value as T ! 1) and B is the activation energy in thermal units (K). For glass within a limited composition region, η 1 is a composition-independent constant, while B is a function of both temperature and composition. For low viscosities, those at η < 10 z Pa s, where z is approximately 2, B is virtually independent of temperature and thus is solely a function of composition [1,2,4,6,7]: where is the composition vector. In Eq. (3), x i (i = 1, 2, . . ., N) is the ith component mass fraction and N is the number of viscosity-affecting components. Generally, B(x) function is nonlinear but can be approximated as linear if the composition region is sufficiently narrow in terms of the component concentration ranges [4,6]: where B i (i = 1, 2, . . ., N) is the ith component coefficient. In the linear model, these coefficients are treated as constants (independent of composition). As shown below (Section 4.4), the component coefficients can be alternatively defined for the composition vector expressed in terms of mole fractions. Mass fractions are favored for technological applications.
If the span of a component concentration, such as that of Na 2 O in LAW glasses, is relatively large, the B(x) nonlinearity is attributed to interactions with other components, such as the mixed-alkali effects, chargecompensation effects, or nonbridging oxygen formation [15]. Interactions can be expressed by means of squared or cross-product terms [16].
Using Eq. (4), Eq. (1) can be expressed as where A ¼ ln η 1 ð Þ: Here, ln(η) is a linear function of the combined composition-temperature variable x i /T. Eq. (5) is a suitable form for determining the A and B i values by means of regression analysis.

Experimental data
The LAW database was created by combining the dataset of 3318 x i /T datapoints for 457 glasses from the Vitreous State Laboratory (VSL) of the Catholic University of America with 569 x i /T datapoints for 97 glasses from the Pacific Northwest National Laboratory (PNNL) [9]. Glass compositions cover the composition region of LAW glasses anticipated for the WTP to support the development of viscosity-composition relationships ( Figure 1). Each glass is represented by 1-8 viscosity values [9,[17][18][19][20]. The temperature range of data in the VSL dataset was 900-1250°C at 50°C steps for most glasses. In the PNNL dataset, 6 viscosity values were measured at 4 temperatures in the order 1150 -1050 -950 -1150 -1250 -1150°C for most glasses. The repeated measurements at 1150°C were taken to check whether the sample was altered by crystallization or volatilization during the tests. Table 1 lists and Figure 1 displays the maximum, minimum, and median values of mass fractions of LAW glass components in the PNNL-VSL combined dataset. In Figure 1, the logarithmic scale on the y-axis was chosen to highlight the compositional data of minor components, which were not statistically designed. The LAW glasses contain nine major components of median mass fractions larger than 0.01 (SiO 2 , Na 2  , neglecting the change of their redox states with temperature and oxygen partial pressure; typically, 4% Fe is reduced to Fe 2+ and Cr 3 + /Cr 6+ ≈ 1 at 1150°C [21]. For the model development, target glass composition was used except for volatile components SO 3 and Cl. Based on data obtained by chemical analysis (not performed for all glasses), the inglass-retained mass fractions of SO 3 and Cl were estimated using the correction functions: x Cl ¼ 0:667x t;Cl ; where x t;i is the ith component target (batched) mass fraction. Eq. (6) is taken from Vienna et al. [9]; Eq. (7) is based on data listed in PNNL reports [17][18][19] and obtained using linear regression (R 2 = 0.962). Note that component volatilization is a complex problem, and Cl retention in melters generally differs from the retention measured in crucible tests, e.g. Cl retention 0:544x t;Cl was reported for laboratory-scale melter [22]. After the approximation functions, Equations (6) to (7), were applied, mass fractions of all components were renormalized to sum to one.  [9]. The effect of this pairwise correlation on the regression analysis is not substantial as seen from small values of standard errors of component coefficients given in Section 4.1. The VSL dataset was not statistically designed to uniformly cover the whole compositional region but consists of viscosity data generated in support of the pilot-scale melter runs. The PNNL dataset was designed to extend the compositional region covered by the VSL database and decrease the correlations between major components. Nevertheless, the combined VSL-PNNL dataset does contain datapoints with a large mass fraction difference from the mean (see individual points in Figure 1).

Model coefficients
The LAW dataset allows us to determine the values of component coefficients only for components that are represented at sufficiently high concentrations. The minor components are grouped into the component Others. The decision which components are included in Others depends on the intention whether to express the effect on viscosity of as many components as possible, or whether to develop a model with as few component coefficients as possible. Below we formulate models with three coefficient sets, two for these extreme aspects and one as a compromise.
To obtain values of coefficients A and B i , Eq. (5) was fitted to data using ordinary least-squares method. The ability of models to predict viscosity of glasses within the LAW composition region was evaluated using a training set consisting of randomly selected 80% of glasses from the whole dataset while the remaining 20% made the testing set. The accuracy metric was defined by the coefficient of determination, R 2 . As stated in Section 2, B is virtually independent of temperature at η < 10 z Pa s, where z ≈ 2. Therefore, 228 datapoints with η > 100 Pa s were removed from both The box shows region from the first quartile, Q 1 , to the third quartile, Q 3 . The upper whisker corresponds to the largest number smaller than 1.5 IQR above Q 3 where IQR ¼ Q 3 À Q 1 is the interquartile region. Similarly, the lower whisker corresponds to the smallest number larger than 1.5 IQR below Q 1 . Individual points show data outside of region delimited by upper and lower whisker. training and testing set, leaving 2921 datapoints in the training set and 738 in the testing set. The choice of z is explored and justified in Section 4.3. Table 2 summarizes and Figure 3 displays the values of fitted B i and their standard errors for three versions of the model. Model A was fitted to the data of glass components with the third quartile of mass fraction Q 3 ≥ 10 −3 , presented in Figure 1 and Table 1; the remaining components, considered as minor, are listed in the Table 1 (Table 3). Model C results from an attempt to reduce the number of variables to a minimum necessary for predicting viscosity with an R 2 ≥ 0.95. As in the case of Model B, the extra components were moved to Others. The number of parameters was reduced to 8 (including parameter A). Table 3 summarizes the fitted values of parameter A and the regression statistics. As expected, R 2 on the training set decreases with decreasing number of independent parameters. R 2 on the testing set is smaller but still close to R 2 on the training set for all three models, indicating that the models represent the data well and thus are qualified to estimate melt viscosity of η < 100 Pa s for other yet untested LAW glasses within the composition region defined in Table 1. The adjusted coefficient of determination, � R 2 , is highest for Model A, implying that the inclusion of F, Cl, and Cr 2 O 3 is beneficial and does not lead to overfitting despite their relatively high standard errors. Figure 4 plots measured viscosity versus estimated viscosity by Model A. The datapoints with η > 100 Pa s were excluded from fitting. Most of these data lie outside of the validity range of the Arrhenius relationship with temperature-independent activation energy. This is further discussed in Sections 4.3 and 5. Of

Component effects
The B i coefficients are partial specific activation energies for the composition region of LAW glasses. They are all independent, but the mass fractions are not because x i ¼ 1 À P j�i x j . Therefore, changing x i can occur only at the expense of one or more other components. Using this identity, Eq. (4) can be written as Generally, the ith component effect is the response of the mixture property to the addition of ith component to the mixture; in the case of the activation energy, β i = dB/dx i . Adding an ith component to the mixture leaves the proportions of mass fractions of all other components unchanged. The mass balance requires that the jth component change is dx j /dx i = -x j /(1-x i ). Using this identity, the above expression for B and the definition of β i , we can arrive at the expression which indicates that components with B i < B decrease B (dB/dx i < 0) and those with B i > B increase B (dB/dx i > 0) when added to the glass, and hence, by Eq. (5), decrease or increase melt viscosity regardless of the melt temperature. Moreover, the ith component has an increasing/decreasing effect on the viscosity of all glasses within the composition region under study if the B i value is larger/smaller than the maximum/minimum B value on the region. Figure 5 shows the values of β i evaluated for the reference composition designated as the average composition of all glasses in the combined dataset. The green dotted line in Fig. 3 represents the reference glass activation energy, B ref = 1.99 × 10 4 K, and the maximum and minimum B values are indicated  by two parallel lines, red and blue. Thus, the addition of Cl, Al 2 O 3 , SO 3 , SiO 2 , ZrO 2 , P 2 O 5 , or SnO 2 to any glass in the studied region results in an increase of melt viscosity. Similarly, addition of Li 2 O, F, Na 2 O, B 2 O 3 , Cr 2 O 3 , CaO, K 2 O, V 2 O 5 , TiO 2 , MgO, or ZnO to any glass in the studied region results in a decrease of melt viscosity. Addition of Fe 2 O 3 or Others to the reference glass results in a slight decrease of melt viscosity, but a subregion exists, for which the addition of Fe 2 O 3 or Others results in a slight increase of melt viscosity. Figure 6 displays the response trace plots.

Linear-nonlinear crossover
Recall that the transition from the near-linear to nonlinear branch of the ln(η) versus T −1 function occurs at the viscosity 10 z Pa s. At η > 10 z Pa s, the Arrhenius model systematically underpredicts viscosity. Thus, the value of z marks the transition from low-viscosity simple Arrhenian liquid with a constant activation energy to non-Arrhenian glass-forming melt with a progressively increasing activation energy [23,24]. As Fig. 4 shows, the transition is gradual because, as mentioned in Section 4.1, the temperature at which η = 10 z Pa s is different for each glass depending on the composition. Yet for practical purposes -mainly for approximating the ln(η) = A + B(x)/T relationship with a minimum number of coefficients -it appears convenient to define a crossover viscosity, η C = 10 z Pa s, that separates out the data for fitting the Arrhenius model. Fig. 7 displays signed squared residuals, sgn ε ð Þε 2 , versus log ðη M ), where ε ¼ ln η M =η E ð Þ=ln η E =η 1 ð Þ is the relative residual, η E is the viscosity estimated by Eq. (5), and η M is the measured viscosity. The value of sgn ε ð Þε 2 ¼ a10 blog η M ð Þ , a function fitted to data, where a and b are constants, becomes increasingly positive at η M > η C = 10 2 Pa s, which was our original estimate for the Arrhenius model validity limit (the z value in Section 2); at η M = 10 2 Pa s, ε = 1.7 × 10 −2 . Fig. 8 shows that the distribution residuals for η M > η C is visibly shifted relative to points in the training and testing sets. The two peaks overlap, indicating that some data that belong to the linear branch of the ln (eta) = f(T −1 ) function were deleted from the model dataset, a price for selecting a single value of the crossover viscosity. This conservative choice possibly avoided "outlying" data from influencing values of the component coefficients.

Molar coefficients
Property models that are intended for application in glass technology are formulated in terms of mass fractions. The reason is that the raw materials are being weighed and the results of chemical analyses are reported in mass percent. Moreover, melter feeds for nuclear waste vitrification are mixtures of radioactive wastes and glassforming and modifying additives, both chemicals and minerals. Wastes and minerals are materials of complex chemistry that is expressed in mass fractions of oxides and elements. To avoid excessive computation in model applications, the component coefficients listed in Table 2

Correlations and causation
As mentioned in Section 4.2, the component coefficients are partial specific or partial molar material properties. Although we call them empirical coefficients or adjustable parameters, they are, in fact, measured properties evaluated by means of regression analysis. They quantify the effects of chemical components on material properties and behavior by means of correlations based on extensive databases from composition variation projects. To predict the material properties of glasses as functions of composition, without a need to design and execute costly and timeconsuming composition variations projects, is an objective of fundamental models based on quantum mechanics, thermodynamics (the modified quasichemical model of Pelton and Blander [7,[25][26][27][28]), or network topology [29][30][31].
The temperature-dependent topological constraint theory, originally developed by Phillips and Thorpe [32,33] and generalized by Gupta and Mauro [34], successfully predicted glass melt viscosity for simple glasses [35][36][37]. These fundamental models claim a general validity, free of restrictions on composition regions or approximation functions. Their ultimate goal, once they are developed to the degree at which viscosity can be predicted accurately for multicomponent glasses, is optimizing glass compositions for manufacturing and application. However, matching the empirically determined coefficients is their litmus test, the crucial challenge and measure of success.
The component effects (Fig. 5) reflect the general trends of network formers, which strengthen glass structure, and network modifiers, which weaken the glass structure [38]. Silica, the major network former, increases viscosity, whereas alkali oxides and alkaline earth oxides decrease viscosity by increasing the fraction of nonbridging oxygens, breaking the linkages between silicon tetrahedra. Moreover, Al 2 O 3 increases viscosity in high alkali glasses, such as the LAW glasses, by bonding alkali and alkaline earth ions through the charge-compensating effect [39,40]. Various other tetra-valent oxides, such as ZrO 2 [41,42] and SnO 2 [43], increase viscosity by their need to charge compensate their six-coordinated ions.
Both B i and P i values of alkali oxides, and, hence, glass viscosity, increase in the order of the ionic potentials: Li 2 O < Na 2 O < K 2 O [44]. Intriguingly, as observed in a wide range of commercial glasses [2,15,45] and high-level waste glasses [2,5,6], alkaline earth oxides exhibit the opposite trend: CaO decreases viscosity significantly more than MgO. In aluminosilicate melts, the anomalous effect of MgO is being attributed to the effect of Mg 2+ on aluminum speciation [46,47]. This agrees with Kim et al. [27], according to whom the Gibbs energy of tetrahedrally coordinated chargecompensated Al in the silica network is higher for NaAl 4+ species than for KAl 4+ species but lower for MgAl 2 4+ species than for CaAl 2 4+ species. However, in the alkali-free system CaO-MgO-SiO 2 at high temperatures (>1400°C) at which the melts exhibit Arrhenius behavior, this effect occurs only in melt with less than 55 mass% SiO 2 [26]. In LAW glasses, high-level waste glasses [4][5][6] and various other glass families [4,15], the predominantly covalent boron weakens the silicate network slightly more than calcium ions when expressed in terms of oxide fractions (see Fig. 6a) [4][5][6]. In the binary B 2 O 3 -CaO system [28], viscosity is nearly independent of the CaO/B 2 O 3 ratio (within 10 to 30 mol% CaO). In E-glasses, soda-lime glasses, and fiber glasses, B 2 O 3 and CaO both decrease high-temperature viscosity, but in various degrees [1].
Fluorine reduces viscosity in LAW glass (Fig 6b), as well as in other silicate and borosilicate glasses, by replacing oxygen bridges with Si-F bonds [48]. Chlorine and SO 3 tend to decrease viscosity [15] in commercial and simple glasses, but, as Fig. 5 shows, these less soluble components increase the viscosity of high-sodium LAW glasses, possibly by trapping alkali ions in nano-inclusions of alkali salts, which also contain chromate [49] and P 2 O 5 [50,51].
Various minor components, Cr 2 O 3 , Fe 2 O 3 , MgO, SnO 2 , TiO 2 , V 2 O 5 , and ZnO, are added to the LAW glass for various reasons. Chromium oxide saturates  the melt with a crucial Monofrax component [52], protecting the refractory lining of the melter against corrosion by molten glass. Vanadium oxide affects the melt surface tension [53], reducing the rate of melt flow to the meniscus at the melt line [54,55]. Moreover, V 2 O 5 increases SO 3 retention in glass melts [56]. Titanium oxide decreases vapor hydration test (VHT) response [9], thus improving the durability of the final glass. Stannic oxide helps to immobilize alkali ions, acting as a charge compensation component. Zinc oxide increases glass resistance against acidic solutions. Both SnO 2 and ZnO moderately decrease VHT response [9]. Magnesium oxide decreases liquidus temperature [57] and iron oxide decreases the viscosity of the slurry feed [58].

Nonlinear ln(η) = f(T −1 ) relationships for interpolation
A range of analytical functions with two or more fitting parameters can be used for interpolation within a narrow temperature interval of η(T) data spread on both sides of the crossover viscosity (see Figures 4 and  7). Whatever approximation function is selected, it faces major problems. First, the number of composition-dependent coefficients increases in proportion to the temperature-independent parameters. Second, a function fitted to a narrow (10 2 -10 3 Pa s) nonlinear segment of the ln(η) versus T −1 relationship does not represent viscosity beyond the temperature interval of data. Thus, nonlinear relationships fitted to the temperature interval form 900°C to 1250°C of the LAW dataset cannot be used for estimating viscosity during melter idling, where temperature may drop as low as 850°C, or for induction-heated melters, where temperature can increase as high as 1450°C. Following Feng et al. [60], Piepel at al [16,61]. used an approximation function where the fitting parameters A and B were treated as polynomial functions of composition, A, as a first-order polynomial and B as either a first-or a second-order polynomial. This model with nearly 2 N coefficients for the first-order version (somewhat less than 2 N because some components have zero B i values) is confined to data within 950°C and 1250°C. It is fully adequate for the LAW glass dataset but breaks down outside its temperature range and composition region. Viscosity temperature relationships that have been developed for the technological viscosity interval spanning 12 orders of magnitude require three or more temperature-independent parameters. The most successful among three-parameter relationships is the Vogel-Tammann-Fulcher (VFT) equation [62][63][64]: where η VFT , B VFT , and T 0 are adjustable coefficients. Alternatively, several authors expressed viscosity as a power-law function of inverse temperature in the form ln(η) = A + (T A /T) α , where A, T A , and α are fitting parameters [65][66][67][68]. By the VFT equation, viscosity approaches infinity as T → T 0 , deeply below the glass transition temperature when glass is rheologically a brittle elastic solid. By the power-law relationship, as well as by the MYEGA equation [69], ln(η) = A + (L/ T)exp(C/T), where A, C, and L are fitting parameters, viscosity approaches infinity only at the absolute zero temperature. All the above equations converge to the Arrhenius equation, but the convergence is overly slow; none approaches Arrhenius behavior at 100 Pa s. The Douglas equation [70] and several others, such as Doremus [71] and Ojovan [72] equations, converge to Arrhenius relationship at both low and high temperatures, but, inevitably, require more than three parameters. Any smooth monotonous function can be used for data interpolation. However, fitting a three-or moreparameter equation to a narrow range of data does not improve model performance. Such a model is overparametrized for the intended application (the formulation of glasses for melters operating at 1150°C) and it breaks down outside the experimental region where it is not supported by data. Vienna et al. [9,11] used the VFT equation while focusing on a single temperature, namely 1150°C, the melter operating temperature, T MO , of WTP melters. The value of η MO = η(T MO ) was obtained through fitting Eq. (10) to data for each glass individually. Polynomial approximation functions were obtained by fitting to η(T MO , x) data. The linear model where h i is the ith component coefficient and N is the number of viscosity-affecting components, estimates the viscosity at a single temperature (T MO ) using N fitted coefficients. Unlike the Arrhenius model, Eq. (5), which has N + 1 fitted coefficients and is based on a subset of data with η < η C , Eq. (11) is based on all measured data on both sides of linear-nonlinear crossover. Table 6 lists and Fig. 9 illustrates the singletemperature coefficients h i from Vienna et al. [11] denoted as Model V16 and Vienna et al. [9] denoted as Model V21 and compares them with coefficients h i = A + B i /T MO obtained from Models A-C for T MO = 1423 K (coefficients A and B i are listed in Tables 2 and 3). Model V21 was created from the same database as Models A-C, except that it used 534 glasses compared to 554 glasses used in this work. Model V16 used a smaller database using 429 LAW glasses. The Model V16 data were on the linear branch of the ln(η) = f(T −1 ) function, the maximum viscosity in the dataset being 20 Pa s, significantly less than the 100 Pa s crossover.
The difference between Model A and Model V21 in h i coefficients is the largest for Cl, SO 3 , and Others, the former two of which are volatile and present in small fractions. The unusually high component coefficient of Others in Model V21 is associated with the exclusion of glasses containing ≥ 0.02 Others (including glasses    Table 7 shows the coefficient of determination for single-temperature models, Eq. (11), applied to the full database of 554 glasses with component coefficients listed in Table 6. Note that the lower R 2 value of Model V21, which, as argued in Section 5.2, was associated with a lower prediction accuracy of the two-step evaluation. The R 2 of Model V21 increased to 0.943 after the outliers mentioned above were excluded.

Linear model recovery from a single-temperature model
The linear ln(η) = f(T −1 ) relationships, Eq. (5), can be recovered from a single-temperature viscosity model, Eq. (11), provided that the A value is known, or at least correctly guessed, and that the viscosity value lies within the low-viscosity range of η < 100 Pa s, which is indeed the case for viscosity at T MO ¼ 1423K, where η MO varies between 0.47 and 35.4 Pa s for LAW glasses in the database. By Eq. (1), Using Equations (4), (11), and (12), we obtain the component coefficients for the activation energy of lowviscosity range as follows:  Table 6 for Model V16 and Model V21. As in the case of coefficients h i , the difference between Model A and Model V21 in B i coefficients is the largest for Others, Cl, and SO 3 . Model V16 and Model V21 were created in two steps by fitting Eq. (10) to data for each glass separately and then by fitting polynomial model, Eq. (11), to η MO (x) values. This two-step process avoided overparameterization caused by treating more than one parameter as a function of composition. The values of R 2 coefficients, listed in Table 9, are close even though Models V16 and V21 used the two-step process, which is generally less favorable than the one-step fitting of temperature-and composition-dependent model.

Extrapolation problem, examples, and resolution
Viscosity-temperature relationships that are designed for application in glass technology from melting to annealing cover viscosity values over 12 orders of magnitude. Although the activation energy is virtually constant for η < η C ≈ 100 Pa s, it may be 10 times higher at the glass transition temperature (T g ). As Figures 4 and 7 indicate, LAW glass viscosity data cover only 1 order of magnitude of the nonlinear segment of the ln(η) versus T −1 relationship (between 10 2 and 10 3 Pa s). One cannot expect to obtain physically meaningful values of parameters from fitting a function designed to cover 12 orders of magnitude to a 1 order of magnitude interval of data. Experimental errors from even a very precise measurement can thus result in unacceptable errors in extrapolating an otherwise physically well-designed function by several orders of magnitude beyond the experimental range of data. To demonstrate this, we selected, in Table 10, five glasses from the LAW dataset [18], three random (LP2-OL-01-3, LP2-OL-12, and LP2-OL-22) and two borderlines (LP2-OL-13 and LP2-OL-05).   Table 11 shows the coefficient for both Arrhenius and VFT equations fitted to viscosity-temperature data of the five selected glasses. Of the two borderline glasses, LP2-OL-13 has an unusually low value of T 0 and a high value of B VFT ; the other of two, LP2-OL-05, has an unusually high values of A VFT = ln(η VFT ) and T 0 .
As for the VFT equation, log(η) versus T −1 does not converge to a linear relationship for η < 100 Pa s (Fig. 11), where most of the data lie. In fact, the VFT curves are bending far below η = 100 Pa s. As a result, A VFT > A for each glass (Table 11). Thus, the values of η ∞ are substantially higher in the VFT fit than those obtained from the Arrhenius fit. Moreover, neither A nor A VFT coefficients fall near to a common value. Because of a narrow temperature range of data and experimental errors, all three parameter values, those of the η VFT (viscosity at infinite temperature), B VFT , and T 0 (temperature at infinite viscosity), are burdened with high uncertainties. This becomes apparent when the parameters are used to estimate glass transition temperature, T g , i.e. temperature for which η = 10 12 Pa s. For the above five Table 10. Viscosities (in Pa s) of five selected LAW glasses [18].  Note that A values are not reasonably close to −12.5 (η ∞ = 3.8 × 10 −6 Pa s) for the Arrhenius relationship, even though the datapoints with η > 100 Pa s (LP2-OL-05 and 13). These deviations were caused by experimental errors. Fig. 10 displays the Arrhenius plots.  parameter sets T g would range from 287 to 715°C, whereas a much narrower range (450-600°C) is realistic. Because glass viscosity at T g is nearly independent of composition, adding the η(T g ) datapoint, either by direct measurement or from a model [29,75], would enable obtaining meaningful VFT parameters.

Conclusion
The Arrhenius relationship with temperatureindependent activation energy is the simplest possible model representing the glass melt viscosity for η < 100 Pa s as a function of temperature and glass composition. It is thus suitable for modeling glass melting furnaces in general and electric melters processing nuclear waste glasses in particular. Using a set containing 3887 viscosity datapoints (3659 for η < 100 Pa s) measured on 554 glasses, three models were developed, a 20-component model (R 2 = 0.972), a 17-component model (R 2 = 0.970), and a 7-component model (R 2 = 0.949). Out of the major constituents, viscosity is most decreased by Li 2 O and Na 2 O and most increased by Al 2 O 3 and SiO 2 . Out of the minor constituents, F decreases viscosity as expected while SO 3 and Cl increase viscosity, likely by bonding alkalis in sulfate-chloride nano-inclusions as discussed in Rong et al. [50,51].
The fraction of LAW viscosity data outside of the validity of the Arrhenius equation (100-3 ×10 3 Pa s) is too small, and the temperature range of data is too narrow, for fitting nonlinear approximation functions, such as the VFT equation. This problem could be mitigated if T g values were available.