Modeling the Viscosity of Anhydrous and Hydrous Volcanic Melts

The viscosity of volcanic melts is a dominant factor in controlling the fluid dynamics of magmas and thereby eruption style. It can vary by several orders of magnitude, depending on temperature, chemical composition, and water content. The experimentally accessible temperature range is restricted by melt crystallization and gas exsolution. Therefore, modeling viscosity as a function of temperature and water content is central to physical volcanology. We present a model that describes these dependencies by combining a physically motivated equation for temperature dependence of viscosity and a glass transition temperature ( Tg ) model for the effects of water. The equation uses the viscosity at infinite temperature η∞ , Tg , and the steepness factor m as fitting parameters. We investigate the effect of leaving η∞ free as a parameter and fixing its value, by fitting anhydrous viscosity data of 45 volcanic melts using the temperature dependent model. Both approaches describe experimental data well. Using a constant η∞ therefore provides a viable route for extrapolating viscosity from data restricted to small temperature intervals. Our model describes hydrous data over a wide compositional range of terrestrial magmas (26 data sets) with comparable or better quality than literature fits. With η∞ constrained, we finally apply our model to viscosities derived by differential scanning calorimetry and find—by comparing to viscometry based data and models—that this approach can be used to reliably describe the dependence of viscosity on temperature and water content. This introduces important implications for modeling the effects of nanostructure formation on viscosity.

A combination of concentric cylinder and falling sphere viscometry is employed to measure melt  above the liquidus T in the low-/high-T regime (L,    Pa s on the timescale of measurements, interpolation between the H and L regimes is required. This is especially critical when the L and H intervals of experiments are reduced as a result of nanostructure formation, primarily nanocrystals and melt demixing, which can lead to a significant increase in  (Di Genova, Brooker, et al., 2020;Di Genova, Kolzenburg, et al., 2017;Di Genova, Zandona, & Deubener, 2020;Liebske et al., 2003). These restrictions on the H range accessible to micropenetration and parallel plate experiments can lead to the virtual absence of data near g T (Al- Mukadam et al., 2020;Chevrel et al., 2013;Dingwell et al., 2004).
Here, we present a new fitting approach for  of volcanic melts motivated by physically based equations that describe the temperature dependence of viscosity (Mauro et al., 2009) and water dependence of g T (Schneider et al., 1997). This represents one of the first attempts to combine physically based equations in order to provide a single formulation for the viscosity of volcanic melts as a function of temperature and water over a large chemical space, with a set of 1,603  data points, containing both multicomponent dry and hydrous systems as indicated in the total alkali-silica (TAS) diagram (Le Bas et al., 1986, Figure 1). To characterize the behavior of anhydrous melts in a systematic way, we order them according to the chemical parameter SM, which is a proxy of the degree of structural polymerization (Giordano & Dingwell, 2003a with x in mol%; for compositions that only report total iron, we distribute it equally between FeO and 2 3 Fe O with an adjustment factor of 1.11 (reflecting the higher molar weight of 2 3 Fe O ) in terms of wt% before conversion.
10.1029/2021GC009918 2 of 26 First we investigate the fit of  for anhydrous samples using a model developed by Mauro et al. (2009) for technical glasses. We discuss the connection between Arrhenian behavior of volcanic melt  and the degree of structural polymerization (SM) as well as the hypothesis of a common viscosity value at infinite T (  ) for glass-forming melts. We do so by significantly expanding previous chemical and experimental data sets of melt viscosity data (Russell et al., 2003). For a given silicate melt, the addition of 2 H O can reduce the viscosity in the H regime by several orders of magnitude (e.g. Richet et al., 1996). We ignore the pressure effect on melt viscosity at fixed water content at shallow conditions typical of volcanic systems (Giordano et al., 2008;Hui & Zhang, 2007;Persikov, 1998;Zhang et al., 2003), but our model implicitly accounts for the pressure effect by varying the water content. Many studies (e.g., Dingwell et al., 1998b;Giordano et al., 2009;Misiti et al., 2011;Robert et al., 2015;Vetere et al., 2006;Whittington et al., 2009) have modeled the influence of 2 H O on  by various differing empirical expressions. We apply a single formulation for water dependence of  and compare our results to published models from the literature. We show that our physically based viscosity equation can perform comparably or better than empirical formulations in the literature. Furthermore, we compare results of our fit with predictions of general chemical models (Duan, 2014;Giordano et al., 2008;Hui & Zhang, 2007). Finally, we apply our model to describe  of hydrous volcanic melts based on differential scanning calorimetry (DSC) measurements, which minimizes or avoids nanocrystallization that can occur during standard viscosity measurement around g T (Di Genova, Zandona, & Deubener, 2020). We implement our model with a constant log  and show that the combination of our fitting approach with DSC data allows both the accurate prediction of high-temperature  and the quantification of the effect nanocrystal formation has on melt viscosity around g T .

Viscosity Models for Anhydrous Systems
The most popular parametrization to describe the viscosity of volcanic melts is the empirical VFT equation named after Vogel (Vogel, 1921), Fulcher (Fulcher, 1925), and Tammann (Tammann & Hesse, 1926). It has been used to fit isochemical  data (e.g. Richet et al., 1996;Whittington et al., 2001) and takes the form is often identified with the Kauzmann temperature ( K T ) (Angell, 1997), at which the liquid and crystalline entropies are equal. At K T , the VFT equation in combination with the Adam-Gibbs equation (Adam & Gibbs, 1965) yields a configurational entropy ( c S ) of zero (Mauro et al., 2009;Scherer, 1992), although c 0 S  is only possible at absolute zero temperature (Avramov & Milchev, 1988;Mauro et al., 2009). On the other hand, Gibbs and DiMarzio (1958) have derived a possible thermodynamic equilibrium glass transition. Overall the physical meaning of the VFT fitting parameters continue to be a subject of discussions (e.g., Hecksher et al., 2008;Schmelzer et al., 2018;Stillinger, 1988). Finally, the VFT equation is also known to break down at low T (Laughlin & Uhlmann, 1972;Mauro et al., 2009;Scherer, 1992).
Therefore, a physically based parametrization of  for glass-forming melts remains an interesting subject of research. For example, the viscosity description given in the model for glass-forming liquids by Adam and Gibbs (1965) (AG) has a physical foundation. It assumes the cooperative rearrangement of independent regions within the liquid and that the potential energy of the system can be expressed by its own partition function. This leads to where AG log A    , and AG B is an effective activation barrier (Adam & Gibbs, 1965;Richet, 1984 Geochemistry, Geophysics, Geosystems Di Genova, Romano, Giordano, & Alletti, 2014;Richet, 1987;Robert et al., 2014;Sehlke & Whittington, 2016;Stebbins et al., 1984;Toplis, 1998;Webb, 2008).
One can avoid fitting c S and measuring ,conf P C by using the MYEGA model by Mauro et al. (2009). It describes c S in the AG expression (Equation 4) using constraint theory and an energy landscape analysis, and takes the form where A, K and C are fitting parameters, with log A    as above. An alternative, physically insightful, parametrization of Equation 5, suggested by Mauro et al. (2009), can be obtained by inserting the definition of g T (Equation 1) and making use of the steepness index m (fragility), which quantifies the deviation of  from Arrhenian behavior at g T (Angell, 1995), Reformulating Equation 5 with respect to these parameters yields: An analogous reformulation can be performed for the VFT model (Equation 3): A comparison between the performance of the MYEGA (Equation 7) and VFT models (Equation 8), using anhydrous simple and multicomponent oxide systems, that is, technical glasses, and molecular liquids covering a wide range of m from 20 to 115, revealed that the MYEGA equation provides a superior fit for  in all systems (Mauro et al., 2009). Moreover, using 568 different technical silicate liquids with widely varying compositions and  data in the range of 1 10 -6 10 Pa s, Mauro et al. (2009) also showed that the MYEGA model predicted the 11 10 Pa s isokom T better. Finally, unlike the VFT parametrization, the MYEGA equation offers a realistic extrapolation of c S to both the high-and low-T limits, with consequences for the estimate of A and the description of the low-T scaling for  (Mauro et al., 2009).

Modeling the Effect of Water on the Viscosity of Silicate Melts
The presence of water in volcanic melts adds complexity to fitting , as even a small amount of 2 H O generally leads to a strong decreases of . While the H and L regimes are usually accessible for anhydrous melts and provide strong constraints on the parametrization over a large  range, the lack of L data for hydrous compositions challenges the quality of the fit. This can lead, for example, to an unphysical crossover of  at different 2 H O content when viscosity is extrapolated to the L domain ( Figure S1). To avoid  (e.g., Giordano et al., 2008Giordano et al., , 2009Romine & Whittington, 2015;Vetere et al., 2013;Whittington et al., 2009). While resulting fits usually provide a good description, there is no systematic approach and no physical interpretation of parameters involved, which results in a plethora of different models based on VFT.
Here we expand the MYEGA parametrization (Equation 7) in a physically motivated way to fit anhydrous and hydrous data for a given volcanic melt with varying 2 H O content. We assume A to be independent of 2 H O content (i.e., fixed by the anhydrous measurements), which reduces the water-dependent parameters to m and g T . We base our  description on a g T model by Schneider et al. (1997), who implemented a power concentration expansion of the Gordon-Taylor equation (Gordon & Taylor, 1952 The parameter d m is the melt fragility of the anhydrous sample.

Fitting the Viscosity of Hydrous Silicate Melts
To fit a set of viscosity data including anhydrous and hydrous measurements of one specific melt composition, we follow these steps: 1. We fit the anhydrous data using the MYEGA model (Equation 7). These data sets often include H and L measurements constraining the values of A, g,d T and d m well. 2. We insert Equations 9 and 12 into the MYEGA equation (Equation 7) and fit the resulting model to the remaining hydrous data. This constrains parameters b, c and d. To evaluate the quality of the fit, we employ the root-mean-square error (RMSE),

Viscosity Database
We use 50 viscosity data sets (1,603 data points) from the literature for fitting (Tables 1-3), displayed in a TAS diagram ( Figure 1). The data sets span a large compositional space with 2 SiO content ranging from 44 wt% to 79 wt% and total alkali content ranges from 0 wt% to 17 wt% (mol% reported in Tables 1-3). Virtually all types of magma erupted on Earth are represented.
Of the 50 data sets, 45 include viscosity measurements in the H and L region for anhydrous melts (Table 1), 26 sets additionally contain data for hydrous compositions (marked by * in Table 1 and listed in Table 2). All anhydrous data are used in Section 4 to explore the parameters A, m and g T ; in Section 5, we explore the quality of our model for the 26 2 H O-bearing liquids (Table 2). Table 3 lists five data sets. One only includes anhydrous measurements, while for the remaining four anhydrous and hydrous measurements are available. The  values for melts in Table 3 are derived from DSC measurements using the approach reviewed in Stabile et al. (2021) (for further discussion see Section 6). They complement viscometry measurements on glasses from eruptions already included in our database (Tables 1 and 2). With DSC,  is determined in the H range only. In Section 6, we use DSC-derived  to illustrate that for high-quality data a reliable and predictive extrapolation of the MYEGA model from the H  to L range is possible, assuming a fixed value for A.

MYEGA Fit
We use the MYEGA (Equation 7) and VFT models (Equation 8) to fit  data from 45 different anhydrous silicate melts, all of which include measurements in the H and L range (MYEGA: Figure 2, VFT: Figure S3). Including H and L measurements provides a good constraint on the fits, as two of the parameters used in the MYEGA model, g,d T and d m , are quantities defined at high  (Equations 1 and 6); A, on the other hand, is a low- quantity for T  . Data and fits are grouped according to increasing SM values (Equation 2) in Figure 2. The often employed structural NBO/T parameter (Mysen, 1988) was not used as it correlates positively with the chemical parameter SM ( Figure S2), and Giordano and Dingwell (2003a) have shown that SM is a valid empirical parameter to infer the degree of structural polymerization of the melt. Moreover, SM is easier to calculate and therefore used here. Figure 2a shows measurements of samples with SM 10  , which are the most polymerized melts with SiO 2 80 x  mol%. Their interval of  measurements ranges from 2 10 to 13 10 Pa s, with 785 °C < T < 1650 °C. For these melts, the 1 / T dependence of  is quasi-linear, that is, they exhibit an Arrhenian behavior. Figure 2b displays  for liquids with 10 ≤ SM < 20. For these less polymerized melts,  and T ranges are 1 14 10 10  Pa s, and 585 1710 C   , respectively. Some melts (e.g., Rhy14, Pho3) display Arrhenian behavior, while others (e.g., Rhy12, And3) exhibit a weak, but significant departure from linearity, that is, behave in a non-Arrhenian fashion. Figure 2c shows  data and fits for relatively depolymerized melts with 20 SM 30   . Viscosity measurements range from 1 10  to 14 10 Pa s, and 615 1570 C   . The majority of these melts exhibit a pronounced non-Arrhenian behavior for , with the exception of the shoshonite sample (Sho), for which the L range appears poorly constrained (see discussion on A below). Finally, Figure 2d shows  for the most depolymerized melts with SM 30  , with 0 10 Pa s 14 10    Pa s and 635 C 1560 C T     . Our results thus agree with the expected scenario that Arrhenian liquids are characterized by a polymerized melt structure due to their high content of network-forming cations (low SM), while liquids with larger values of SM exhibit non-Arrhenian behavior (e.g., Angell, 1995;Mysen, 1988;Ni et al., 2015).   (Tables 1-3). Data sets are color coded according to dry only data (orange squares), those including dry and hydrous data (blue) and differential scanning calorimetry (DSC)-derived viscosities (red triangles). Open blue circles denote samples that are used to illustrate the combined fits of the MYEGA (Equation 7) and 2 H O model (Equations 9-12) in Section 5.

Geochemistry, Geophysics, Geosystems
LANGHAMMER ET AL.  SiO -rich counterparts. Rhy14 is a peralkaline rhyolite (pantellerite), characterized by an excess of alkali and alkaline earth cations over 2 3 Al O which induces a dramatic depolymerization of the melt structure within rhyolite chemistry (Di Genova et al., 2013;Dingwell et al., 1998a), leading to relatively low  ( Figure 2b). As expected from Figure 2, melt fragility (m) positively correlates with SM (Figure 3c). In particular, we find that the strongest melt ( 20.4 m  ) is Rhy3 with SM 7.6  , the most fragile melt is Di ( 61.1 m  ) with SM 56.2  (Table 1).
Finally, the parameter A increases significantly from −9.6 for Rhy3 to −1.9 for Tep with SM (Figure 3a, Table 1). We find the largest variation of A for SM 10  , and a relatively constant value of 3 A   for SM 20  . The low values of A for the polymerized melts with SM 10  is likely caused by the limited  range accessible for measurements in the laboratory. For example, the viscosity of the polymerized melt Rhy3 (SM 7.6  , 9.6 A   ) that follows an Arrhenian behavior (Figure 2a), was measured in the range of 3.24 log 11.15 It is not possible to extend measurements to significantly lower  values for such polymerized melts with T becoming too high for the measuring system and causing volatilization of alkalis from the melt. Therefore, A is not well constrained by this measurement interval. The sample Sho deviates from the expected behavior with SM 29.3  and 9.4 A   . This is a very low value of A compared to melts with similar SM. For Sho, only three data points exist in the L range with the lowest measured viscosity log 1.27   (Vetere et al., 2007). This restricted L range may not permit an accurate determination of the T dependence in the L region and thus a reliable estimate of A.

The Viscosity at Infinite Temperature
A common assumption is that the viscosity of glass-forming melts converge to constant value of A as T   (Angell et al., 2000), an assumption that can be integrated into the fitting by fixing the parameter A (Section 2.1). Maxwell's equation G     provides an order-of-magnitude estimate. G  is the shear modulus at infinite frequency and  the relaxation time. For silicate melts at infinite T , they are estimated as 10 10 G   Pa (Dingwell & Webb, 1989) and 14 10     s (Angell, 1997;Börjesson et al., 1987;Fujimori & Oguni, 1995) content in mol%) and contains information on the data, the third block information on references. In the second block the fitting parameters for the constrained hydrous MYEGA model (Equation 7 with 2.9 A   , 9, and 10) and the RMSE are given. "Comment" indicates the sample name in the respective publication. Measurements mentioned to have crystallized/lost water and so on in the respective reference are excluded from fitting. Bas, Basalt; Di, Diopside; Lat, Latite; Rhy, Rhyolite; RMSE, root-mean-square error; Tra, Trachyte.

Geochemistry, Geophysics, Geosystems
The VFT (Equation 3) and AG models (Equation 4) have been used in the literature to explore the range of A values for volcanic melts. Russell et al. (2003) obtained an average 4.3 0.7 A    (VFT) and 3.2 0.7 A    (AG) for a compilation of 20 silicate melts. Subsequent work by Giordano et al. (2008) Mauro et al. (2009) show that the MYEGA model results in a larger value for A than VFT. We observe a larger A for MYEGA than VFT with MYEGA 4.3 1.9 A    and VFT 5.1 1.5 A    , respectively. The difference between them is consistent with the results of Zheng et al. (2011). The trend to low values of A that we observe stems largely from the 11 Arrhenian data sets with SM 10  for which the quasi-linear extrapolation of  to high T yields very low values of A (Figure 3). When the eleven A values for melts with SM 10  are excluded from averaging, VFT 4.6 1.2 A    , in agreement with the value found by Giordano et al. (2008) and close to that of Russell et al. (2003). Nine of the 11 melts in Table 1 with SM 10  were not used in these two studies, but we assume they would have a similar influence on the values of A. A significant-but smaller-difference in A remains compared to the technical data set of Zheng et al. (2011 Figure S3). Abbreviations and references for the different data sets can be found in Table 1: * denotes samples for which hydrous measurements are also reported (Table 2). Symbols are assigned as follows: X for rhyolites, empty circles for HPG8, triangles to the left for trachytes, squares for dacites, pentagons for phonolites, empty crosses for andesites, empty X for latites, diamonds for basaltic andesites, stars for tephriphonolites, octagons for shoshonite, hexagons for basalts, upwards triangles for phono-tephrites, triangles to the right for tephrites, tripods for foidite, crosses for diopside. Low values of A also correlate with low values of the steepness factor (Figure 3), highlighting a difference between the current data set and that of Zheng et al. (2011). In their database, all 25.9 m  . If we restrict averaging of A to melts with such m values, we obtain MYEGA 3.2 1.0 A    , in excellent agreement with Zheng et al. (2011). This underlines the observation that the measurable T interval for highly polymerized melts (low SM/low m) often is to narrow to constrain A.

Fitting With a Constant Value of A
In order to explore differences in the MYEGA fitting parameters when A is fixed or left as a free parameter, we refit the anhydrous data sets (Table 1) using 2.9 A   (Zheng et al., 2011). This may also be important for cases where only a small number of measurements over a limited H range are available, including DSC measurements which we address in Section 6. The RMSE values reported in Table 1 show an expected increase due to the reduction in fitting parameters, but overall the fitting quality is still high.
Values for g T (Table 1 and Figure 3b) are very similar to the fits with free A since g T is generally well constrained by measurements in the vicinity of Similarly, m values for fixed 2.9 A   in the interval SM 10  are systematically larger. This is readily rationalized by reversing the argument given in Section 4.2 that an Arrhenian behavior of  leads to small A. With 2.9 A   constrained, the fit is forced to become more non-Arrhenian, increasing the curvature near g T . For 10 SM 20   , the majority of m values associated with fixed A are larger but the deviation is less pronounced. In the interval SM 20  , deviations are generally small and not systematic. A notable difference is Sho, for which L data are scarce as discussed in Section 4.1, with 35.39 m  for 2.9 A   , compared to 25.47 m  for a fitted 9.44 A   . General trends discussed for the MYEGA fit with variable A are preserved for fixed 2.9 A   , and become more systematic: g T decreases with SM, and the fragility m increases with SM. Fixing A leads to a narrower distribution of m and indicates a quasi-linear correlation with SM.

Hydrous Silicate Melts
After fitting anhydrous viscosity data using the MYEGA model (Figure 2), we explore the 2 H O-dependent model of Equations 9-12 for the 26 samples with hydrous data ( Table 2). As examples, we show two compositions in Figure 4 that are also highlighted in Figures 1 and 3: a Basaltic Andesite (BasAnd2) (Robert et al., 2013) and a Phonolite (Pho1) (Giordano et al., 2009) Our model describes the  measurements for BasAnd2 by Robert et al. (2013) significantly better than the literature model ( Figure 4a)-with the exception of the two L falling sphere data-which is most clearly visible for 12 mol% 2 H O. In addition, our model shows a tendency toward larger curvature in log -1 / T (stronger non-Arrhenian behavior, larger m). For Pho1 (Figure 4d), the data are well described by both our fit and the model used in Giordano et al. (2009), with the exception of the highest 2 H O content (14.39 mol%), which neither of the models match. With a high alkaline content (Figure 1 and Table 2  The steepness parameter m deviates between our model and literature fits (Figure 4) for the non-Arrhenian melt BasAnd2, which is already apparent in the fits themselves. d m reported by Robert et al. (2013) is slightly higher than the value calculated here, and their m shows a steeper decrease with 2 H O, resulting in an increasing deviation between the two models. For Pho1, our model formulation leads to lower values of m with 2 H O compared to the fit by Giordano et al. (2009). The initial decrease is more pronounced than for BasAnd2. This behavior reflects that BasAnd2 has lower degree of polymerization, with the SM 29.3  and SiO 2 56.9 x  mol% (Table 2), an effect that is not clearly visible in the models from the literature.
In some cases-illustrated by Pho1 for our model (Figure 4f), but also apparent in some trends from the literature-m extrapolates to negative values at high 2 H O content, which constitutes unphysical behavior. Such behavior should serve as warning against extrapolating models of melt viscosity far beyond the 2 H O content actually measured in the experiments used for fitting. Figure 5 shows a comparison of our fit calculation with RMSE = 0.17 against the measured viscosities as well as prediction of three general chemical viscosity models for these compositions (Duan, 2014;Giordano et al., 2008;Hui & Zhang, 2007). The model by Duan (2014) is the only viscosity model that accounts for the pressure effect on melt viscosity, which we fixed to 1 bar. Also, this model requires the partitioning of the total iron in FeO and 2 3 Fe O . Here, for the melts for which iron partitioning was not provided, we assigned 1 / 2 of the total iron (always given as tot FeO ) as FeO and 1.11 / 2 as 2 3 Fe O . The RMSE across all calculations is 1.95. The models by Giordano et al. (2008) and Hui and Zhang (2007) have RMSE values of 0.74 and 0.69 respectively. Table 2 documents the RMSE values for all three general chemical models and literature models for the individual compositions. Compared to the latter our model performs with comparable or better quality ( Figure S5). However, previously published models differ in their formulations of 2 H O dependence, while we use the same model for all melts (Equations 9-12). In the Supporting Information we provide an excel file to calculate viscosities for the melts referenced here.
Parameters c and d in Equation 9 obtained for six samples (Rhy8,Dac2,Tra3,Pho4,Pho5,Pho6) show strong deviations from the other values ( 19 c  and 24 d   , Table 2). This leads to unphysical extrapolations of g T and-via Equation 12-m, that is, to an increase of g T with 2 H O content ( Figure S4). Nevertheless, our model accurately reproduces the measured  data with RMSE = 0.09−0.35 for these six compositions. The anomalous behavior of g T and m with 2 H O appears to result from minimizing the residuals during the fit process. The unphysical extrapolation behavior serves as reminder to use our model-like any other model-not to extrapolate far beyond the experimental 2 H O range.

Using DSC for Modeling Melt Viscosity
During viscometry experiments in the H regime volcanic melts can be subjected to nanostructural modification (i.e., crystallization and demixing) (Di Genova, Zandona, & Deubener, 2020), and DSC measurements provide an alternative route to obtain  data (e.g., Stabile et al., 2021). DSC measurements require a few mg of glass, which is exposed to g T T  for a few minutes only (Di Genova, Zandona, & Deubener, 2020;Stabile et al., 2021;Zheng et al., 2019). This is in stark contrast to experiments using micropenetration and parallel plate techniques that require large and double-polished samples (ideally with a thickness of 3  mm) and expose the melt to g T T  for significantly longer periods of time (Douglas et al., 1965) which can lead to severe chemical and textural changes in anhydrous and hydrous samples (Bouhifd et al., 2004;Di Genova, Zandona, & Deubener, 2020;Liebske et al., 2003;Richet et al., 1996). However, only temperatures around g T can be probed using DSC, leaving the L range unexplored, complicating  fitting. In Sections 4.2 and 4.3, we have explored the role of A for the  model, and found that using 2.9 A   (Zheng et al., 2011)constraining the high T behavior-provides a systematic and good description of melt viscosity in the L range. Using 2.9 A   in the MYEGA fit and applying our description of 2 H O dependence to DSC-derived  can therefore provide an alternative route to attain high-quality and reliable predictions.

Diopside: A Test Case
We test this approach for DSC-based data of a diopside melt (Di), an Fe-free system that is a good proxy of volcanic melt not prone to crystallization around g T , and for which a large number of viscometry T marks the sudden drop in heat flow measured in DSC, and peak T corresponds to the (endothermic) minimum of the heat flow undershoot of the glass transformation interval. onset T and peak T were measured at five heating rates, leading to 10 data points. We use the approach of Scherer (1984) where K is the chemically independent parallel shift factor and   q c, h the heating rate in 1 K s  for onset/peak T (Di Genova, Zandona, & Deubener, 2020).
Here we fit both the DSC-based values, that is, 10 data points with 9 12 10 10    Pa s ( Figure 6) as well as the viscometric measurements compiled by Al-Mukadam et al. (2020), using the MYEGA expression (Equation 7) and assuming 2.9 A   (Zheng et al., 2011). Our fit and that by Al-Mukadam et al. (2020)-which leaves A free-to viscometry data show good agreement overall. The deviation at high T stems from the differing values in A. The MYEGA model based on DSC-derived viscosities (at H) predicts the L viscometry data well. Our approach shows that a predictive extrapolation from the H regime over more than 10 orders of magnitudes is reliably possible, spanning the entire  range relevant to volcanic eruptions.

Predicting Viscosities Using DSC
After testing this fitting approach on Di, we move to natural melts with fewer DSC data points and more complex oxide chemistry, which can lead to nanocrystallization even in the DSC experiments (Di Genova, Zandona, & Deubener, 2020). We compare the results from the fit to DSC-derived data with models that are based on viscometry measurement on melts of the same eruptions (Table 3) Our results for this set of examples indicate that hydrous DSC-derived  can be used to calibrate the model developed here (Equation 7 with 2.9 A   and Equations 9-12). Viscosity values of different 2 H O concentration can not only be described well, but accurately predicted (Figure 7). Resulting  at eruptive T are well behaved with 2 H O for all DSC-derived models. However, to fully validate this approach and explain the deviations between viscometry and DSC-derived models comprehensively, more DSC and viscometry measurements carried out on samples of equivalent compositions are necessary. As we have pointed out explicitly for Bas1, the formation of nanostructures appears to not only affect viscometry measurements, but also DSC experiments, albeit to a much smaller extent. Careful analysis of samples after experiments, for example, by Raman spectroscopy or TEM, is necessary to check for the formation of nanostructures (Di Genova, Zandona, & Deubener, 2020).

Conclusions
We present a new approach to fit the temperature and water dependence of viscosity for volcanic melts. It is based on a combination of the physically motivated MYEGA model (Mauro et al., 2009) (Equation 7) for an isochemical fit to anhydrous data and a two-component model (Schneider et al., 1997) to describe the influence of water. In the MYEGA model, the fitting parameters are the viscosity at infinite T ( log A    ), the glass transition temperature g T , and the steepness factor m. In the two-component model, we formulate a dependence of g T only between the endmembers of the anhydrous melt composition and that of water (Equations 9 and 10). For the dependence of m on water content, we derive an analytical expression dependent on  (Table 1), we show that the MYEGA model describes the data comparably to-or better than-the more commonly used VFT fit. We further explore the performance of the MYEGA model by assuming a global constant value of 2.9 A   (Zheng et al., 2011); naturally, the misfit to the data increases, but the fits remain good overall. We also find that highly polymerized Arrhenian melts tend to yield smaller values of A due to the experimental inaccessibility of higher T measurements for these types of melts. For 26 data sets with both anhydrous and hydrous measurements, we apply the MYEGA model in combination with the 2 H O-dependent description of g T . We find that our model performs with comparable or better quality than various differing literature models (Table 2)  H O dependence of differential scanning calorimetry (DSC) and viscometry derived models at eruptive T : 945 C  for Tra3, 1225 C  for Bas1, 900 C  for Lat, and 750 C  for Rhy14. Water content is given in mol%. The black ticks are set in 1 wt% intervals of excel file to calculate viscosities of all melts considered here using our model is provided as Supporting Information.
We further investigate and fit viscosities derived from DSC which is an attractive experimental approach that avoids or reduces nanocrystallization and demixing of samples during the measurements compared to viscometric methods. The lack of low viscosity data due to DSC only probing T around g T is compensated by using a constrained 2.9 A   . For a small set of five examples (Table 3), we illustrate that such a fit extrapolates well to high T when compared to viscometry measurements. We apply the 2 H O dependent model with 2.9 A   to hydrous DSC-derived viscosities, and find the model to show good fitting and predictive capabilities. Investigating these models at eruptive T also shows well behaved functions; viscosities monotonically decrease with 2 H O content. This underlines the viability of determining  with DSC.
Since nanostructures have been shown to significantly influence  of volcanic melts (Di Genova, Brooker, et al., 2020;Di Genova, Kolzenburg, et al., 2017;Di Genova, Zandona, & Deubener, 2020), understanding and quantifying their impact on magma transport is an important task in physical volcanology. The characterization of samples exposed to DSC and viscometry measurements by Raman spectroscopy and transmission electron microscopy gives insight into the structural and textural impact of nanostructures. In combination with fitting the DSC-derived viscosities with 2.9 A   as well as viscometric measurements, this opens up the possibility to quantify the impact of nanostructure formation on the viscosity of volcanic melts. This in turn may improve our understanding of the eruptive dynamics of volcanoes.

Data Availability Statement
Data can be found in the cited references (Tables 1 and 3). An Excel file to compute viscosities with our model using fitting parameters of Table 2 is supplied as Supporting Information.