Determination of cooling rates of glasses over four orders of magnitude

Volcanic materials can experience up to eleven orders of magnitude of cooling rate (qc) starting from 10–5 K s−1. The glassy component of volcanic material is routinely measured via differential scanning calorimeter (DSC) to obtain qc through the determination of the glass fictive temperature (Tf). Conventional DSC (C-DSC), which has been employed for decades, can only access a relatively small range of qc (from ~ 10–2 to ~ 1 K s−1). Therefore, extrapolations up to six orders of magnitude of C-DSC data are necessary to derive qc of glasses quenched both at extremely low and high qc. Here, we test the reliability of such extrapolations by combining C-DSC with the recently introduced flash calorimetry (F-DSC). F-DSC enables to extend the qc exploration up to 104 K s−1. We use three synthetic glasses as analogs of volcanic melts. We first apply a normalization procedure of heat flow data for both C-DSC and F-DSC to derive Tf as a function of experimental qc, following the “unified area-matching” approach. The obtained Tf–qc relationship shows that Arrhenius models, widely adopted in previous studies, are only valid for qc determination within the calibration range. In contrast, a non-Arrhenius model better captures qc values, especially when a significant extrapolation is required. We, therefore, present a practical “how-to” protocol for estimating qc using DSC.

Here we present a combined conventional and flash differential scanning calorimetry (C-DSC and F-DSC) calibration by subjecting for the first time silicate melts of interest for volcanic processes to q c ranging between 0.08 and 30,000 K s −1 . We significantly exceed the interval of q c previously investigated in geospeedometry studies (~ 10 -2 to ~ 1 K s −1 ) and hence explore the validity of previous extrapolations from the relatively slow experimental q c to the realm (> 10 K s −1 ) of the submarine and explosive volcanism (e.g., Potuzak et al. 2008;Nickols et al. 2009). Our study is also of particular interest for Communicated by Mark S Ghiorso.

Theoretical background and experimental challenges
Two approaches are commonly employed to estimate q c of the glass: (i) the first relies on the quantification of the molecular H 2 O and OH − species in the glass (Zhang et al. 1997(Zhang et al. , 2000Benne and Behrens 2003;Behrens and Nowak 2003;Bauer et al. 2015;Behrens 2020) and requires knowledge of the chemical dependence of the absorption coefficients (e.g., Behrens 2020), whereas (ii) the second requires the knowledge of the limiting fictive temperature [T f , Tool (1946)] of the glass and can be applied both on dry and hydrous glasses. Here, we focus on the latter method that lies at the heart of the enthalpy relaxation of the melt (Wilding et al. 1995(Wilding et al. , 1996a(Wilding et al. , b, 2000(Wilding et al. , 2004Dingwell 2001a, b, 2002;Potuzak et al. 2008;Nichols et al. 2009;Helo et al. 2013;Hui et al. 2018).
The enthalpy relaxation approach is based on the observation that the glass structure depends on the cooling rate at which the parental melt crossed the glass transition interval (Tool and Eichlin 1931;DeBolt et al. 1976;Moynihan et al. 1976). Once the transition is crossed, the melt structure appears frozen at the observation timescale. Τhe glass retains the structure and properties (e.g., enthalpy, H, and volume, V) of the melt at the characteristic limiting fictive temperature, T f (Tool 1946).
Because T f correlates positively with q c (e.g., Tool 1946;Moynihan et al. 1976), the knowledge of T f enables the derivation of q c , provided that the T f vs. q c relationship is known (Zheng et al. 2019). The T f vs. q c relationship has been observed to be Arrhenian (i.e., linear in the -log 10 (q c ) vs. 1/T f space) at T close to T g (e.g., Webb et al. 1992;Knoche et al. 1994;Potuzak et al. 2008;Zheng et al. 2019), where T g is defined as the temperature at which the shear viscosity (η) of the melt is 10 12 Pa s: The Arrhenius correlation between q c and T f is described by the Tool (1946) equation: where A q and E q a are fitting parameters describing the preexponential factor and the activation energy for structural relaxation, respectively, and R is the gas constant. E q a scales with the activation energy of viscous flow ( E a ; Moynihan (1) T g = 10 12 Pa s al. 1976, 1996) that, for a relatively small temperature range, can be also approximated as Arrhenian: where A η is the pre-exponential factor corresponding to the viscosity at infinite T (Russell et al. 2003). However, both descriptions of q c (Eq. 2) and η (Eq. 3) are non-Arrhenian when considered across the whole T range of silicate melts (Angell 1985;Hui and Zhang 2007;Giordano et al. 2008;Mauro et al. 2009;Zheng et al. 2017). The degree of deviation from the Arrhenian behavior can be described by the fragility index m vis , also known as kinetic melt fragility: Similarly, the degree of deviation from the Arrhenian behavior for q c follows the Moynihan et al. (1996) formulation: Zheng et al. (2017) found that m vis correlates linearly and positively with the calorimetric fragility (m DSC ) over a wide range of kinetic fragility (varying from 26 to 108).
The TNM approach relates T f to four parameters (A q , E q a , β and ξ). It requires first multiple heat capacity measurements of the rejuvenated glass at different but matching cooling (DSC downscan) and heating (DSC upscan) cycles that allow the derivation of A q and E q a (Eq. 2). Afterwards, two empirical parameters (β and ξ) ranging between 0 and 1 are fitted by tweaking them to minimize the root square mean error to derive T f and thereby the unknown cooling rate (q c ). Kenderes and Whittington (2021) have recently provided a Matlab code to derive the four kinetic parameters without the need to perform multiple heat capacity measurements. However, the TNM approach cannot model the broad exothermic enthalpy relaxation upon DSC upscan before the glass transition interval typical of hyperquenched glasses (Yue et al. 2002;Potuzak et al. 2008;Nichols et al. 2009;Zheng et al. 2019).
The "area-matching" approaches offer a route to obtain T f of glasses quenched at any cooling rate. These approaches are generally based on comparing two DSC upscans (see Zheng et al. 2019 for a review). Here the glass with unknown cooling rate is heated (upscan 1) above the glass transition interval until the liquid state (i.e., T f = T) to erase its thermal history. Afterwards, the rejuvenated melt is cooled (q c ) across glass transition interval below T f and finally heated up (upscan 2) at the same rate (q h ). Because here q c = q h , the onset of the calorimetric glass transition (T onset , Al- Mukadam et al. 2020) corresponds to T f Sherer 1984;Moynihan 1993Moynihan , 1995Yue et al. 2004;Zheng et al. 2019;Di Genova et al. 2020a). T onset corresponds to the temperature of the crossing point between the tangent to the heat flow curve of the glass and the tangent to the inflection point on the following increase in heat flow (see "Treatment of DSC measurements and T f determination") (Al-Mukadam et al. 2020;Di Genova et al. 2020a;Stabile et al. 2021). Finally, upscan 1 and 2 are compared, and T f of the upscan 1 is obtained by a geometric reconstruction. This routine was recently refined by Guo et al. (2011) in the "unified area-matching" approach (see also "Treatment of DSC measurements and T f determination") for C-DSC measurements. However, this approach cannot be applied when the sample undergoes physicochemical modifications (degassing and/or crystallization) occurring in the glass transition interval during the upscan of a few K per minute typical of C-DSC (Richet et al. 1996;Liebske et al. 2003;Di Genova et al. 2020a). Although there are no studies of this effect, we hypothesize that modifications are particularly likely in iron-bearing samples prone to FeO nanocrystallization whose formation changes melt chemistry, increasing the melt relaxation time (Di Genova et al. 2017;2020b). Therefore, the first aspect to be explored in this study is the applicability of the "unified area-matching" approach for calorimetric measurements performed with the new flash DSC (F-DSC, q c up to 5 × 10 4 K s −1 ) that nowadays allows the study of the relaxation processes of extremely depolymerized melts through the suppression of crystallization (Schawe and Hess 2019;Al-Mukadam et al. 2020, 2021a. Figure 1 shows a compilation of cooling rates of volcanic glasses derived by conventional Differential Scanning Calorimetry (C-DSC) measurements by both TNM and area matching approaches. These glasses experienced more than ten orders of magnitude of q c ranging from ~ 10 -5 K s −1 (agglutinate/welded materials; Wilding et al. 1995Wilding et al. , 1996bDingwell 2001a, 2002) to ~ 10 6 K s −1 (explosive submarine products; Potuzak et al. 2008;Helo et al. 2013). Glasses from a large spectrum of volcanic environments (effusive, explosive, and extra-terrestrial) experienced intermediate q c (Wilding et al. 1995(Wilding et al. , 1996aWilding et al. 2000;Dingwell 2001a, b, 2002;Gottsmann et al. 2004;Wilding et al. 2004;Potuzak et al. 2008;Nickols et al. 2009;Kueppers et al. 2012;Helo et al. 2013;Lavallée et al. 2015;Hui et al. 2018).  Wilding et al. (1995Wilding et al. ( , 1996b, Dingwell (2001a, b, 2002), Gottsmann et al. (2004), and Lavallée et al. (2015) for subaerial quenched glass; Wilding et al. (2004) for subglacial quenched glass; Wilding et al. (1996a), and Hui et al. (2018) for extra-terrestrial quenched glass. The horizontal and colored areas represent the range of experimental cooling rates for C-DSC and F-DSC The accessible interval of q c,h of C-DSC equipment ranges from ~ 10 -2 to ~ 1 K s −1 (Wilding et al. 1995(Wilding et al. , 1996bDingwell 2001a, 2002;Helo et al. 2013;Lavallée et al. 2015;Hui et al. 2018;Al-Mukadam et al. 2020; Fig. 1). Consequently, the T f -q c relationship calibration (Eq. 2) may require a significant Arrhenian extrapolation up to four orders of magnitude in the slow-cooling rate realm (e.g., agglutinated/welded pyroclasts), or even up to six orders of magnitude for the explosive submarine environment. Although this assumption may hold for nearly Arrhenian melts (i.e., "strong" melts after Angell 1985) where m vis (Eq. 4) is ~ 15, the increase in melt fragility questions the reliability of cooling rate estimates through the Arrhenian extrapolation of Eq. 2.

Cooling rates ranges of volcanic glasses and literature assumptions
As such, the second aspect to be explored in this study is the use of the F-DSC by subjecting glasses to high cooling rates and thus avoiding the extrapolation of C-DSC data to q c typical of the submarine and the subaerial explosive environments ( Fig. 1). The significant q c expansion enables us to compare Arrhenian and non-Arrhenian models to describe the T f -q c relation, within and outside the calibration range. The non-Arrhenian temperature dependence of viscosity ( Fig. 2a) is described by VFT equations (Vogel 1921;Fulcher 1925;Tamman and Hesse 1926):

Starting materials and their properties
where A, B, and C are fitting parameters provided by Meerlender (1974) for DGG-1, by Reinsch et al. (2008) for Di, and by fitting viscometry data from Al-Mukadam et al. (2020) for N-PK52A (Supplementary Table 2). N-PK52A sample shows the lowest viscosity independently on T 2a). A crossover of viscosity is observed for DGG-1, and Di melts. The DGG-1 exhibits the higher viscosity only for T higher than 1334 K (~ 7.5 × 10 4 K −1 on the x-axis; Fig. 2a). This behavior results from the interplay between different melt fragility and T g of the melts (Fig. 2b).

Differential scanning calorimetry
The T f was determined by measuring the heat flow as a function of temperature using i) a C-DSC (404 F3 Pegasus) under N 2 5.0 atmosphere (flow rate 30 ml min −1 ) and ii) a F-DSC (Flash DSC 2 +) equipped with the UFH 1 sensor, under constant gas flow (40 ml min −1 ) of Ar 5.0. The C-DSC was calibrated using melting temperature and enthalpy of fusion of reference materials (pure metals of In, Sn, Bi, Zn, Al, Ag, and Au) up to 1337 K. The mass of the glasses measured with C-DSC was 28.20 mg for DGG-1, 22.90 mg for Di, and 26.60 mg for N-PK52A. The F-DSC was calibrated using the melting temperature of aluminum (melting temperature T m = 933.6 K) and indium (T m = 429.8 K). F-DSC measurements were conducted on glass chips with mass ranging from ~ 10 to ~ 300 ng depending on sample density (Al-Mukadam et al. 2020) to obtain an optimal signal-tonoise ratio.
For all measurements, 4 C-DSC and up to 14 F-DSC cycles of upscan and downscan were carried out at a fixed heating rate (q h ) and different cooling (q c ) rates. Heating rates imposed were ~ 0.17 K s −1 (10 K min −1 ) and 1000 K s −1 (6 × 10 4 K min −1 ) for C-DSC and F-DSC, respectively. Each upscan was followed by a downscan at a cooling rate ranging from ~ 0.08 to ~ 0.5 K s −1 (from 5 to 30 K min −1 ) for C-DSC and from 3 to 30,000 K s −1 (from 180 to 1.8 × 10 6 K min −1 ) for F-DSC. Consequently, one pair of upscan and downscan was performed at the same q c,h of ~ 0.17 K s −1 (10 K min −1 ) and 1000 K s −1 for the C-DSC and F-DSC, respectively. We refer to this pair of scans as a "matching cycle".

Treatment of DSC measurements and T f determination
The "unified area-matching" approach (Guo et al. 2011) for T f determination requires baseline correction and normalization. Figure 3a, c, e shows the heat flow output of C-DSC. We derived the specific heat capacity (c P in J g −1 K −1 ) using a standard sapphire (21.21 mg; Archer 1993) (Potuzak et al. 2008;Nichols et al. 2009;Hui et al. 2018). The c P error of the glassy contribution is ± 1%, whereas it is ± 3% for the liquid state (e.g., Potuzak et al. 2008). The glassy contribution (c Pg ) below the glass transition interval of matching cycles (q c,h ) at 10 K min −1 (black thick curves in Fig. 3a, c, e) was fitted using the Maier-Kelly empirical expression (Maier and Kelly 1932) to obtain the description of c Pg also in the glass transition interval and liquid state. We first calculated the difference between c P of the liquid state and c Pg at T onset , which here corresponds to T f only for matching cycles (q c,h = 10 K min −1 for C-DSC and 1000 K s −1 for F-DSCC) and specifically to T g when q c,h = 10 K min −1 Sherer 1984;Moynihan 1993Moynihan , 1995Yue et al. 2004;Zheng et al. 2019;Stabile et al. 2021). This difference thus corresponds to the configuration heat capacity of the melt at T g [Δc P (T g )] for C-DSC. We obtained Δc P (T g ) equal to 0.23 J g −1 K −1 for DGG-1, 0.45 J g −1 K −1 for Di, and 0.43 J g −1 K −1 for N-PK52A.
Subsequently, the Maier-Kelly fit of c Pg was subtracted from all c P measurements which were normalized using the Δc P (T g ) (Fig. 3b, d, f).
The procedure described so far was not applied to F-DSC measurements due to (1) the inability to acquire a proper background and sapphire measurements and (2) the extremely low sample mass (~ 10 -8 -10 -7 g) which would otherwise make the conversion of heat flow to heat capacity inaccurate (Schawe and Pogatscher 2016). Here we first applied a linear baseline over the heat flow of the glass region (Fig. 4a, c, e) and the heat flow data was subsequently converted to heat capacity (Fig. 4b, d, f) using Δc P (T g ) from C-DSC measurements (Fig. 3) after Schawe and Pogatscher (2016).
The fictive temperature T f was estimated from the normalized c P measurements. While T f ≡ T onset for matching cycles (Fig. 5a), T f of unmatching cycles was estimated according to the "unified area-matching" approach (Guo et al. 2011) as follows: where c P2 and c P1 are the normalized excess heat capacity of the matching and the unmatching cycle, respectively, and  Fig. 3 C-DSC measurements (a, c, e) and treatment (b, d, f). Measured heat flow (a, c, e) and normalized excess c P (b, d, f) of DGG-1, N-PK52A and Di samples measured at q h = 10 K min −1 and variable q c (from 5 to 30 K min −1 ). The matching cycle at q c,h = 10 K min −1 is shown as a thick black curve. Δc P in panels b, d, f refers to to the configuration heat capacity of the melt at T g (see text for details)

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Δc P is the configurational heat capacity at T g . The area difference between c P2 and c P1 corresponds to the rectangle whose base is defined by T f -T onset and height by Δc P (Fig. 5b). T f is thus the only unknown parameter. to 30,000 K s −1 ). The matching cycle at q c,h = 1000 K s −1 is showed as a thick black curve (from ~ 0.08 K s −1 up to 5 × 10 3 K s −1 ) using the same samples, C-DSC and F-DSC adopted here. We thus use their results to externally validate our strategy to derive T f of unmatching cycles.

Al
We first focus on the standard glass DGG-1 whose viscosity and T fq c relationships are well characterized (Meerlender 1974;Al-Mukadam et al. 2020;2021b). Figure 6a shows the temperature dependence -log 10 (q c ) vs.
1/T f provided by Al-Mukadam et al. (2020, 2021b. They found that the -log 10 (q c ) vs. 1/T f relationship is shifted from the VFT viscosity (left axis in Fig. 6a) by a factor of K = 11.20 ± 0.10 (log 10 Pa K) (see "Non-Arrhenian approximations" for discussion). As expected (e.g., Webb 2021), the values of T onset of our unmatching cycles (Table 2; triangles in Fig. 6a) strongly deviate from the main trend, which is crossed only when the matching cycle is considered (i.e., 0.17 K s −1 for C-DSC and 1000 K s −1 for F-DSC). Here T onset ≡ T f as discussed above (see "Treatment of DSC measurements and T f determination").
After the normalization procedure and the application of the area-matching method, the derived T f from unmatching cycles (Table 2; circles in Fig. 6a) are in basic agreement with the literature data (red line in Fig. 6a) when q c < 1000 K s −1 . Only a small deviation of up to -0.2 log units is evident in case of F-DSC. Conversely, when q c > 1000 K s −1 is considered (Fig. 6a), our results are systematically lower than the literature data. We speculate that the absence of the exothermic enthalpy relaxation event before the glass transition region (Fig. 4a, c, e), typical of hyper-quenched glasses (i.e., q c ≫ q h ; Yue et al. 2002;Dingwell et al. 2004), affects the area calculation of Eq. 7 that in turn leads to an underestimation of T f values. It thus appears that the F-DSC normalization procedure employed here (i.e., heating rate of 1000 K s −1 ) does not allow the accurate estimation of exothermic enthalpy relaxation before the glass transition interval for q c > 1000 K s −1 . Therefore, T f values when q c > 1000 K s −1 were discarded and not used for the T f -q c modeling. Overall, our results demonstrate that the "unified area-matching" strategy holds for determining T f when q c ranges between 0.08 and 1000 K s −1 using unmatching cycles. We thus conclude that this methodology can be employed also at fast cooling rates typical of F-DSC. Table 2 lists the estimated T f from DSC measurements for all samples considered here. As expected ), T f increases with increasing cooling rate. The N-PK52A sample exhibit the lowest T f that ranges from 746.9 K (0.08 K s −1 , C-DSC) to 804.5 K (20,000 K s −1 , F-DSC). Di sample exhibits the highest T f that ranges from 992.5 K (0.08 K s −1 , C-DSC) to 1082.0 K (30,000 K s −1 , F-DSC). The T f of DGG-1 melt lies in between N-PK52A and Di samples and varies between 811.8 K (0.08 K s −1 , C-DSC) and 936.7 K (30,000 K s −1 , F-DSC). Figure 6b shows the comparison between our T f and T onset from Al-Mukadam et al. (2020) for the three samples (see Table 2 for details). The figure illustrates an excellent agreement between the two datasets derived by C-DSC measurements, where the RMSE (Root Mean Square Error) estimated is equal to 3.0 K. Conversely, T f here derived by F-DSC agrees with T onset from Al-Mukadam et al. (2020) when q c ≤ 1000 K s −1 (RMSE = 11.4 K). For q c > 1000 K s −1 , T f data from N-PK52A and Di glasses show a smaller departure from the 1:1 line with respect to DGG-1 data. However, in analogy to what observed for DGG-1, we decided to conservatively discard these data from further analysis. Application of Eq. 7 to determine T f for the unmatching cycle (blue line, q c = 3 K s −1 ; q h = 1000 K s −1 ). The T f of the matching cycle corresponds to T onset (see text for details) Figure 7 shows the Arrhenian plot of -log 10 (q c ) vs. 10 4 /T f of our samples derived using both the unmatching approach employed in this study and the matching approach of Al-Mukadam et al. (2020). We initially fit only C-DSC data using the Arrhenian approximation of Eq. 2 (Arr. C-DSC Model). By doing this, we simulate the situation where F-DSC data (high q c ) are unavailable (e.g., Hui et al. 2018). Afterwards, we use Eq. 2 to fit both C-DSC and F-DSC data (Arr. C-DSC + F-DSC Model). Finally, we consider three non-Arrhenian models (non-Arr η Μοdel, non-Arr C-DSC Model and non-Arr C-DSC + F-DSC Model), based on the analogy between the temperature-dependence of cooling rate and viscosity (e.g., Scherer 1984;Yue et al. 2004;Al-Mukadam et al. 2020;Stabile et al. 2021). The non-Arr η Μοdel requires a priori knowledge of the viscosity-temperature relation, whereas, in non-Arr C-DSC and non-Arr C-DSC + F-DSC Models, this relationship is internally calibrated based on DSC measurements.

Non-Arrhenian approximations
Here we use the analogy between the -log 10 (q c ) vs. 1/T f and the log 10 (η) vs. 1/T to derive q c when the temperature Fig. 6 Validation of strategy to calculate T f from unmatching cycles. a DGG-1 sample. Left axis: Arrhenius plot of viscosity (blue line) using the VFT description (Eq. 6). Right axis: Arrhenius plot of the reciprocal cooling rate. Οnsets of the calorimetric glass transition (T onset ; triangles) and characteristic fictive temperatures (T f ; circles) obtained by DSC unmatching cycles. The onsets of the calorimetric glass transition (T onset ; squares) from Al-Mukadam et al. (2020, 2021b were evaluated from matching cycles and the model proposed by the authors relating viscosity and cooling rate (Eq. 8, with a shift factor of K = 11.20 ± 0.10 log 10 Pa K) are reported for reference. b Comparison between T f from this study and T onset from matching cycles by Al-Mukadam et al. (2020) for the three studied glasses (DGG-1, Di, N-PK52A). Note that only pairs of T f and T onset from measurements at the same q c are plotted. Results agree when q c < 1000 K s −1 . Data are reported in Table 2 dependence of viscosity is known. This analogy builds on the equivalence between the activation energy for viscous flow ( E a ) and the activation energy for structural relaxation E q a determined by DSC (Scherer 1984;Moynihan et al. 1996;Al-Mukadam et al. 2020, 2021aDi Genova et al. 2020a). Scherer (1984) first proposed a parallel shift factor correlating viscosity and q c following the equation: where η(T f ) is the viscosity at T f .
In the following, we examine two cases: (i) log 10 (η) at T f is independently known (non-Arr. η Model) and (ii) log 10 (η) at T f is derived from C-DSC and F-DSC data (non-Arr.

Viscosity is independently known (non-Arr. η Model)
The viscosity-temperature dependence of melts used here is known (Al-Mukadam et al. 2020) over a large interval (8) K = log 10 | | q c | | + log 10 T f (~ 10 1 < η < ~ 10 12 Pa s) and described by the non-Arrhenian VFT equation (Eq. 6; Supplementary Table 2). Therefore, T f can be used to calculate η(T f ) and obtain K (Eq. 8; Supplementary Fig. 1). We use either the C-DSC subset, the F-DSC subset, or the entire dataset (C-DSC + F-DSC). Slightly different K values are obtained for different samples and DSC devices (Supplementary Tables 2 and 3). The lowest mean value (11.11 ± 0.06 log 10 Pa K) is calculated for Di (only C-DSC data), while the highest (11.49 ± 0.09 log 10 Pa K) for DGG-1 (only F-DSC). By combining C-DSC and F-DSC data, we obtain K = 11.40 ± 0.15 log 10 Pa K for DGG, K = 11.34 ± 0.11 log 10 Pa K for N-PK52A and K = 11.30 ± 0.20 log 10 Pa K for Di. We thus provide a global value of K = 11.35 ± 0.16 log 10 Pa K that agrees with previous data within the standard deviation (11.20 ± 0.10 < K < 11.35 ± 0.10; Scherer 1984; Yue et al. 2004;Chevrel et al. 2013;Shawe and Hess, 2019;Al-Mukadam et al. 2020, 2021bDi Genova et al. 2020b;Stabile et al. 2021). The K mean = 11.35 log 10 Pa K allows the estimation of q c up to 1000 K s −1 with a RMSE of 0.15 log 10 Table 2 Οnsets of the calorimetric glass transition (T onset ) and characteristic fictive temperatures (T f ) of the studied glasses obtained by DSC unmatching cycles Οnsets of the calorimetric glass transition (T onset ) from DSC matching cycles by Al-Mukadam et al. (2020) are reported for comparison. Stars (*) indicate data excluded from subsequent analyses (see text for details)  Red lines correspond to Εq. 8 (where K mean = 11.35 log 10 Pa K as derived in this study). Shaded areas correspond to the K mean uncertainty (± 0.16 log 10 Pa K). b, d, f Experimentally imposed (q c ) vs. modelled (q cm ) cooling rates for the different models: b Arr. C-DSC Model (Arr. C-DSC + F-DSC Model is reported in the inset); d non-Arr. η Model; e non-Arr. C-DSC Model (non-Arr. C-DSC + F-DSC Model is reported in the inset). Dotted lines represent the 1:1 line; all calculated data are reported with relative standard deviations K s −1 for DGG-1, 0.20 log 10 K s −1 for Di, and 0.11 log 10 K s −1 for N-PK52A. For instance, the non-Arr. η Model returns q cm = 655 K s −1 (2.82 log 10 K s −1 ) for DGG-1, q cm = 1443 K s −1 (3.16 log 10 K s −1 ) for Di, and q cm = 705 K s −1 (2.85 log 10 K s −1 ) for N-PK52A, when the experimental q c is 1000 K s −1 (Table 3; Fig. 7d).
Notably, the non-Arr η Model excellently fits the T onset from Al-Mukadam et al. (2020) up to 30,000 K s −1 within the error (Fig. 7c).

Viscosity is internally calibrated by DSC measurements (non-Arr. C-DSC and non-Arr. C-DSC + F-DSC models)
Natural and synthetic melts (including those of volcanological interest) can undergo crystallization or exsolution of volatiles on the timescale of viscosity measurements near T g (Liebske et al. 2003;Di Genova et al. 2017, 2020a. Because crystallization and dehydration around T g lead to the absence of pure melt viscosity data near T g (Dingwell Table 3 Cooling rates estimated (q cm ) according to the different model proposed in this study Bold numbers highlight matching cycle measurements Sample DSC-type log 10 q c (K s −1 ) Arr. C-DSC Model Arr. C-DSC + F-DSC Model non-Arr. η Model non-Arr. C-DSC Model non-Arr. C-DSC + F-DSC Model log 10 q cm (K s −1 ) log 10 q cm (K s −1 ) log 10 q cm (K s −1 ) log 10 q cm (K s −1 ) log 10 q cm (K s −1 )  Chevrel et al. 2013), the use of DSC offers a viable route to derive melt viscosity due to its lower impact on melt chemistry and texture Chevrel et al. 2013;Di Genova et al. 2020b;Langhammer et al. 2021;Stabile et al. 2021).
Here we use the viscosity-temperature description provided by the Mauro-Yue-Ellison-Gupta-Allan equation (MYEGA; Mauro et al. 2009): where log 10 (η ∞ ) is the limit of viscosity logarithm at infinite T (− 2.93 log 10 Pa s, Zheng et al., 2011), T g is the glass transition temperature derived by DSC via a matching measurement at 10 K min −1 ( T g ≡ T onset ), and m vis is the liquid fragility index (Eq. 4). Because η and q c are related to each other via Eq. 8 and K mean = 11.35 log 10 Pa K, from Eq. 9 we can derive: where m DSC is the calorimetric fragility (Zheng et al. 2017) according to Eq. 5. m vis linearly correlates (~ 1:1) with m DSC (derived by C-DSC) over a wide range of kinetic fragility (e.g., Chen et al. 2014;Zheng et al. 2017;Al-Mukadam et al. 2021b).
We consider two cases based on C-DSC data (non-Arr. C-DSC Model) and C-DSC + F-DSC data (non-Arr.

C-DSC + F-DSC Model).
For non-Arr. C-DSC model, we use E q a (Supplementary Table 1 Figure 2).
Finally, Fig. 7e shows that the non-Arr. C-DSC + F-DSC Model better captures the T onset related to the ultra-fast cooling rates from Al-Mukadam et al. (2020) due to the increased number of data considered. This reflects on a better constrain to the ultra-fast cooling region.

Comparison between models within and outside the calibration range
The proposed models return q c of our melts to different extents (Fig. 7). In general, with the exception of the Arr. C-DSC Model, all the models can predict the experimentally imposed cooling rate with a good degree of approximation.
The Arr. C-DSC Model, based only on C-DSC, fails to predict F-DSC data due to the narrow range of q c employed by C-DSC (~ 0.08 ≤ q c ≤ 0.5 K s −1 ). This results in the unreliable extrapolation of q c to higher values (RMSE = 0.37 log 10 K s −1 ). Conversely, the Arr. C-DSC + F-DSC Model, based on C-DSC and F-DSC, allows the estimation of q c up to 1000 K s −1 with an RMSE = 0.16 log 10 K s −1 ). The Arr. C-DSC + F-DSC Model well predicts q cm in a η(T f ) range varying between ~ 10 8 and ~ 10 12 Pa s (Supplementary Fig. 1 and Supplementary Table 1). This viscosity interval is analogous to that commonly investigated through micropenetration viscometry, where silicate melts approximate Arrhenian fluids (e.g., Hess et al. 1995).
All non-Arrhenian models proved to successfully predict q c in the investigated cooling rates interval. This can be achieved using the chemically independent shift factor (K = 11.35 ± 0.16 log 10 Pa K, Eq. 8). If the viscosity-temperature relationship is known, the non-Arr. η Model represents the most robust procedure for obtaining q c values (Table 3, RMSE of 0.15 log 10 K s −1 ) also capturing the T onset related to ultra-fast q c from Al- Mukadam et al. (2020). Alternatively, the non-Arr. C-DSC and non-Arr. C-DSC + F-DSC Models (mean RMSE of 0.20 log 10 K s −1 and 0.31 log 10 K s −1 , respectively) offer a viable route when viscosity data are unavailable. Figure 8 depicts the models' performances in the entire cooling range typical of volcanic materials. The models substantially diverge, both in the slow-cooling rate (q c < 10 -1 K s −1 ) and fast-cooling rate (q c > 10 3 K s −1 ) regimes. It is therefore apparent that, when extensive extrapolation is needed (e.g., for the study of fast-quenched submarine materials or slow-quenched lava flows and vitrophyres), the choice of the models becomes non-trivial.
The analogy between E q a and E a (Scherer 1984;Moynihan et al. 1996;Al-Mukadam et al. 2020, 2021aDi Genova et al. 2020a, b) makes it plausible to hypothesize the T f -q c relationship as non-Arrhenian in the entire cooling range typical of volcanic environments ( Fig. 8; 10 -5 K s −1 ≤ q c ≤ 10 6 K s −1 ), as for viscosity in the whole temperature range. As a consequence, in the case of ultrafast and ultra-slow cooling rates, the employment of a non-Arrhenian model is expected to provide more reliable data and should be preferred. In contrast, the constant E q a of Arrhenian models is expected to produce an overestimation of q c for fast-and slow-quenched glasses (Fig. 8).
Geospeedometry recipes: a "how-to" protocol for cooling rate determination Figure 9 summarizes a practical "how-to" guide to determine the cooling rate of glasses, depending on the available dataset.
The choice of the appropriate calorimeter should be dictated in the first place by the expected range of cooling rates, based on the geological setting of the studied glass (Fig. 1). In addition, F-DSC is more suited when small amounts of glass material are available for the analysis and when the sample is prone to rapid modifications at high temperatures. Is η(T f ) known?

non-Arr. η Model
Calculate m DSC and T g non-Arr. DSC Model "Unified area-matching" approach for T f determination