## Abstract

Volcanic materials can experience up to eleven orders of magnitude of cooling rate (*q*_{c}) starting from 10^{–5} K s^{−1}. The glassy component of volcanic material is routinely measured via differential scanning calorimeter (DSC) to obtain *q*_{c} through the determination of the glass fictive temperature (*T*_{f}). Conventional DSC (C-DSC), which has been employed for decades, can only access a relatively small range of *q*_{c} (from ~ 10^{–2} to ~ 1 K s^{−1}). Therefore, extrapolations up to six orders of magnitude of C-DSC data are necessary to derive *q*_{c} of glasses quenched both at extremely low and high *q*_{c}. 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 *q*_{c} exploration up to 10^{4} 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 *T*_{f} as a function of experimental *q*_{c}, following the “unified area-matching” approach. The obtained *T*_{f}–*q*_{c} relationship shows that Arrhenius models, widely adopted in previous studies, are only valid for *q*_{c} determination within the calibration range. In contrast, a non-Arrhenius model better captures *q*_{c} values, especially when a significant extrapolation is required. We, therefore, present a practical “how-to” protocol for estimating *q*_{c} using DSC.

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## Introduction

### Motivation and aims

The estimation of the cooling rate (*q*_{c}) embedded in volcanic glasses is pivotal for the reconstruction of their petrogenesis (Wilding et al. 1995, 1996a, b, 2000, 2004; Gottsmann and Dingwell 2001a, b, 2002; Gottsmann 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). Indeed, the *q*_{c} experienced by the magma controls the texture of volcanic rocks via the modulation of crystal and bubble growth with time, as well as the timescale of post-depositional processes such as sintering, welding, agglutination, rheomorphism, degassing, and crystallization (Quane et al. 2009; Shea et al. 2010; Gonnermann and Manga 2012; Di Genova et al. 2013; Vasseur et al. 2013; Vetere et al. 2015; Iezzi et al. 2017; Wadsworth et al. 2019; Cashman 2020; Giuliani et al. 2020; Di Fiore et al. 2021).

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 those melts prone to crystallization and dehydration during C-DSC measurements (Richet et al. 1996; Liebske et al. 2003; Di Genova et al. 2020a). Finally, we provide a practical “how-to” guide to calculating the cooling rate of volcanic glasses.

### 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, 2000; Benne 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, 1996a, b, 2000, 2004; Gottsmann and Dingwell 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 \({E}_{a}^{q}\) are fitting parameters describing the pre-exponential factor and the activation energy for structural relaxation, respectively, and *R* is the gas constant*.* \({E}_{a}^{q}\) scales with the activation energy of viscous flow (\({E}_{a}^{\eta }\) ; Moynihan et 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).

Different approaches have been proposed to derive *T*_{f} of glasses. One is the Tool-Narayanaswamy-Moynihan (TNM) enthalpy relaxation geospeedometer (e.g., Tool 1946; Narayanaswamy 1971, 1988; DeBolt et al. 1976; Moynihan et al. 1976; Wilding et al. 1995), whereas others belong to the “area-matching” (or “enthalpy-matching”) methodologies (Moynihan et al. 1976; Yue et al. 2002; Potuzak et al. 2008; Nichols et al. 2009; Guo et al. 2011).

The TNM approach relates *T*_{f} to four parameters (*A*_{q}, \({E}_{a}^{q}\) , *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 \({E}_{a}^{q}\) (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 hyper-quenched 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} (Moynihan et al. 1976; Sherer 1984; Moynihan 1993, 1995; Yue 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, b).

### Cooling rates ranges of volcanic glasses and literature assumptions

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. 1995, 1996b; Gottsmann and Dingwell 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, 1996a, b; Wilding et al. 2000; Gottsmann and 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).

The accessible interval of *q*_{c,h} of C-DSC equipment ranges from ~ 10^{–2} to ~ 1 K s^{−1} (Wilding et al. 1995, 1996b; Gottsmann and Dingwell 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.

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.

## Starting materials, experimental and analytical methods

### Starting materials and their properties

We use three glasses: a sodium-calcium silicate (DGG-1, Meerlender 1974), a diopside (Di, Fanara and Behrens 2011), and a fluorophosphate (N-PK52A, Schott 2015). The chemical composition of the samples is listed in Table 1. The viscosity (Fig. 2a) and melt fragility (Fig. 2b) of these systems were characterized by Al-Mukadam et al. (2020). The melt fragility (*m*_{vis}) of our samples ranges from 37.8 (DGG-1) to 72.3 (N-PK52A), with the Di sample having *m*_{vis} = 62.7. Both the viscosity–temperature space and melt fragility of our melts well match the spectrum of viscosity and fragility of anhydrous and hydrous volcanic melts (shaded areas in Fig. 2) recently compiled by Langhammer et al. (2021).

The non-Arrhenian temperature dependence of viscosity (Fig. 2a) is described by VFT equations (Vogel 1921; Fulcher 1925; Tamman and Hesse 1926):

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* (Fig. 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-to-noise 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} (Moynihan et al. 1976; Sherer 1984; Moynihan 1993, 1995; Yue 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 *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.

## Results

Al-Mukadam et al. (2020) determined the *T*_{onset} (i.e., *T*_{f}) via matching cycles over four orders of magnitude of *q*_{c,h} (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.

We first focus on the standard glass DGG-1 whose viscosity and *T*_{f} – *q*_{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 (Moynihan et al. 1976), *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.

## Discussion

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.

### Arrhenian approximations (Arr. C-DSC and Arr. C-DSC + F-DSC models)

When only C-DSC data are considered (*Arr. C-DSC Model*), we obtain activation energy (\({E}_{a}^{q}\) ) of 619 kJ mol^{−1} for DGG-1, 1085 kJ mol^{−1} for N-PK52A, and 1407 kJ mol^{−1} for Di (Supplementary Table 1). While this model can obviously reproduce C-DSC data (Fig. 7a), it cannot accurately reproduce F-DSC data. The estimated RMSE is 0.17 log_{10} K s^{−1} for DGG-1, 0.53 log_{10} K s^{−1} for N-PK52A, and 0.43 log_{10} K s^{−1} for Di. The difference (*Δ*_{q}) between measured (*q*_{c}) and calculated (*q*_{cm}) cooling rates depends on the melt fragility (Fig. 7b; Table 3). For instance, when the experimental *q*_{c} was 1000 K s^{−1} we obtain *q*_{cm} = 1727 K s^{−1} (3.24 log_{10} K s^{−1};) for DGG-1 sample, *q*_{cm} = 17,019 K s^{−1} (4.23 log_{10} K s^{−1}) for Di and *q*_{cm} = 3832 K s^{−1} (3.58 log_{10} K s^{−1}) for N-PK52A.

When C-DSC and F-DSC are considered (*Arr. C-DSC* + *F-DSC Model*), lower activation energies are obtained (581 kJ mol^{−1} for DGG-1, 923 kJ mol^{−1} for N-PK52A and 1121 kJ mol^{−1} for Di; Supplementary Table 1). This results in a more accurate estimation of *q*_{cm} and thereby lower RSME (Fig. 7b inset; RMSE_{DGG-1} = 0.14 log_{10} K s^{−1}; RMSE_{N-PK52A} = 0.08 log_{10} K s^{−1}; RMSE_{Di} = 0.24 log_{10} K s^{−1}). Here, cooling rates of *q*_{cm} = 1278 K s^{−1} (3.11 log_{10} K s^{−1}) for DGG-1, 2281 K s^{−1} (3.36 log_{10} K s^{−1}) for Di, and 839 K s^{−1} (2.92 log_{10} K s^{−1}) for N-PK52A are derived when the experimental *q*_{c} is 1000 K s^{−1} (Table 3).

Figure 7 also shows *T*_{onset} from Al-Mukadam et al. (2020) for comparison. Except for N-PK52A sample, both Arrhenian models fail to reproduce this dataset when *q*_{c} > 1000 K s^{−1} (Fig. 7a).

### 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 dependence of viscosity is known. This analogy builds on the equivalence between the activation energy for viscous flow (\({E}_{a}^{\eta }\) ) and the activation energy for structural relaxation \({E}_{a}^{q}\) determined by DSC (Scherer 1984; Moynihan et al. 1996; Al-Mukadam et al. 2020, 2021a, b; Di 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. C-DSC Model*, and *non-Arr. C-DSC* + *F-DSC Model*).

#### 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 (~ 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, 2021b; Di 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} 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, b). Because crystallization and dehydration around *T*_{g} lead to the absence of pure melt viscosity data near *T*_{g} (Dingwell et al. 2004; 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 (Dingwell et al. 2004; 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 \({E}_{a}^{q}\) (Supplementary Table 1) derived from the *Arr. C-DSC Model* and Eq. 5 and find *m*_{DSC} = 39.5 for DGG-1, *m*_{DSC} = 75.5 for N-PK52A and *m*_{DSC} = 73.8 for Di. These values well match the *m*_{vis} provided by Al-Mukadam et al. (2020) (cf. Figure 2).

We estimate (Eq. 10) *q*_{c} up to 1000 K s^{−1} with a RMSE of 0.20 log_{10} K s^{−1} for DGG-1, 0.24 log_{10} K s^{−1} for Di, and 0.15 log_{10} K s^{−1} for N-PK52A (Table 3; Fig. 7f). The *non-Arr. C-DSC Model* predicts *q*_{cm} = 637 K s^{−1} (2.80 log_{10} K s^{−1}) for DGG-1, *q*_{cm} = 3418 K s^{−1} (3.53 log_{10} K s^{−1}) for Di, and *q*_{cm} = 981 K s^{−1} (2.99 log_{10} K s^{−1}) for N-PK52A, when the experimental *q*_{c} is 1000 K s^{−1} (Table 3; Fig. 7f).

The *non-Arr. C-DSC* + *F-DSC Model* finds slightly lower fragility values. For instance, *m*_{DSC} = 37.1 for DGG-1, *m*_{DSC} = 64.2 for N-PK52A and *m*_{DSC} = 58.8 for Di, according to a lower \({E}_{a}^{q}\) (Supplementary Table 1) derived from the *Arr. C-DSC* + *F-DSC Model* and Eq. 5. Here, cooling rates of *q*_{cm} = 414 K s^{−1} (2.62 log_{10} K s^{−1}) for DGG-1, 598 K s^{−1} (2.78 log_{10} K s^{−1}) for Di, and 314 K s^{−1} (2.50 log_{10} K s^{−1}) for N-PK52A are derived (Table 3) when the experimental *q*_{c} is 1000 K s^{−1}. The model underestimates F-DSC cooling rates, leading to a general RMSE of 0. 29 log_{10} K s^{−1} for DGG-1, 0.37 log_{10} K s^{−1} for Di, and 0.26 log_{10} K s^{−1} for N-PK52A.

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 \({E}_{a}^{q}\) and \({E}_{a}^{\eta }\) (Scherer 1984; Moynihan et al. 1996; Al-Mukadam et al. 2020, 2021a, b; Di 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 ultra-fast 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 \({E}_{a}^{q}\) 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.

The “unified area-matching” holds for glasses subjected to *q*_{c} between ~ 0.08 and ~ 0.5 K s^{−1} in the C-DSC and between 3 and 1000 K s^{−1} in the F-DSC. The method requires two DSC upscans (unmatching and matching cycles) to estimate *T*_{f} of the glass with unknown thermal history (Eq. 7).

A separate set of matching cycles (*q*_{c2,h2}, *q*_{c3,h3}, *q*_{cn,hn}, …) is required for the calibration of the *T*_{f}–*q*_{c} relationship. Arrhenian models (Eq. 2) should be only used when *T*_{f} of the glass with unknown cooling history falls within or close to the DSC (C-DSC or F-DSC) calibration range. For C-DSC measurements this conditions corresponds to *T*_{f} close to *T*_{g} (as defined in Eq. 1).

When a significant extrapolation is required, a non-Arrhenian model represents the best protocol to estimate *q*_{c}. Notably, the *non-Arr η Model* does not require the calibration of the *T*_{f}–*q*_{c} relationship through matching cycles. This approach only requires two DSC measurements (i.e., the first upscan on the glass with an unknown cooling history and the rejuvenated glass matching cycle). Once *T*_{f} is determined, the application of Eq. 8 allows the estimation of the unknown *q*_{c}, provided the viscosity of the sample is independently known.

However, crystallization and dehydration of melts may hamper the viscosity measurements (Richet et al. 1996; Liebske et al. 2003; Zheng et al. 2017, 2019; Di Genova et al. 2020a, b) and pure melt viscosity data near *T*_{g} may be lacking. Here, the non-Arrhenian DSC-based models offer an alternative procedure for *q*_{c} determination. The application of these models requires a C-DSC matching cycle at *q*_{c,h} = 10 K min^{−1} to constrain *T*_{g} (Eq. 1; *Τg ≡ T*_{onset}) and a set of matching cycles at variable *q*_{c,h} (C-DSC and/or F-DSC) to constrain the calorimetric fragility (*m*_{DSC}; Eq. 5). Once these parameters are determined, the unknown *q*_{c} is computed via Eq. 10. It should be noted that these models are not constrained by data in the low-viscosity/high-cooling rates regime (a constant value of *log*_{10}(*η*_{∞}) = –2.93 is used in Eq. 9 and 10). Consequently, when extrapolation is needed, they are expected to perform better in the slow cooling rate regime and may progressively becomes less accurate at fast cooling rates. In addition, since the determination of *m*_{DSC} is slightly affected by the matching calibration interval over which is averaged, the use of C-DSC should be preferred for extrapolation towards slow cooling rates, whereas the use of a F-DSC is expected to reduce the extrapolation uncertainty at very fast cooling rates (i.e., *q*_{c} > 10^{3} K s^{−1}).

## Concluding remarks

This study combined conventional and flash calorimetry to estimate the cooling rate (*q*_{c}) of glass-forming melts over four orders of magnitude, from 0.08 K s^{−1} (5 K min^{−1}) to 1000 K s^{−1}. Our study is based on the relationship between *q*_{c} and the glass’s limiting fictive temperature (*T*_{f}).

We demonstrate that the “unified area-matching” approach for calculating *T*_{f} proposed by Guo et al. (2011) holds for *q*_{c} up to 1000 K s^{−1}. This is central for obtaining the *T*_{f} of the glass quenched at an unknown cooling rate without the need to perform empirical fitting of heat capacity measurements.

We show that the Arrhenian extrapolation of *–log*_{10}(*q*_{c}) over *1/T*_{f} can overestimate the unknown *q*_{c}. The non-Arrhenian description of *–log*_{10}(*q*_{c}) over *1/T*_{f} offers the most accurate estimation of *q*_{c}*,* especially when *q*_{c} falls outside the calibration range. The non-Arrhenian description requires, however, the knowledge of the melt viscosity. Because volcanic melt can be prone to crystallization and dehydration during viscosity measurements, we present an alternative route to derive *q*_{c} without performing viscosity measurements.

## Change history

### 20 July 2022

Missing Open Access funding information has been added in the Funding Note.

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## Acknowledgements

This research was funded by MIUR Grants to Roma Tre PhD School in Earth Sciences (XXXIV doctoral cycle). The Grant of Excellence Departments, MIUR-Italy (ARTICOLO 1, COMMI 314–337 LEGGE 232/2016) is also gratefully acknowledged. D. Di Genova acknowledges funding by Deutsche Forschungsgemeinschaft (DFG) projects DI 2751/2-1.

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Scarani, A., Vona, A., Di Genova, D. *et al.* Determination of cooling rates of glasses over four orders of magnitude.
*Contrib Mineral Petrol* **177**, 35 (2022). https://doi.org/10.1007/s00410-022-01899-5

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DOI: https://doi.org/10.1007/s00410-022-01899-5