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Reconciling calorimetric and kinetic fragilities of glass-forming liquids
NOC-18087; No of Pages 6 Journal of Non-Crystalline Solids xxx (2016) xxx–xxx
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Reconciling calorimetric and kinetic fragilities of glass-forming liquids Qiuju Zheng a,c, John C. Mauro a,b,⁎, Yuanzheng Yue a,c,⁎ a b c
Shandong Key Laboratory of Glass and Ceramics, Qilu University of Technology, 250353 Jinan, China Science and Technology Division, Corning Incorporated, Corning, NY 14831, USA Department of Chemistry and Bioscience, Aalborg University, DK-9000 Aalborg, Denmark
a r t i c l e
i n f o
Article history: Received 7 October 2016 Received in revised form 9 November 2016 Accepted 11 November 2016 Available online xxxx Keywords: Viscosity Fragility Differential scanning calorimetry (DSC)
a b s t r a c t The liquid fragility index (mvis) describes the rate of viscosity change of a glass-forming liquid with temperature at the glass transition temperature (Tg), which is very important for understanding liquid dynamics and the glass transition itself. Fragility can be directly determined using viscosity measurements. However, due to various technical complications with determining viscosity, alternative methods to obtain fragility are needed. One simple method is based on measurement of the calorimetric fragility index (mDSC), i.e., the changing rate of fictive temperature (Tf) with heating (cooling) rate in a small Tf range around Tg. The crucial question is how mDSC is quantitatively related to mvis. Here, we establish this relation by performing both dynamic and calorimetric measurements on some selected glass compositions covering a wide range of liquid fragilities. The results show that mDSC deviates systematically from mvis. The deviation is attributed to the Arrhenian approximation of the log(1/qc) ~ Tg/Tf relationship in the glass transition range. We have developed an empirical model to quantify the deviation, by which mvis can be well predicted from mDSC across a large range of fragilities. Combined with the high-T viscosity limit (10–2.93 Pa·s), we are able to obtain the entire viscosity curve of a glass-forming liquid by only performing DSC measurements. © 2016 Published by Elsevier B.V.
1. Introduction The shear viscosity of glass-forming liquids is of great importance in all stages of industrial glass production [1–3]. Since viscosity is very sensitive to temperature and composition, we need to have accurate knowledge of the scaling of viscosity with both of these parameters. Viscosity is also critical for understanding the glass transition and the relaxation characteristics of liquids and their corresponding glasses. In the well-known Angell plot [4,5], the logarithm of viscosity, log10 η, is plotted as a function of the Tg-scaled inverse temperature, Tg/T, where T is absolute temperature. With this scaling, Angell was able to compare the viscous flow behavior of all glass-forming liquids in a single universal plot. According to the Angell plot, there are three important parameters [6]: (i) the glass transition temperature, Tg(x); (ii) the fragility, m(x); and (iii) the extrapolated infinite temperature viscosity, η∞(x). For any composition x, the glass transition temperature is defined as the temperature at which the shear viscosity is equal to 1012 Pa·s [7],
⁎ Corresponding authors. E-mail addresses: [email protected] (J.C. Mauro), [email protected] (Y. Yue).
i.e., η(Tg(x),x) = 1012 Pa·s. Fragility [8] is defined as the slope of the logη versus Tg/T at Tg: mðxÞ ¼
∂log10 ηðT; xÞ ∂ T g ðxÞ=T
:
ð1Þ
T¼T g ðxÞ
The fragility index describes the rate of change in the liquid dynamics upon cooling through the glass transition. Liquids can be classified as either “strong” or “fragile” depending on whether they exhibit an Arrhenius or super-Arrhenius scaling of viscosity with temperature, respectively. While the fragility index itself is a first-derivative property of the viscosity curve, the degree of non-Arrhenius scaling reflects the second derivative of the viscosity curve with respect to inverse temperature. Following Angell [4,8], this non-Arrhenius scaling of liquid viscosity can be quantified directly through the fragility index (mvis in this work) with the assumption of a universal high temperature limit of viscosity, i.e., η∞(x) = η∞. In our previous work [9], we have analyzed the viscosity data of 946 silicate liquids and other 31 non-silicate liquids; the results imply that the silicate liquids have a universal high temperature viscosity limit of around 10–2.93 Pa·s. Thus, we have validated Angell's assumption, which enables this direct connection between first- and second-derivative properties.
http://dx.doi.org/10.1016/j.jnoncrysol.2016.11.014 0022-3093/© 2016 Published by Elsevier B.V.
Please cite this article as: Q. Zheng, et al., Reconciling calorimetric and kinetic fragilities of glass-forming liquids, J. Non-Cryst. Solids (2016), http:// dx.doi.org/10.1016/j.jnoncrysol.2016.11.014
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Given the great importance of the fragility parameter, it has been the subject of extensive interest in both experimental and theoretical studies for many years [10–15]. While fragility is a kinetic property, it is closely correlated with various thermodynamic quantities of glassforming liquids. For example, the change in heat capacity at the glass transition, width of the glass transition range, and configurational entropy loss at the glass transition all exhibit linear correlations with kinetic fragility [16–23]. An accurate determination of fragility is thus of great significance for understanding liquid physics near glass transition and for understanding the glass transition itself. Following the definition of fragility in Eq. (1), it should be directly determined using viscosity measurements. However, viscosity measurements are difficult for glass-forming systems, especially those systems with strong crystallization tendency and high liquidus temperatures. Liquid viscosity varies by over twelve orders of magnitude and requires specialized equipment in different viscosity and temperature regimes. Crystallization and volatilization of the melts can hinder high temperature viscosity measurements, while the specific sample size/shape demands and long measurement time impede the low temperature viscosity experiments. Therefore, alternative methods are needed to give an indirect quantification of fragility. Various methods have been proposed to calculate fragility using differential scanning calorimetry (DSC). Moynihan and his co-workers found that the activation energy for structural relaxation determined by DSC is an accurate estimate of the activation energy for shear viscosity [24–27]. The activation energy for structural relaxation in the glass transition region can be determined from the cooling rate (qc) dependence of the fictive temperature Tf measured using DSC. E d lnqc ¼− g: R d 1=T f
ð2Þ
where Eg is the activation energy for equilibrium viscous flow in the glass transition region and R is the ideal gas constant. The fictive temperature, Tf, is defined here as the temperature at which the configurational enthalpy of the glass equals that of the corresponding liquid state [28]. This is obtained by an enthalpy-matching integral method using the heat capacity (Cp) curve during reheating [29]. It is found that when the prior cooling (qc) and reheating rates (qh) are the same, the Tf obtained using the integration method from the Cp reheating curve is very close to the onset glass transition temperature, Tg,onset [24–26, 30]. For all the DSC measurements performed in this work, the reheating rates are equal to the preceding cooling rates. Therefore the Tg,onset has been used as the fictive temperature Tf for the studied glasses in this paper. Kissinger derived another equation to calculate activation energy for glass transition using DSC analysis at different scan rates [31]: ln
q T f2
! ¼−
Eg þ constant: RT f
ð3Þ
After obtaining the activation energy Eg, the calorimetric fragility could then be calculated from Eg and Tg as Eg : m¼ 2:303RT g
ð4Þ
Wang et al. [22,32] have deduced another equation to calculate calorimetric fragility in a more straightforward way: Q Ts log ¼ m−m f : Qs Tf
ð5Þ
Q is the DSC scan rate, Qs and Tsf correspond to a standard scan rate, and a standard fictive temperature Tf. The fictive temperatures for runs of different cooling rates are assessed by an enthalpy differencing procedure.
This is done by quantifying the enthalpy difference between the reheating scan of the glass previously cooled at the standard rate and the reheating scans of glass samples having different cooling rates. By incorporating Eq. (4) into Eq. (2), fragility can be obtained directly from the slope of the reduced cooling rate versus reciprocal reduced fictive temperature (or simply from the intercept) [22]. This is a modification of Moynihan's fictive temperature method, but the advantage is that the activation energy does not need to be obtained first to calculate fragility. Yue et al. have investigated the physical correlation between the cooling rate dependence of the calorimetric fictive temperature and the temperature dependence of the equilibrium liquid viscosity [30,33,34]. The difference between both dependences is described by the equation log(qc) = 11.35 − logη(Tf), This means that both dependences differ by a shifting factor of 11.35, but their slopes at Tg are the same. The above equation indicates that the dependence of the cooling rate on the fictive temperature can be well described using a suitable viscosity model. Over a rather small range of the cooling rates, e.g., between 2 and 40 K/min, the slope of log(1/qc) ~ Tg/Tf near Tg can be approximated as the calorimetric fragility (mDSC). We have compared the fragility indices calculated by the several abovementioned methods, which produce generally the same values with the error range of ~1. Actually Moynihan, Wang, and Yue's methods yield exactly the same value of fragility since they are all derived based on Arrhenian approximation, although they appear differently in equation form. Only Kissinger's equation gives a slightly different value, but the difference is no N 1. In this work, we use the slope of log(1/qc) ~ Tg/Tf near Tg as the mDSC since it is the most direct and simplest method without calculating activation energy first. Each of these calorimetric methods for determining fragility has an inherent assumption that the correlation of log(1/q c ) ~ 1/T f is based on Arrhenius behavior near Tg. However, the actual scaling is non-Arrhenius across the whole temperature range. Therefore the Arrhenian approximation in the calculation may lead to the deviation between calorimetric fragility and kinetic fragility. There have been a few studies attempting to compare the two kinds of fragilities, and some deviation is found in organic and metallic glass-forming liquids [22,23]. However, a comprehensive evaluation of the deviation between kinetic fragility and calorimetric fragility for oxide glasses is not yet available. In this paper, we have determined and collected both the kinetic and calorimetric fragility values of 20 boroaluminosilicate glasses, 6 sodalime borate glasses, two fiber glass compositions (Rockwool (RW) and glass wool (GW) compositions), and 6 vanadium tellurite glasses (Table 1). The kinetic fragility (mvis) is obtained by fitting the measured viscosity data to the Mauro-Yue-Ellison-Gupta-Allan (MYEGA) equation [6]. The mDSC is determined as the slope of log(1/qc) ~ Tg/Tf [30,33]. The calorimetric fragility of the studied glasses is compared with the kinetic fragility. If the quantitative link between kinetic and calorimetric fragilities is established, the viscosity of glass forming liquids could be determined solely by performing DSC measurements. Since viscosity measurements can be very challenging for some glass compositions, it is useful to supply an alternative way to determine viscosity. 2. Experimental procedure All glasses were prepared by using the melt-quenching method. The details of sample preparation can be found in [35–41]. In order to determine the liquid fragility index, viscosity measurements were performed. The low viscosities (approximately 100 − 103 Pa·s) were measured using a concentric cylinder viscometer, while the high viscosities (approximately 1010–1013 Pa·s) were determined by micro-penetration viscometry. For some of the compositions, we also performed beam bending and parallel plate compressing experiments. More information on the viscosity measurements is supplied in [36,38, 40]. The measured viscosity data are fitted to the MYEGA model to obtain Tg,vis and mvis [6].
Please cite this article as: Q. Zheng, et al., Reconciling calorimetric and kinetic fragilities of glass-forming liquids, J. Non-Cryst. Solids (2016), http:// dx.doi.org/10.1016/j.jnoncrysol.2016.11.014
Q. Zheng et al. / Journal of Non-Crystalline Solids xxx (2016) xxx–xxx
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Table 1 Glass transition temperature determined from viscosity (Tg,vis) and DSC measurements (Tg,DSC), liquid fragility index from viscosity (mvis) and DSC measurements (mDSC) of 20 boroaluminosilicate glasses, 6 soda-lime borate glasses, two fiber glass compositions, and 6 vanadium tellurite glasses. The deviation of the Tg,vis and Tg,DSC and that of the mvis and mDSC of the investigated glasses are also listed. The uncertainty of Tg,vis and Tg,DSC are ±2 and ±1 K, respectively [42]. (Tg,vis and mvis of the boroaluminosilicate glasses has been published in [36]. Tg,DSC, mDSC and mvis of the borate glasses has been published in Refs. [37,38]. Tg,vis, Tg,DSC, mDSC and mvis of the vanadium tellurite glasses has been collected from Refs. [39–41].)
Glass ID a
Al0* Al1* Al2.5* Al5* Al7.5* Al10* Al12.5* Al15* Al17.5* Al20* Al0 Al1 Al2.5 Al5 Al7.5 Al10 Al12.5 Al15 Al17.5 Al20 GW RW b NCBF5 NCBF10 NCBF15 NCBF20 NCBF25 NCBF30 c VT_10 VT_20 VT_30 VT_40 VT_50 VT_65
Tg,vis (K)
Tg,DSC (K)
(Tg,vis − Tg,DSC)/Tg,vis (%)
mvis (−)
mDSC (−)
(mvis − mDSC)/mvis (%)
807 807 816 825 846 850 864 905 922 936 809 814 822 837 851 871 887 899 956 966 831 946 721 755 766 765 751 733 552 538 524 511 502 489
807 804 816 823 839 849 856 891 914 919 803 807 816 832 842 861 866 877 936 951 815 937 693 756 771 768 756 740 559 544 530 518 507 500
0.0 0.4 0.0 0.1 0.8 0.1 0.9 1.4 0.9 1.8 0.7 0.9 0.7 0.6 1.1 1.1 2.4 2.4 2.1 1.4 1.9 1.0 3.9 −0.1 −0.7 −0.4 −0.7 −1.0 −1.3 −1.1 −1.1 −1.4 −1.0 −2.2
36 37 37 37 37 34 29 26 28 30 35 36 37 35 38 36 36 31 28 31 46 43 49 59 63 67 74 65 84 85 82 83 88 108
29 25 30 28 28 29 23 22 28 21 29 29 29 28 30 37 29 33 20 29 39 36 45 49 54 58 65 56 66 65 64 64 69 93
19 32 19 24 24 15 21 19 0 27 17 19 22 20 21 0 19 −6 29 6 15 16 8 17 14 13 12 14 21 24 22 22 20 14
a The glass ID is based on the concentration of Al2O3 in the Na2O-B2O3-Al2O3-SiO2 glasses. The names with * denote the Fe-containing glasses, while the glass ID without * corresponds to the Fe-free glasses [35]. b The numbers in the glass ID means the Na2O contents in Na2O-CaO-B2O3-Fe2O3 glasses [38]. c The numbers in the glass ID represent the V2O5 contents in V2O5-TeO2 glasses [40].
Fig. 1. The heat capacity (Cp) curves of one boroaluminosilicate glass (Al12.5) measured at different reheating rates (qh), which equal their prior cooling rates (qc). Tf is determined as the onset temperature of the glass transition peak (as shown in the inset). Tg,DSC is defined as the Tf which is measured with the reheating rate 10 K/min [42]. The numbers in the figure refer to the reheating rates in K/min.
reheating rates. The reheating rate (qh) always equals the prior cooling rate (qc); therefore, the heating rate dependence of the onset Tg may be taken as the cooling rate dependence of the Tf [30,42–43]. In this work, the log(1/qh) ~ Tg/Tf data is used to describe the log(1/qc) ~ Tg/Tf relationship. The shift of the Cp curve, and hence the change in the onset Tg with the reheating rate, is plotted in Fig. 1, showing that the value of Tf increases with increasing reheating rate. The inset shows how Tf is determined with the reheating scan of the Cp curve. With a reheating rate of 10 K/min, the Tf is defined as the standard glass transition temperature [42], i.e., Tg,DSC. The dependence of the reciprocal rate of cooling log(1/qc) on the Tgscaled fictive temperature (Tg/Tf) for Al12.5, Al12.5*, RW, and GW is displayed in Fig. 2. The Tf values obtained with different cooling rates are fitted linearly to get the calorimetric fragility (mDSC) [30]. In other words, the slope near Tg in the log(1/qc) ~ Tg/Tf plot is taken as the mDSC. As the slope become steeper from Al12.5* to GW, the fragility index increases from 23 to 39 as marked in the plot. The glass transition temperature determined from viscosity (Tg,vis) and DSC measurements (Tg,DSC), and the liquid fragility index from viscosity (mvis) and DSC measurements (mDSC) of the glasses under study are listed in Table 1. Here, Tg,vis and mvis are obtained by fitting the viscosity data to the MYEGA equation, and Tg,DSC and mDSC are calculated by using the heat capacity curves as demonstrated in Figs. 1 and 2. We
To obtain the Tf values, the heat capacities (Cp) of the glasses were measured using a differential scanning calorimeter (DSC 404 C, Netzsch). The measurements were conducted under a flow of argon at 40 ml/min. Before each DSC scan, a baseline was measured using two empty platinum crucibles to exclude the instrument uncertainties. To determine the specific heat capacity, a sapphire sample as reference was measured after the measurement of the baseline. The samples were subjected to several runs of DSC heating and cooling scans. The reheating and the prior cooling rates were always equal, and the reheating rate qh was varied between 2 K/min and 40 K/min. At each reheating scan, the calorimetric fictive temperature (Tf) is defined as the temperature at the intersection point between the extrapolated straight line of the glass Cp curve and the tangent line at the inflection point of the sharp rising Cp curve. The standard Tg obtained by DSC is denoted Tg,DSC, which corresponds to the Tf value measured at the standard reheating rate of 10 K/min after a prior cooling scan at the same rate [42]. 3. Results Fig. 1 shows the heat capacity (Cp) curves of one boroaluminosilicate glass (Al12.5) as a function of temperature obtained at different
Fig. 2. Dependence of the reciprocal rate of cooling log(1/qc) on the Tg-scaled fictive temperature (Tg/Tf) for Al12.5, Al12.5*, RW and GW. The experimental values are fitted linearly to obtain mDSC.
Please cite this article as: Q. Zheng, et al., Reconciling calorimetric and kinetic fragilities of glass-forming liquids, J. Non-Cryst. Solids (2016), http:// dx.doi.org/10.1016/j.jnoncrysol.2016.11.014
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have also reported the percentage difference between Tg,vis and Tg,DSC and between mvis and mDSC. Excellent consistency can be seen with regard to the glass transition temperature determined by viscosity and DSC measurements for the studied glasses, as depicted in Fig. 3. This equivalence between the Tg,DSC and the Tg,vis values agrees with the results of previous studies [7,42] and indicates that the calorimetric glass transition temperatures can be used as the kinetic glass transition temperature. The kinetic fragility values cover a wide range from 26 to 108. There is deviation between the calorimetric and kinetic fragilities for all the studied glasses, with the kinetic fragility generally higher than the corresponding calorimetric fragility. The offset between the kinetic and calorimetric fragilities becomes larger with an increase of the fragility magnitude, as depicted in Fig. 4. The boroaluminosilicate glasses have the lowest fragility, showing only a small offset between the calorimetric and kinetic fragilities. The fragilities of the two fiber glass compositions and the borate glasses are larger, and the offset between the calorimetric and kinetic fragilities increases. The vanadium tellurite glasses [40,41] possess the largest fragility, which exhibit the greatest offset among these systems. 4. Discussion In this work, we use the slope of log(1/qc) ~ Tg/Tf near Tg as the calorimetric fragility (mDSC) [24–26,30]. However, over a broad range of cooling rates, the log(1/qc) ~ 1/Tf relationship would deviate from the linear assumption made in this approach. The deviation between mvis and mDSC reveals the shortcomings of the Arrhenian approximation for modeling the real non-Arrhenius behavior [24–26,30]. Ideally, the calorimetric and kinetic fragilities would be identical [30], but the Arrhenius approximation when calculating mDSC leads to deviation, especially for high values of fragility. However, it is difficult to determine the log(1/qc) ~ 1/Tf relation over a broad enough range of cooling rates to capture the true non-Arrhenius scaling. High cooling rates are especially difficult to measure experimentally, and have been typically found using the molecular dynamics simulations. Given these challenges for determining the log(1/qc) ~ 1/Tf relationship over a large range of cooling rates to obtain the true non-Arrhenius scaling, here we propose an empirical correction to circumvent this problem and gain a numerically accurate value of fragility from standard DSC measurements. The empirical model is based on applying a correction factor to account for the non-Arrhenius scaling of the relaxation time. In the limit of mDSC → m0 (m0 is the fragility of a perfectly strong glass which
Fig. 4. Comparison between the calorimetric and kinetic fragility (mDSC vs. mvis) for the 20 boroaluminosilicate glasses [36], 6 soda-lime borate glasses [37,38], two fiber glass compositions, and 6 vanadium tellurite glasses [40,41]. The dashed line is added to show the result of the corrected model in Eq. (10).
exhibits Arrhenius behavior, m0 = 14.97), there is no error in mDSC, and therefore no correction is needed. As fragility becomes larger than m0 (i.e., for mDSC − m0 N 0), a correction factor is needed. Since the Arrhenius assumption becomes worse at higher values of fragility, a greater correction factor is needed. We can describe the correction factor using a power series expansion in mDSC − m0: mvis −m0 ¼ ðmDSC −m0 Þ½1 þ f ðmDSC −m0 Þ:
ð6Þ
In the above equation, the correction factor, f (mDSC − m0) is an unknown function of mDSC − m0. Invoking the power series expansion: mvis −m0 ¼ ðmDSC −m0 Þ½1 þ a0 þ a1 ðmDSC −m0 Þ
ð7Þ
þ a2 ðmDSC −m0 Þ2 þ a3 ðmDSC −m0 Þ3 þ :::: The goal of this expansion is to use the minimum number of terms necessary to capture the function accurately. We first try the simplest possible power series correction, truncating at the a0 (zeroth order) term. This gives the following equation: mvis −m0 ¼ ðmDSC −m0 Þð1 þ a0 Þ:
ð8Þ
a0 is optimized as a0 = 0.289 with the lowest error using the data in Table 1, which yield the following: mvis −m0 ¼ ðmDSC −m0 Þð1 þ 0:289Þ:
ð9Þ
or, equivalently: mvis ¼ 1:289ðmDSC −m0 Þ þ m0 :
Fig. 3. Comparison of the glass transition temperatures obtained from viscosity measurements (Tg,vis) and DSC measurements (Tg,DSC) for 20 boroaluminosilicate glasses [36], 6 soda-lime borate glasses [37,38], two fiber glass compositions, 6 vanadium tellurite glasses [39], and 10 aluminosilicate glasses [44].
ð10Þ
We use this approach to calculate the kinetic fragility (mcal) and compare with the kinetic fragility determined by viscosity data (mvis). As shown in Fig. 5, the corrected fragility agrees very well with the experimental fragility over the full range of values. Even the fragility of the highly non-Arrhenius vanadium tellurite glasses is accurately predicted, which is strong evidence that the model works over a very wide range of fragility values. Our approach enables the calculation of viscosity curves using DSC measurements alone. This is important because it is challenging to perform viscosity measurements on high temperature melts due to temperature constraints and potential for volatilization. Moreover, low temperature measurements are particularly time consuming [45]. In addition, some compositions can only be made into glass fibers or thin glass samples due to their poor glass-forming ability [46]. In this case, the sample size cannot meet the requirements of low temperature
Please cite this article as: Q. Zheng, et al., Reconciling calorimetric and kinetic fragilities of glass-forming liquids, J. Non-Cryst. Solids (2016), http:// dx.doi.org/10.1016/j.jnoncrysol.2016.11.014
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Fig. 5. Kinetic fragility calculated by the model (mcal), including the empirical correction factor, is compared with the kinetic fragility obtained by fitting viscosity data to the MYEGA equation (mvis).
viscosity measurements, e.g., some measurements need cylindrical geometries with certain diameter and height. More importantly, some glasses have strong tendency to crystallize as they are heated, which makes accurate viscosity measurements impossible. There exist several viscosity models to describe the temperature dependence of shear viscosity [6,34,47,48]. Three-parameter viscosity models can be expressed in terms of the aforementioned parameters: Tg(x), m(x), and η∞(x). The Mauro-Yue-Ellison-Gupta-Allan (MYEGA) model is the most accurate model thus far, and is physically derived in [6]: Tg Tg m log10 ηðT Þ ¼ log10 η∞ þ 12−log10 η∞ −1 exp −1 : ð11Þ 12−log10 η∞ T T
Once the three parameters are determined, the η-T relationship can be described. In our previous work [9], we have determined the value of η∞(x) to be a constant 10–2.93 Pa·s based on a comprehensive experimental and theoretical analysis. Our findings here imply that the calorimetric glass transition temperature can be used as the kinetic glass transition temperature. Moreover, our empirical model can be used to calculate mvis through mDSC across the full range of fragilities. Therefore we can obtain Tg and m from DSC measurements and combine the highT viscosity limit to obtain the viscosity curve across the full range of temperatures. As an example, we have applied this method to calculate the viscosity curve of a borate glass and compare with the experimental viscosity data. It is seen from Fig. 6 that the predicted viscosity agrees very well with the measured viscosity across the full temperature range. The agreement in the low-T range is especially good, where the two viscosity curves nearly overlap each other. Our result implies that this approach could be applied as a standard method for determining the temperature dependence of viscosity using DSC measurements without performing viscosity measurements. The DSC method can be simpler and more effective compared to direct viscosity measurements, since only one small sample (e.g., 20–50 mg) is needed to conduct the DSC measurements. Moreover, the Tf values can be acquired by performing several DSC up and down scans [49]. The Tg,DSC is directly obtained from the DSC reheating scan at 10 K/min equal to the prior cooling rate, while mDSC can be calculated easily based on the Tf values measured at different reheating rates. 5. Summary We have determined the glass transition temperature and fragility from viscosity and DSC measurements of 20 boroaluminosilicate glasses, 6 soda-lime borate glasses, two fiber glass compositions, and 6 vanadium tellurite glasses. The glass transition temperatures obtained
5
Fig. 6. The logarithmic viscosity (log10 η) as a function of the Tg,DSC scaled inverse temperature (Tg,DSC/T) for one borate glass composition. The red dots are the measured viscosity data, and the red line represents the MYEGA fitting of the viscosity data. The blue line is the viscosity data calculated from this work. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
from viscosity measurements (Tg,vis) are consistent with those obtained from DSC measurements (Tg,DSC); thus Tg,DSC can be used interchangeably with Tg,vis. However, there is a deviation between the calorimetric and kinetic fragilities for all the studied glasses. The kinetic fragility is generally higher than the corresponding calorimetric fragility, and larger deviation is associated with more fragile glass compositions. The Arrhenian approximation of the log(1/qc) ~ Tg/Tf in the temperature range near Tg is the source of the deviation between kinetic and calorimetric fragilities. We have developed an approach to predict kinetic fragility (mvis) using calorimetric fragility (mDSC) across a wide range of fragilities. The predicted fragility values agree well with the experimental fragility. By combining the high-T viscosity limit (10–2.93 Pa·s) with the MYEGA equation, the entire viscosity curve of oxide glasses can be well predicted just using DSC measurements without performing viscosity measurements. Acknowledgements The authors thank Mette Solvang for the support in the data acquirement of the two fiber glass compositions and Jonas Kjeldsen for data supplement of the vanadium tellurite glasses. References [1] A.K. Varshneya, Fundamentals of Inorganic Glasses, Academic, Boston, MA, U.S.A., 1994 [2] P.K. Gupta, J.C. Mauro, Composition dependence of glass transition temperature and fragility. I. A topological model incorporating temperature-dependent constraints, J. Chem. Phys. 130 (2009) 094503. [3] J.C. Mauro, D.C. Allan, M. Potuzak, Nonequilibrium viscosity of glass, Phys. Rev. B 80 (2009) 094204. [4] C.A. Angell, Spectroscopy, simulation, and the medium range order problem in glass, J. Non-Cryst. Solids 73 (1985) 1. [5] C.A. Angell, K.L. Ngai, G.B. McKenna, P.F. McMillan, S.W. Martin, Relaxation in glassforming liquids and amorphous solids, J. Appl. Phys. 88 (2000) 3113. [6] J.C. Mauro, Y.Z. Yue, A.J. Ellison, P.K. Gupta, D.C. Allan, Viscosity of glass-forming liquids, Proc. Natl. Acad. Sci. U. S. A. 106 (2009) 19780. [7] Y.Z. Yue, The iso-structural viscosity, configurational entropy and fragility of oxide liquids, J. Non-Cryst. Solids 355 (2009) 737. [8] C.A. Angell, Formation of glasses from liquids and biopolymers, Science 267 (1995) 1924. [9] Q.J. Zheng, J.C. Mauro, A.J. Ellison, M. Potuzak, Y.Z. Yue, Universality of the high-temperature viscosity limit of silicate liquids, Phys. Rev. B 83 (2011) 212202. [10] C.A. Angell, B.E. Richards, V. Velikov, Simple glass-forming liquids: their definition, fragilities, and landscape excitation profiles, J. Phys. Condens. Matter 11 (1999) A75. [11] C.A. Angell, Relaxation in liquids, polymers and plastic crystals - strong/fragile patterns and problems, J. Non-Cryst. Solids 131 (1991) 13. [12] C.A. Angell, J.L. Green, K. Ito, P. Lucas, B.E. Richards, Glassformer fragilities and landscape excitation profiles by simple calorimetric and theoretical methods, J. Therm. Anal. Calorim. 57 (1999) 717.
Please cite this article as: Q. Zheng, et al., Reconciling calorimetric and kinetic fragilities of glass-forming liquids, J. Non-Cryst. Solids (2016), http:// dx.doi.org/10.1016/j.jnoncrysol.2016.11.014
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[13] J.C. Mauro, P.K. Gupta, R.J. Loucks, Composition dependence of glass transition temperature and fragility. II. A topological model of alkali borate liquids, J. Chem. Phys. 130 (2009) 234503. [14] C. Hermansen, X.J. Guo, R.E. Youngman, J.C. Mauro, M.M. Smedskjaer, Y.Z. Yue, Structure-topology-property correlations of sodium phosphosilicate glasses, J. Chem. Phys. 143 (2015) 064510. [15] G.L. Paraschiv, F. Muñoz, L.R. Jensen, Y.Z. Yue, M.M. Smedskjaer, Impact of nitridation of metaphosphate glasses on liquid fragility, J. Non-Cryst. Solids 441 (2016) 22. [16] L.M. Martinez, C.A. Angell, A thermodynamic connection to the fragility of glassforming liquids, Nature 410 (2001) 663. [17] J.C. Mauro, R.J. Loucks, Impact of fragility on enthalpy relaxation in glass, Phys. Rev. E 78 (2008) 021502. [18] K. Ito, C.T. Moynihan, C.A. Angell, Thermodynamic determination of fragility in liquids and a fragile-to-strong liquid transition in water, Nature 398 (1999) 492. [19] G.B. McKenna, A brief discussion: thermodynamic and dynamic fragilities, non-divergent dynamics and the Prigogine–Defay ratio, J. Non-Cryst. Solids 355 (2009) 663. [20] D. Cangialosi, A. Alegría, J. Colmenero, A thermodynamic approach to the fragility of glass-forming polymers, J. Chem. Phys. 124 (2006) 024906. [21] H. Ma, H.J. Fecht, Thermodynamic and kinetic fragilities of Mg-based bulk metallic glass-forming liquids, J. Mater. Res. 23 (2008) 10. [22] L.M. Wang, V. Velikov, C.A. Angell, Direct determination of kinetic fragility indices of glassforming liquids by differential scanning calorimetry: kinetic versus thermodynamic fragilities, J. Chem. Phys. 117 (2002) 10184. [23] Z.M. Chen, Z.J. Li, Y.Q. Zhang, R.P. Liu, Y.J. Tian, L.M. Wang, Calorimetric determination of fragility in glass forming liquids: Tf vs. Tg-onset methods, Eur. Phys. J. E 37 (2014) 52. [24] T. Moynihan, A.J. Easteal, M.A. DeBolt, J. Tucker, Dependence of the fictive temperature of glass on cooling rate, J. Am. Chem. Soc. 59 (1976) 12. [25] C.T. Moynihan, Structural relaxation and the glass transition, Rev. Mineral. 32 (1995) 1. [26] C.T. Moynihan, S.K. Lee, M. Tatsumisago, T. Minami, Estimation of activation energies for structural relaxation and viscous flow from DTA and DSC experiments, Thermochim. Acta 280 (1996) 153. [27] A.Q. Tool, Relation between inelastic deformability and thermal expansion of glass in its annealing range, J. Am. Ceram. Soc. 29 (1946) 240. [28] J.C. Mauro, R.J. Loucks, P.K. Gupta, Fictive temperature and the glassy state, J. Am. Ceram. Soc. 92 (2009) 75. [29] X.J. Guo, M. Potuzak, J.C. Mauro, D.C. Allan, T.J. Kiczenski, Y.Z. Yue, Unified approach for determining the enthalpic fictive temperature of glasses with arbitrary thermal history, J. Non-Cryst. Solids 357 (2011) 3230. [30] Y.Z. Yue, R. von der Ohe, S.L. Jensen, Fictive temperature, cooling rate, and viscosity of glasses, J. Chem. Phys. 120 (17) (2004); Y.Z. Yue, R. von der Ohe, S.L. Jensen, Erratum: “Fictive temperature, cooling rate, and viscosity of glasses” J. Chem. Phys. 120, 8053 2004, J. Chem. Phys. 121 (2004) 11508.
[31] H.E. Kissinger, Reaction kinetics in differential thermal analysis, Anal. Chem. 29 (1957) 1702. [32] V. Velikov, S. Borick, C.A. Angell, The glass transition of water, based on hyperquenching experiments, Science 294 (2001) 2335. [33] Y.Z. Yue, J.D. Christiansen, S.L. Jensen, Determination of the fictive temperature for a hyperquenched glass, Chem. Phys. Lett. 357 (2002) 20. [34] I. Avramov, A. Milchev, Effect of disorder on diffusion and viscosity in condensed systems, J. Non-Cryst. Solids 104 (1988) 253. [35] Q.J. Zheng, R.E. Youngman, C.L. Hogue, J.C. Mauro, M. Potuzak, M.M. Smedskjaer, Y.Z. Yue, Structure of boroaluminosilicate glasses: impact of [Al2O3]/[SiO2] ratio on the structural role of sodium, Phys. Rev. B 86 (2012) 054203. [36] Q.J. Zheng, M. Potuzak, J.C. Mauro, M.M. Smedskjaer, R.E. Youngman, Y.Z. Yue, Composition-structure-property relationships in boroaluminosilicate glasses, J. NonCryst. Solids 358 (2012) 993. [37] M.M. Smedskjaer, J.C. Mauro, S. Sen, Y.Z. Yue, Quantitative design of glassy materials using temperature-dependent constraint theory, Chem. Mater. 22 (2010) 5358. [38] Q.J. Zheng, J.C. Mauro, M.M. Smedskjaer, R.E. Youngman, M. Potuzak, Y.Z. Yue, Glassforming ability of soda lime borate liquids, J. Non-Cryst. Solids 358 (2012) 658. [39] J. Kjeldsen and Y. Z. Yue (Unpublished results). [40] J. Kjeldsen, A.C.M. Rodrigues, S. Mossin, Y.Z. Yue, Critical V2O5/TeO2 ratio inducing abrupt property changes in vanadium tellurite glasses, J. Phys. Chem. B 118 (2014) 14942. [41] J. Kjeldsen, Mechanical and Dynamic Properties of Mixed Modifier Silicate Glasses and Vanadium-Tellurite Glasses(Ph.D. Thesis) Aalborg University, Aalborg, Denmark, 2014. [42] Y.Z. Yue, Characteristic temperatures of enthalpy relaxation in glass, J. Non-Cryst. Solids 354 (2008) 1112. [43] P. Ducroux, S.M. Rekhson, F.L. Merat, Structural relaxation in thermorheologically complex materials, J. Non-Cryst. Solids 172 (1994) 541. [44] Q.J. Zheng, M.M. Smedskjaer, R.E. Youngman, M. Potuzak, J.C. Mauro, Y.Z. Yue, Influence of aluminum speciation on the stability of aluminosilicate glasses against crystallization, Appl. Phys. Lett. 101 (2012) 041906. [45] J.E. Shelby, Introduction to Glass Science and Technology, second ed. The Royal Society of Chemistry, Cambridge, U.K., 2005 [46] J.W.E. Drewitt, L. Hennet, A. Zeidler, S. Jahn, P.S. Salmon, D.R. Neuville, H.E. Fischer, Structural transformations on vitrification in the fragile glass-forming system CaAl2O4, Phys. Rev. Lett. 109 (2012) 235501. [47] G. Adam, J.H. Gibbs, On the temperature dependence of cooperative relaxation properties in glass-forming liquids, J. Chem. Phys. 43 (1965) 139. [48] F. Stickel, E.W. Fischer, R. Richert, Dynamics of glass-forming liquids. I. Temperaturederivative analysis of dielectric relaxation data, J. Chem. Phys. 102 (1995) 6251. [49] T.A. Waniuk, R. Busch, A. Masuhr, W.L. Johnson, Equilibrium viscosity of the Zr41.2Ti13.8Cu12.5Ni10Be22.5 bulk metallic glass-forming liquid and viscous flow during relaxation, phase separation, and primary crystallization, Acta Mater. 46 (1998) 5229.
Please cite this article as: Q. Zheng, et al., Reconciling calorimetric and kinetic fragilities of glass-forming liquids, J. Non-Cryst. Solids (2016), http:// dx.doi.org/10.1016/j.jnoncrysol.2016.11.014
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Reconciling calorimetric and kinetic fragilities of glass-forming liquids Qiuju Zheng a,c, John C. Mauro a,b,⁎, Yuanzheng Yue a,c,⁎ a b c
Shandong Key Laboratory of Glass and Ceramics, Qilu University of Technology, 250353 Jinan, China Science and Technology Division, Corning Incorporated, Corning, NY 14831, USA Department of Chemistry and Bioscience, Aalborg University, DK-9000 Aalborg, Denmark
a r t i c l e
i n f o
Article history: Received 7 October 2016 Received in revised form 9 November 2016 Accepted 11 November 2016 Available online xxxx Keywords: Viscosity Fragility Differential scanning calorimetry (DSC)
a b s t r a c t The liquid fragility index (mvis) describes the rate of viscosity change of a glass-forming liquid with temperature at the glass transition temperature (Tg), which is very important for understanding liquid dynamics and the glass transition itself. Fragility can be directly determined using viscosity measurements. However, due to various technical complications with determining viscosity, alternative methods to obtain fragility are needed. One simple method is based on measurement of the calorimetric fragility index (mDSC), i.e., the changing rate of fictive temperature (Tf) with heating (cooling) rate in a small Tf range around Tg. The crucial question is how mDSC is quantitatively related to mvis. Here, we establish this relation by performing both dynamic and calorimetric measurements on some selected glass compositions covering a wide range of liquid fragilities. The results show that mDSC deviates systematically from mvis. The deviation is attributed to the Arrhenian approximation of the log(1/qc) ~ Tg/Tf relationship in the glass transition range. We have developed an empirical model to quantify the deviation, by which mvis can be well predicted from mDSC across a large range of fragilities. Combined with the high-T viscosity limit (10–2.93 Pa·s), we are able to obtain the entire viscosity curve of a glass-forming liquid by only performing DSC measurements. © 2016 Published by Elsevier B.V.
1. Introduction The shear viscosity of glass-forming liquids is of great importance in all stages of industrial glass production [1–3]. Since viscosity is very sensitive to temperature and composition, we need to have accurate knowledge of the scaling of viscosity with both of these parameters. Viscosity is also critical for understanding the glass transition and the relaxation characteristics of liquids and their corresponding glasses. In the well-known Angell plot [4,5], the logarithm of viscosity, log10 η, is plotted as a function of the Tg-scaled inverse temperature, Tg/T, where T is absolute temperature. With this scaling, Angell was able to compare the viscous flow behavior of all glass-forming liquids in a single universal plot. According to the Angell plot, there are three important parameters [6]: (i) the glass transition temperature, Tg(x); (ii) the fragility, m(x); and (iii) the extrapolated infinite temperature viscosity, η∞(x). For any composition x, the glass transition temperature is defined as the temperature at which the shear viscosity is equal to 1012 Pa·s [7],
⁎ Corresponding authors. E-mail addresses: [email protected] (J.C. Mauro), [email protected] (Y. Yue).
i.e., η(Tg(x),x) = 1012 Pa·s. Fragility [8] is defined as the slope of the logη versus Tg/T at Tg: mðxÞ ¼
∂log10 ηðT; xÞ ∂ T g ðxÞ=T
:
ð1Þ
T¼T g ðxÞ
The fragility index describes the rate of change in the liquid dynamics upon cooling through the glass transition. Liquids can be classified as either “strong” or “fragile” depending on whether they exhibit an Arrhenius or super-Arrhenius scaling of viscosity with temperature, respectively. While the fragility index itself is a first-derivative property of the viscosity curve, the degree of non-Arrhenius scaling reflects the second derivative of the viscosity curve with respect to inverse temperature. Following Angell [4,8], this non-Arrhenius scaling of liquid viscosity can be quantified directly through the fragility index (mvis in this work) with the assumption of a universal high temperature limit of viscosity, i.e., η∞(x) = η∞. In our previous work [9], we have analyzed the viscosity data of 946 silicate liquids and other 31 non-silicate liquids; the results imply that the silicate liquids have a universal high temperature viscosity limit of around 10–2.93 Pa·s. Thus, we have validated Angell's assumption, which enables this direct connection between first- and second-derivative properties.
http://dx.doi.org/10.1016/j.jnoncrysol.2016.11.014 0022-3093/© 2016 Published by Elsevier B.V.
Please cite this article as: Q. Zheng, et al., Reconciling calorimetric and kinetic fragilities of glass-forming liquids, J. Non-Cryst. Solids (2016), http:// dx.doi.org/10.1016/j.jnoncrysol.2016.11.014
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Given the great importance of the fragility parameter, it has been the subject of extensive interest in both experimental and theoretical studies for many years [10–15]. While fragility is a kinetic property, it is closely correlated with various thermodynamic quantities of glassforming liquids. For example, the change in heat capacity at the glass transition, width of the glass transition range, and configurational entropy loss at the glass transition all exhibit linear correlations with kinetic fragility [16–23]. An accurate determination of fragility is thus of great significance for understanding liquid physics near glass transition and for understanding the glass transition itself. Following the definition of fragility in Eq. (1), it should be directly determined using viscosity measurements. However, viscosity measurements are difficult for glass-forming systems, especially those systems with strong crystallization tendency and high liquidus temperatures. Liquid viscosity varies by over twelve orders of magnitude and requires specialized equipment in different viscosity and temperature regimes. Crystallization and volatilization of the melts can hinder high temperature viscosity measurements, while the specific sample size/shape demands and long measurement time impede the low temperature viscosity experiments. Therefore, alternative methods are needed to give an indirect quantification of fragility. Various methods have been proposed to calculate fragility using differential scanning calorimetry (DSC). Moynihan and his co-workers found that the activation energy for structural relaxation determined by DSC is an accurate estimate of the activation energy for shear viscosity [24–27]. The activation energy for structural relaxation in the glass transition region can be determined from the cooling rate (qc) dependence of the fictive temperature Tf measured using DSC. E d lnqc ¼− g: R d 1=T f
ð2Þ
where Eg is the activation energy for equilibrium viscous flow in the glass transition region and R is the ideal gas constant. The fictive temperature, Tf, is defined here as the temperature at which the configurational enthalpy of the glass equals that of the corresponding liquid state [28]. This is obtained by an enthalpy-matching integral method using the heat capacity (Cp) curve during reheating [29]. It is found that when the prior cooling (qc) and reheating rates (qh) are the same, the Tf obtained using the integration method from the Cp reheating curve is very close to the onset glass transition temperature, Tg,onset [24–26, 30]. For all the DSC measurements performed in this work, the reheating rates are equal to the preceding cooling rates. Therefore the Tg,onset has been used as the fictive temperature Tf for the studied glasses in this paper. Kissinger derived another equation to calculate activation energy for glass transition using DSC analysis at different scan rates [31]: ln
q T f2
! ¼−
Eg þ constant: RT f
ð3Þ
After obtaining the activation energy Eg, the calorimetric fragility could then be calculated from Eg and Tg as Eg : m¼ 2:303RT g
ð4Þ
Wang et al. [22,32] have deduced another equation to calculate calorimetric fragility in a more straightforward way: Q Ts log ¼ m−m f : Qs Tf
ð5Þ
Q is the DSC scan rate, Qs and Tsf correspond to a standard scan rate, and a standard fictive temperature Tf. The fictive temperatures for runs of different cooling rates are assessed by an enthalpy differencing procedure.
This is done by quantifying the enthalpy difference between the reheating scan of the glass previously cooled at the standard rate and the reheating scans of glass samples having different cooling rates. By incorporating Eq. (4) into Eq. (2), fragility can be obtained directly from the slope of the reduced cooling rate versus reciprocal reduced fictive temperature (or simply from the intercept) [22]. This is a modification of Moynihan's fictive temperature method, but the advantage is that the activation energy does not need to be obtained first to calculate fragility. Yue et al. have investigated the physical correlation between the cooling rate dependence of the calorimetric fictive temperature and the temperature dependence of the equilibrium liquid viscosity [30,33,34]. The difference between both dependences is described by the equation log(qc) = 11.35 − logη(Tf), This means that both dependences differ by a shifting factor of 11.35, but their slopes at Tg are the same. The above equation indicates that the dependence of the cooling rate on the fictive temperature can be well described using a suitable viscosity model. Over a rather small range of the cooling rates, e.g., between 2 and 40 K/min, the slope of log(1/qc) ~ Tg/Tf near Tg can be approximated as the calorimetric fragility (mDSC). We have compared the fragility indices calculated by the several abovementioned methods, which produce generally the same values with the error range of ~1. Actually Moynihan, Wang, and Yue's methods yield exactly the same value of fragility since they are all derived based on Arrhenian approximation, although they appear differently in equation form. Only Kissinger's equation gives a slightly different value, but the difference is no N 1. In this work, we use the slope of log(1/qc) ~ Tg/Tf near Tg as the mDSC since it is the most direct and simplest method without calculating activation energy first. Each of these calorimetric methods for determining fragility has an inherent assumption that the correlation of log(1/q c ) ~ 1/T f is based on Arrhenius behavior near Tg. However, the actual scaling is non-Arrhenius across the whole temperature range. Therefore the Arrhenian approximation in the calculation may lead to the deviation between calorimetric fragility and kinetic fragility. There have been a few studies attempting to compare the two kinds of fragilities, and some deviation is found in organic and metallic glass-forming liquids [22,23]. However, a comprehensive evaluation of the deviation between kinetic fragility and calorimetric fragility for oxide glasses is not yet available. In this paper, we have determined and collected both the kinetic and calorimetric fragility values of 20 boroaluminosilicate glasses, 6 sodalime borate glasses, two fiber glass compositions (Rockwool (RW) and glass wool (GW) compositions), and 6 vanadium tellurite glasses (Table 1). The kinetic fragility (mvis) is obtained by fitting the measured viscosity data to the Mauro-Yue-Ellison-Gupta-Allan (MYEGA) equation [6]. The mDSC is determined as the slope of log(1/qc) ~ Tg/Tf [30,33]. The calorimetric fragility of the studied glasses is compared with the kinetic fragility. If the quantitative link between kinetic and calorimetric fragilities is established, the viscosity of glass forming liquids could be determined solely by performing DSC measurements. Since viscosity measurements can be very challenging for some glass compositions, it is useful to supply an alternative way to determine viscosity. 2. Experimental procedure All glasses were prepared by using the melt-quenching method. The details of sample preparation can be found in [35–41]. In order to determine the liquid fragility index, viscosity measurements were performed. The low viscosities (approximately 100 − 103 Pa·s) were measured using a concentric cylinder viscometer, while the high viscosities (approximately 1010–1013 Pa·s) were determined by micro-penetration viscometry. For some of the compositions, we also performed beam bending and parallel plate compressing experiments. More information on the viscosity measurements is supplied in [36,38, 40]. The measured viscosity data are fitted to the MYEGA model to obtain Tg,vis and mvis [6].
Please cite this article as: Q. Zheng, et al., Reconciling calorimetric and kinetic fragilities of glass-forming liquids, J. Non-Cryst. Solids (2016), http:// dx.doi.org/10.1016/j.jnoncrysol.2016.11.014
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Table 1 Glass transition temperature determined from viscosity (Tg,vis) and DSC measurements (Tg,DSC), liquid fragility index from viscosity (mvis) and DSC measurements (mDSC) of 20 boroaluminosilicate glasses, 6 soda-lime borate glasses, two fiber glass compositions, and 6 vanadium tellurite glasses. The deviation of the Tg,vis and Tg,DSC and that of the mvis and mDSC of the investigated glasses are also listed. The uncertainty of Tg,vis and Tg,DSC are ±2 and ±1 K, respectively [42]. (Tg,vis and mvis of the boroaluminosilicate glasses has been published in [36]. Tg,DSC, mDSC and mvis of the borate glasses has been published in Refs. [37,38]. Tg,vis, Tg,DSC, mDSC and mvis of the vanadium tellurite glasses has been collected from Refs. [39–41].)
Glass ID a
Al0* Al1* Al2.5* Al5* Al7.5* Al10* Al12.5* Al15* Al17.5* Al20* Al0 Al1 Al2.5 Al5 Al7.5 Al10 Al12.5 Al15 Al17.5 Al20 GW RW b NCBF5 NCBF10 NCBF15 NCBF20 NCBF25 NCBF30 c VT_10 VT_20 VT_30 VT_40 VT_50 VT_65
Tg,vis (K)
Tg,DSC (K)
(Tg,vis − Tg,DSC)/Tg,vis (%)
mvis (−)
mDSC (−)
(mvis − mDSC)/mvis (%)
807 807 816 825 846 850 864 905 922 936 809 814 822 837 851 871 887 899 956 966 831 946 721 755 766 765 751 733 552 538 524 511 502 489
807 804 816 823 839 849 856 891 914 919 803 807 816 832 842 861 866 877 936 951 815 937 693 756 771 768 756 740 559 544 530 518 507 500
0.0 0.4 0.0 0.1 0.8 0.1 0.9 1.4 0.9 1.8 0.7 0.9 0.7 0.6 1.1 1.1 2.4 2.4 2.1 1.4 1.9 1.0 3.9 −0.1 −0.7 −0.4 −0.7 −1.0 −1.3 −1.1 −1.1 −1.4 −1.0 −2.2
36 37 37 37 37 34 29 26 28 30 35 36 37 35 38 36 36 31 28 31 46 43 49 59 63 67 74 65 84 85 82 83 88 108
29 25 30 28 28 29 23 22 28 21 29 29 29 28 30 37 29 33 20 29 39 36 45 49 54 58 65 56 66 65 64 64 69 93
19 32 19 24 24 15 21 19 0 27 17 19 22 20 21 0 19 −6 29 6 15 16 8 17 14 13 12 14 21 24 22 22 20 14
a The glass ID is based on the concentration of Al2O3 in the Na2O-B2O3-Al2O3-SiO2 glasses. The names with * denote the Fe-containing glasses, while the glass ID without * corresponds to the Fe-free glasses [35]. b The numbers in the glass ID means the Na2O contents in Na2O-CaO-B2O3-Fe2O3 glasses [38]. c The numbers in the glass ID represent the V2O5 contents in V2O5-TeO2 glasses [40].
Fig. 1. The heat capacity (Cp) curves of one boroaluminosilicate glass (Al12.5) measured at different reheating rates (qh), which equal their prior cooling rates (qc). Tf is determined as the onset temperature of the glass transition peak (as shown in the inset). Tg,DSC is defined as the Tf which is measured with the reheating rate 10 K/min [42]. The numbers in the figure refer to the reheating rates in K/min.
reheating rates. The reheating rate (qh) always equals the prior cooling rate (qc); therefore, the heating rate dependence of the onset Tg may be taken as the cooling rate dependence of the Tf [30,42–43]. In this work, the log(1/qh) ~ Tg/Tf data is used to describe the log(1/qc) ~ Tg/Tf relationship. The shift of the Cp curve, and hence the change in the onset Tg with the reheating rate, is plotted in Fig. 1, showing that the value of Tf increases with increasing reheating rate. The inset shows how Tf is determined with the reheating scan of the Cp curve. With a reheating rate of 10 K/min, the Tf is defined as the standard glass transition temperature [42], i.e., Tg,DSC. The dependence of the reciprocal rate of cooling log(1/qc) on the Tgscaled fictive temperature (Tg/Tf) for Al12.5, Al12.5*, RW, and GW is displayed in Fig. 2. The Tf values obtained with different cooling rates are fitted linearly to get the calorimetric fragility (mDSC) [30]. In other words, the slope near Tg in the log(1/qc) ~ Tg/Tf plot is taken as the mDSC. As the slope become steeper from Al12.5* to GW, the fragility index increases from 23 to 39 as marked in the plot. The glass transition temperature determined from viscosity (Tg,vis) and DSC measurements (Tg,DSC), and the liquid fragility index from viscosity (mvis) and DSC measurements (mDSC) of the glasses under study are listed in Table 1. Here, Tg,vis and mvis are obtained by fitting the viscosity data to the MYEGA equation, and Tg,DSC and mDSC are calculated by using the heat capacity curves as demonstrated in Figs. 1 and 2. We
To obtain the Tf values, the heat capacities (Cp) of the glasses were measured using a differential scanning calorimeter (DSC 404 C, Netzsch). The measurements were conducted under a flow of argon at 40 ml/min. Before each DSC scan, a baseline was measured using two empty platinum crucibles to exclude the instrument uncertainties. To determine the specific heat capacity, a sapphire sample as reference was measured after the measurement of the baseline. The samples were subjected to several runs of DSC heating and cooling scans. The reheating and the prior cooling rates were always equal, and the reheating rate qh was varied between 2 K/min and 40 K/min. At each reheating scan, the calorimetric fictive temperature (Tf) is defined as the temperature at the intersection point between the extrapolated straight line of the glass Cp curve and the tangent line at the inflection point of the sharp rising Cp curve. The standard Tg obtained by DSC is denoted Tg,DSC, which corresponds to the Tf value measured at the standard reheating rate of 10 K/min after a prior cooling scan at the same rate [42]. 3. Results Fig. 1 shows the heat capacity (Cp) curves of one boroaluminosilicate glass (Al12.5) as a function of temperature obtained at different
Fig. 2. Dependence of the reciprocal rate of cooling log(1/qc) on the Tg-scaled fictive temperature (Tg/Tf) for Al12.5, Al12.5*, RW and GW. The experimental values are fitted linearly to obtain mDSC.
Please cite this article as: Q. Zheng, et al., Reconciling calorimetric and kinetic fragilities of glass-forming liquids, J. Non-Cryst. Solids (2016), http:// dx.doi.org/10.1016/j.jnoncrysol.2016.11.014
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have also reported the percentage difference between Tg,vis and Tg,DSC and between mvis and mDSC. Excellent consistency can be seen with regard to the glass transition temperature determined by viscosity and DSC measurements for the studied glasses, as depicted in Fig. 3. This equivalence between the Tg,DSC and the Tg,vis values agrees with the results of previous studies [7,42] and indicates that the calorimetric glass transition temperatures can be used as the kinetic glass transition temperature. The kinetic fragility values cover a wide range from 26 to 108. There is deviation between the calorimetric and kinetic fragilities for all the studied glasses, with the kinetic fragility generally higher than the corresponding calorimetric fragility. The offset between the kinetic and calorimetric fragilities becomes larger with an increase of the fragility magnitude, as depicted in Fig. 4. The boroaluminosilicate glasses have the lowest fragility, showing only a small offset between the calorimetric and kinetic fragilities. The fragilities of the two fiber glass compositions and the borate glasses are larger, and the offset between the calorimetric and kinetic fragilities increases. The vanadium tellurite glasses [40,41] possess the largest fragility, which exhibit the greatest offset among these systems. 4. Discussion In this work, we use the slope of log(1/qc) ~ Tg/Tf near Tg as the calorimetric fragility (mDSC) [24–26,30]. However, over a broad range of cooling rates, the log(1/qc) ~ 1/Tf relationship would deviate from the linear assumption made in this approach. The deviation between mvis and mDSC reveals the shortcomings of the Arrhenian approximation for modeling the real non-Arrhenius behavior [24–26,30]. Ideally, the calorimetric and kinetic fragilities would be identical [30], but the Arrhenius approximation when calculating mDSC leads to deviation, especially for high values of fragility. However, it is difficult to determine the log(1/qc) ~ 1/Tf relation over a broad enough range of cooling rates to capture the true non-Arrhenius scaling. High cooling rates are especially difficult to measure experimentally, and have been typically found using the molecular dynamics simulations. Given these challenges for determining the log(1/qc) ~ 1/Tf relationship over a large range of cooling rates to obtain the true non-Arrhenius scaling, here we propose an empirical correction to circumvent this problem and gain a numerically accurate value of fragility from standard DSC measurements. The empirical model is based on applying a correction factor to account for the non-Arrhenius scaling of the relaxation time. In the limit of mDSC → m0 (m0 is the fragility of a perfectly strong glass which
Fig. 4. Comparison between the calorimetric and kinetic fragility (mDSC vs. mvis) for the 20 boroaluminosilicate glasses [36], 6 soda-lime borate glasses [37,38], two fiber glass compositions, and 6 vanadium tellurite glasses [40,41]. The dashed line is added to show the result of the corrected model in Eq. (10).
exhibits Arrhenius behavior, m0 = 14.97), there is no error in mDSC, and therefore no correction is needed. As fragility becomes larger than m0 (i.e., for mDSC − m0 N 0), a correction factor is needed. Since the Arrhenius assumption becomes worse at higher values of fragility, a greater correction factor is needed. We can describe the correction factor using a power series expansion in mDSC − m0: mvis −m0 ¼ ðmDSC −m0 Þ½1 þ f ðmDSC −m0 Þ:
ð6Þ
In the above equation, the correction factor, f (mDSC − m0) is an unknown function of mDSC − m0. Invoking the power series expansion: mvis −m0 ¼ ðmDSC −m0 Þ½1 þ a0 þ a1 ðmDSC −m0 Þ
ð7Þ
þ a2 ðmDSC −m0 Þ2 þ a3 ðmDSC −m0 Þ3 þ :::: The goal of this expansion is to use the minimum number of terms necessary to capture the function accurately. We first try the simplest possible power series correction, truncating at the a0 (zeroth order) term. This gives the following equation: mvis −m0 ¼ ðmDSC −m0 Þð1 þ a0 Þ:
ð8Þ
a0 is optimized as a0 = 0.289 with the lowest error using the data in Table 1, which yield the following: mvis −m0 ¼ ðmDSC −m0 Þð1 þ 0:289Þ:
ð9Þ
or, equivalently: mvis ¼ 1:289ðmDSC −m0 Þ þ m0 :
Fig. 3. Comparison of the glass transition temperatures obtained from viscosity measurements (Tg,vis) and DSC measurements (Tg,DSC) for 20 boroaluminosilicate glasses [36], 6 soda-lime borate glasses [37,38], two fiber glass compositions, 6 vanadium tellurite glasses [39], and 10 aluminosilicate glasses [44].
ð10Þ
We use this approach to calculate the kinetic fragility (mcal) and compare with the kinetic fragility determined by viscosity data (mvis). As shown in Fig. 5, the corrected fragility agrees very well with the experimental fragility over the full range of values. Even the fragility of the highly non-Arrhenius vanadium tellurite glasses is accurately predicted, which is strong evidence that the model works over a very wide range of fragility values. Our approach enables the calculation of viscosity curves using DSC measurements alone. This is important because it is challenging to perform viscosity measurements on high temperature melts due to temperature constraints and potential for volatilization. Moreover, low temperature measurements are particularly time consuming [45]. In addition, some compositions can only be made into glass fibers or thin glass samples due to their poor glass-forming ability [46]. In this case, the sample size cannot meet the requirements of low temperature
Please cite this article as: Q. Zheng, et al., Reconciling calorimetric and kinetic fragilities of glass-forming liquids, J. Non-Cryst. Solids (2016), http:// dx.doi.org/10.1016/j.jnoncrysol.2016.11.014
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Fig. 5. Kinetic fragility calculated by the model (mcal), including the empirical correction factor, is compared with the kinetic fragility obtained by fitting viscosity data to the MYEGA equation (mvis).
viscosity measurements, e.g., some measurements need cylindrical geometries with certain diameter and height. More importantly, some glasses have strong tendency to crystallize as they are heated, which makes accurate viscosity measurements impossible. There exist several viscosity models to describe the temperature dependence of shear viscosity [6,34,47,48]. Three-parameter viscosity models can be expressed in terms of the aforementioned parameters: Tg(x), m(x), and η∞(x). The Mauro-Yue-Ellison-Gupta-Allan (MYEGA) model is the most accurate model thus far, and is physically derived in [6]: Tg Tg m log10 ηðT Þ ¼ log10 η∞ þ 12−log10 η∞ −1 exp −1 : ð11Þ 12−log10 η∞ T T
Once the three parameters are determined, the η-T relationship can be described. In our previous work [9], we have determined the value of η∞(x) to be a constant 10–2.93 Pa·s based on a comprehensive experimental and theoretical analysis. Our findings here imply that the calorimetric glass transition temperature can be used as the kinetic glass transition temperature. Moreover, our empirical model can be used to calculate mvis through mDSC across the full range of fragilities. Therefore we can obtain Tg and m from DSC measurements and combine the highT viscosity limit to obtain the viscosity curve across the full range of temperatures. As an example, we have applied this method to calculate the viscosity curve of a borate glass and compare with the experimental viscosity data. It is seen from Fig. 6 that the predicted viscosity agrees very well with the measured viscosity across the full temperature range. The agreement in the low-T range is especially good, where the two viscosity curves nearly overlap each other. Our result implies that this approach could be applied as a standard method for determining the temperature dependence of viscosity using DSC measurements without performing viscosity measurements. The DSC method can be simpler and more effective compared to direct viscosity measurements, since only one small sample (e.g., 20–50 mg) is needed to conduct the DSC measurements. Moreover, the Tf values can be acquired by performing several DSC up and down scans [49]. The Tg,DSC is directly obtained from the DSC reheating scan at 10 K/min equal to the prior cooling rate, while mDSC can be calculated easily based on the Tf values measured at different reheating rates. 5. Summary We have determined the glass transition temperature and fragility from viscosity and DSC measurements of 20 boroaluminosilicate glasses, 6 soda-lime borate glasses, two fiber glass compositions, and 6 vanadium tellurite glasses. The glass transition temperatures obtained
5
Fig. 6. The logarithmic viscosity (log10 η) as a function of the Tg,DSC scaled inverse temperature (Tg,DSC/T) for one borate glass composition. The red dots are the measured viscosity data, and the red line represents the MYEGA fitting of the viscosity data. The blue line is the viscosity data calculated from this work. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
from viscosity measurements (Tg,vis) are consistent with those obtained from DSC measurements (Tg,DSC); thus Tg,DSC can be used interchangeably with Tg,vis. However, there is a deviation between the calorimetric and kinetic fragilities for all the studied glasses. The kinetic fragility is generally higher than the corresponding calorimetric fragility, and larger deviation is associated with more fragile glass compositions. The Arrhenian approximation of the log(1/qc) ~ Tg/Tf in the temperature range near Tg is the source of the deviation between kinetic and calorimetric fragilities. We have developed an approach to predict kinetic fragility (mvis) using calorimetric fragility (mDSC) across a wide range of fragilities. The predicted fragility values agree well with the experimental fragility. By combining the high-T viscosity limit (10–2.93 Pa·s) with the MYEGA equation, the entire viscosity curve of oxide glasses can be well predicted just using DSC measurements without performing viscosity measurements. Acknowledgements The authors thank Mette Solvang for the support in the data acquirement of the two fiber glass compositions and Jonas Kjeldsen for data supplement of the vanadium tellurite glasses. References [1] A.K. Varshneya, Fundamentals of Inorganic Glasses, Academic, Boston, MA, U.S.A., 1994 [2] P.K. Gupta, J.C. Mauro, Composition dependence of glass transition temperature and fragility. I. A topological model incorporating temperature-dependent constraints, J. Chem. Phys. 130 (2009) 094503. [3] J.C. Mauro, D.C. Allan, M. Potuzak, Nonequilibrium viscosity of glass, Phys. Rev. B 80 (2009) 094204. [4] C.A. Angell, Spectroscopy, simulation, and the medium range order problem in glass, J. Non-Cryst. Solids 73 (1985) 1. [5] C.A. Angell, K.L. Ngai, G.B. McKenna, P.F. McMillan, S.W. Martin, Relaxation in glassforming liquids and amorphous solids, J. Appl. Phys. 88 (2000) 3113. [6] J.C. Mauro, Y.Z. Yue, A.J. Ellison, P.K. Gupta, D.C. Allan, Viscosity of glass-forming liquids, Proc. Natl. Acad. Sci. U. S. A. 106 (2009) 19780. [7] Y.Z. Yue, The iso-structural viscosity, configurational entropy and fragility of oxide liquids, J. Non-Cryst. Solids 355 (2009) 737. [8] C.A. Angell, Formation of glasses from liquids and biopolymers, Science 267 (1995) 1924. [9] Q.J. Zheng, J.C. Mauro, A.J. Ellison, M. Potuzak, Y.Z. Yue, Universality of the high-temperature viscosity limit of silicate liquids, Phys. Rev. B 83 (2011) 212202. [10] C.A. Angell, B.E. Richards, V. Velikov, Simple glass-forming liquids: their definition, fragilities, and landscape excitation profiles, J. Phys. Condens. Matter 11 (1999) A75. [11] C.A. Angell, Relaxation in liquids, polymers and plastic crystals - strong/fragile patterns and problems, J. Non-Cryst. Solids 131 (1991) 13. [12] C.A. Angell, J.L. Green, K. Ito, P. Lucas, B.E. Richards, Glassformer fragilities and landscape excitation profiles by simple calorimetric and theoretical methods, J. Therm. Anal. Calorim. 57 (1999) 717.
Please cite this article as: Q. Zheng, et al., Reconciling calorimetric and kinetic fragilities of glass-forming liquids, J. Non-Cryst. Solids (2016), http:// dx.doi.org/10.1016/j.jnoncrysol.2016.11.014
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[13] J.C. Mauro, P.K. Gupta, R.J. Loucks, Composition dependence of glass transition temperature and fragility. II. A topological model of alkali borate liquids, J. Chem. Phys. 130 (2009) 234503. [14] C. Hermansen, X.J. Guo, R.E. Youngman, J.C. Mauro, M.M. Smedskjaer, Y.Z. Yue, Structure-topology-property correlations of sodium phosphosilicate glasses, J. Chem. Phys. 143 (2015) 064510. [15] G.L. Paraschiv, F. Muñoz, L.R. Jensen, Y.Z. Yue, M.M. Smedskjaer, Impact of nitridation of metaphosphate glasses on liquid fragility, J. Non-Cryst. Solids 441 (2016) 22. [16] L.M. Martinez, C.A. Angell, A thermodynamic connection to the fragility of glassforming liquids, Nature 410 (2001) 663. [17] J.C. Mauro, R.J. Loucks, Impact of fragility on enthalpy relaxation in glass, Phys. Rev. E 78 (2008) 021502. [18] K. Ito, C.T. Moynihan, C.A. Angell, Thermodynamic determination of fragility in liquids and a fragile-to-strong liquid transition in water, Nature 398 (1999) 492. [19] G.B. McKenna, A brief discussion: thermodynamic and dynamic fragilities, non-divergent dynamics and the Prigogine–Defay ratio, J. Non-Cryst. Solids 355 (2009) 663. [20] D. Cangialosi, A. Alegría, J. Colmenero, A thermodynamic approach to the fragility of glass-forming polymers, J. Chem. Phys. 124 (2006) 024906. [21] H. Ma, H.J. Fecht, Thermodynamic and kinetic fragilities of Mg-based bulk metallic glass-forming liquids, J. Mater. Res. 23 (2008) 10. [22] L.M. Wang, V. Velikov, C.A. Angell, Direct determination of kinetic fragility indices of glassforming liquids by differential scanning calorimetry: kinetic versus thermodynamic fragilities, J. Chem. Phys. 117 (2002) 10184. [23] Z.M. Chen, Z.J. Li, Y.Q. Zhang, R.P. Liu, Y.J. Tian, L.M. Wang, Calorimetric determination of fragility in glass forming liquids: Tf vs. Tg-onset methods, Eur. Phys. J. E 37 (2014) 52. [24] T. Moynihan, A.J. Easteal, M.A. DeBolt, J. Tucker, Dependence of the fictive temperature of glass on cooling rate, J. Am. Chem. Soc. 59 (1976) 12. [25] C.T. Moynihan, Structural relaxation and the glass transition, Rev. Mineral. 32 (1995) 1. [26] C.T. Moynihan, S.K. Lee, M. Tatsumisago, T. Minami, Estimation of activation energies for structural relaxation and viscous flow from DTA and DSC experiments, Thermochim. Acta 280 (1996) 153. [27] A.Q. Tool, Relation between inelastic deformability and thermal expansion of glass in its annealing range, J. Am. Ceram. Soc. 29 (1946) 240. [28] J.C. Mauro, R.J. Loucks, P.K. Gupta, Fictive temperature and the glassy state, J. Am. Ceram. Soc. 92 (2009) 75. [29] X.J. Guo, M. Potuzak, J.C. Mauro, D.C. Allan, T.J. Kiczenski, Y.Z. Yue, Unified approach for determining the enthalpic fictive temperature of glasses with arbitrary thermal history, J. Non-Cryst. Solids 357 (2011) 3230. [30] Y.Z. Yue, R. von der Ohe, S.L. Jensen, Fictive temperature, cooling rate, and viscosity of glasses, J. Chem. Phys. 120 (17) (2004); Y.Z. Yue, R. von der Ohe, S.L. Jensen, Erratum: “Fictive temperature, cooling rate, and viscosity of glasses” J. Chem. Phys. 120, 8053 2004, J. Chem. Phys. 121 (2004) 11508.
[31] H.E. Kissinger, Reaction kinetics in differential thermal analysis, Anal. Chem. 29 (1957) 1702. [32] V. Velikov, S. Borick, C.A. Angell, The glass transition of water, based on hyperquenching experiments, Science 294 (2001) 2335. [33] Y.Z. Yue, J.D. Christiansen, S.L. Jensen, Determination of the fictive temperature for a hyperquenched glass, Chem. Phys. Lett. 357 (2002) 20. [34] I. Avramov, A. Milchev, Effect of disorder on diffusion and viscosity in condensed systems, J. Non-Cryst. Solids 104 (1988) 253. [35] Q.J. Zheng, R.E. Youngman, C.L. Hogue, J.C. Mauro, M. Potuzak, M.M. Smedskjaer, Y.Z. Yue, Structure of boroaluminosilicate glasses: impact of [Al2O3]/[SiO2] ratio on the structural role of sodium, Phys. Rev. B 86 (2012) 054203. [36] Q.J. Zheng, M. Potuzak, J.C. Mauro, M.M. Smedskjaer, R.E. Youngman, Y.Z. Yue, Composition-structure-property relationships in boroaluminosilicate glasses, J. NonCryst. Solids 358 (2012) 993. [37] M.M. Smedskjaer, J.C. Mauro, S. Sen, Y.Z. Yue, Quantitative design of glassy materials using temperature-dependent constraint theory, Chem. Mater. 22 (2010) 5358. [38] Q.J. Zheng, J.C. Mauro, M.M. Smedskjaer, R.E. Youngman, M. Potuzak, Y.Z. Yue, Glassforming ability of soda lime borate liquids, J. Non-Cryst. Solids 358 (2012) 658. [39] J. Kjeldsen and Y. Z. Yue (Unpublished results). [40] J. Kjeldsen, A.C.M. Rodrigues, S. Mossin, Y.Z. Yue, Critical V2O5/TeO2 ratio inducing abrupt property changes in vanadium tellurite glasses, J. Phys. Chem. B 118 (2014) 14942. [41] J. Kjeldsen, Mechanical and Dynamic Properties of Mixed Modifier Silicate Glasses and Vanadium-Tellurite Glasses(Ph.D. Thesis) Aalborg University, Aalborg, Denmark, 2014. [42] Y.Z. Yue, Characteristic temperatures of enthalpy relaxation in glass, J. Non-Cryst. Solids 354 (2008) 1112. [43] P. Ducroux, S.M. Rekhson, F.L. Merat, Structural relaxation in thermorheologically complex materials, J. Non-Cryst. Solids 172 (1994) 541. [44] Q.J. Zheng, M.M. Smedskjaer, R.E. Youngman, M. Potuzak, J.C. Mauro, Y.Z. Yue, Influence of aluminum speciation on the stability of aluminosilicate glasses against crystallization, Appl. Phys. Lett. 101 (2012) 041906. [45] J.E. Shelby, Introduction to Glass Science and Technology, second ed. The Royal Society of Chemistry, Cambridge, U.K., 2005 [46] J.W.E. Drewitt, L. Hennet, A. Zeidler, S. Jahn, P.S. Salmon, D.R. Neuville, H.E. Fischer, Structural transformations on vitrification in the fragile glass-forming system CaAl2O4, Phys. Rev. Lett. 109 (2012) 235501. [47] G. Adam, J.H. Gibbs, On the temperature dependence of cooperative relaxation properties in glass-forming liquids, J. Chem. Phys. 43 (1965) 139. [48] F. Stickel, E.W. Fischer, R. Richert, Dynamics of glass-forming liquids. I. Temperaturederivative analysis of dielectric relaxation data, J. Chem. Phys. 102 (1995) 6251. [49] T.A. Waniuk, R. Busch, A. Masuhr, W.L. Johnson, Equilibrium viscosity of the Zr41.2Ti13.8Cu12.5Ni10Be22.5 bulk metallic glass-forming liquid and viscous flow during relaxation, phase separation, and primary crystallization, Acta Mater. 46 (1998) 5229.
Please cite this article as: Q. Zheng, et al., Reconciling calorimetric and kinetic fragilities of glass-forming liquids, J. Non-Cryst. Solids (2016), http:// dx.doi.org/10.1016/j.jnoncrysol.2016.11.014