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An experimental critique on the existence of fragile-to-strong transition in glass-forming liquids
Journal of Non-Crystalline Solids 495 (2018) 102–106
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Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol
Letter to the editor
An experimental critique on the existence of fragile-to-strong transition in glass-forming liquids W. Zhua, M.A.T. Marplea, M.J. Lockhartb, B.G. Aitkenb, S. Sena, a b
T
⁎
Dept. of Materials Science & Engineering, University of California, Davis, CA 95616, USA Science & Technology Division, Corning Incorporated, Corning, NY 14831, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: Viscosity Fragile-to-strong transition Oscillation viscometry Chalcogenide Super-liquidus
A number of recent studies in the literature have reported the existence of unusual temperature dependence of the shear viscosity for a variety of glass-forming liquids that cannot be described with a single fragility index. Rather, these liquids display an apparent fragile-to-strong transition (FST) with lowering of temperature that was suggested to be indicative of a structural transition. In the present work we critique the accuracy of the hightemperature oscillation viscometry technique that was used almost exclusively for viscosity measurements of these liquids above their liquidus temperature. Viscosity measurements are carried out on binary Ge-Se chalcogenide liquids in their supercooled and stable states using conventional capillary and parallel-plate rheometry techniques and are compared with the oscillation viscometry data in the literature. Such comparisons conclusively demonstrate that the latter measurement technique often underestimates the viscosity by nearly an order of magnitude, especially at high temperatures. These results, when taken together, indicate that the observation of FST in glass-forming liquids is likely an artifact of the inaccuracy of the oscillation viscometry technique. Moreover, these results cast serious doubt on the reliability of the high-temperature oscillation viscometry data for metals and alloys that are prevalent in the literature.
1. Introduction Viscosity is a key transport property that is at the heart of the processing of a wide variety of liquids, including metals, alloys, glasses and polymers. The remarkable increase in viscosity of glass-forming liquids by more than twelve orders of magnitude with decreasing temperature in the supercooled regime holds the clue for an atomistic understanding of the phenomenon of glass transition [1]. The temperature dependence of the shear viscosity η of glass-forming liquids displays behaviors ranging from approximately Arrhenius, with an activation energy that is nearly independent of temperature, to strongly non-Arrhenius, where the activation energy increases rapidly with decreasing temperature. The degree of departure from an Arrhenius behavior has been used by Angell to classify glass-forming liquids into “strong” vs. “fragile” via the use of the fragility index m which is defined as [2,3]:
m=
dlog10 η d
( ) Tg T
T = Tg
(1)
where Tgand T are the glass transition temperature and the absolute ⁎
Corresponding author. E-mail address: [email protected] (W. Zhu).
https://doi.org/10.1016/j.jnoncrysol.2018.05.009 Received 10 March 2018; Received in revised form 4 May 2018; Accepted 5 May 2018 0022-3093/ © 2018 Elsevier B.V. All rights reserved.
temperature, respectively. In most cases, the temperature dependence of the viscosity of a glass-forming liquid can be described reasonably well, over the entire range of supercooling, using the Vogel–Fulcher–Tammann (VFT) equation [1]:
log10 η = A +
B T − T0
(2)
where A, B and T0 are material-dependent constants. In this case, the fragility index can be expressed in terms of the VFT parameters as:
() m= ⎡1 − ( ) ⎤⎦ ⎣ B Tg
T0 Tg
2
(3)
Since, at T = Tg the viscosity of any glass-forming liquid typically T reaches a value of η ≈ 1012Pa · s, the log10η vs. g behavior of a glassT forming liquid over the entire range of supercooling is expected to be characterized by a single value of m. In a recent study Zhang et al. reported experimental viscosity data on a variety of metallic glassforming liquids that displayed a violation of this behavior [4]. As these liquids are prone to crystallization at intermediate temperatures, the
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and alloys, can be found in the literature [12–14]. Here we focus on the most commonly adopted technique of oscillation viscometry, since all reports in the literature on the fragile-to-strong transition in glassforming chalcogenide and metallic liquids have exclusively used the high-temperature viscosity data generated by this technique. In the application of the oscillation technique the liquid under consideration is typically contained in a cylindrical vessel which is vertically suspended using a torsion wire and is set into oscillatory motion about the vertical axis. This motion is damped by the presence of the liquid due to its absorption and dissipation of the frictional energy and the viscosity of the liquid is determined from its relation to the time period of the oscillatory motion and its decrement which can be measured with relative ease. Relatively small amounts of liquid can be used with this technique, which allows for the attainment of stable thermal equilibrium and extremely low viscosities (1 mPa·s or less) can be measured [12–14]. In spite of the ease with which the technique can be used at very high temperatures under controlled atmosphere conditions and the parameters of the oscillatory motion can be measured, this method suffers from some serious disadvantages [12–14]. Most important among these is the fact that the viscosity is derived from the measured parameters through the solution of a second order differential equation of oscillatory motion of the system which also involves the approximate (linearized) solutions of the Navier-Stokes equations for the motion of the liquid in the container. In the case of a liquid in a cylindrical container the equation of motion for its damped oscillatory motion is [12,14]:
viscosity data were obtained near Tg using either calorimetric or threepoint beam bending techniques and near the liquidus temperature with an oscillating viscometer. It was shown that the temperature dependence of the shear viscosity of these liquids cannot be explained with a single fragility index [4]. Rather, these liquids are characterized by a low value of m i.e. behave as strong liquids at low temperatures near Tg, while their viscosity near liquidus is characteristic of that of a fragile liquid with high m. This apparent fragile-to-strong transition (FST) with lowering of temperature was ascribed to a polyamorphic transformation of the structure of these liquids with temperature. It was hypothesized that such a transformation is accompanied by the destruction of the medium-range and even short-range structural order in these liquids with increasing temperature [4]. It is to be noted here that such transitions were predicted by molecular dynamics simulations to be prevalent in tetrahedral glasses-forming liquids such as H2O and SiO2 [5,6]. A recent study by Lucas et al. reported fragility measurements on two other tetrahedral glass-forming liquids namely, ZnCl2 and GeSe2 using differential scanning calorimetry [7]. It was shown that an FST scenario, similar to that mentioned above, needs to be invoked when near Tg fragility data for both systems are compared to the high-temperature viscosity data available in the literature [7]. Conjectures regarding the existence of such a transition have also been made previously in the literature for other Ge-Se liquids as well as phase change telluride liquids (e.g. Ag-In-Sb-Te, Ge-Te), based on similar comparisons between the DSC-based fragility and the high-temperature viscosity data [8,9]. The ratio F of the high- and low-temperature fragilities has been used as an indicator of the strength of this transition which typically ranges between 2 ≤ F ≤ 8, with metallic liquids being characterized by some of the highest F values [4]. It is clear from Eq. (1) that fragility m is defined at T = Tg. However, for many of the glass-forming liquids that display the FST, viscosity measurements are often not feasible near Tg due to their instability against crystallization or their extreme hygroscopic nature. Under these circumstances, any comparison between the fragility derived from fitting high-temperature viscosity data over a limited viscosity range and that obtained from calorimetric methods near Tg, has its caveats. In the case of ZnCl2, the high-temperature viscosity was fitted to an empirical equation under the inherent assumption of a viscosity of 1012 Pa·s at T = Tg. While this assumption is quite accurate for oxide liquids, in the case of chalcogenides and halides the high-frequency shear modulus G∞ is lower than 1010 Pa, which, on the basis of the Maxwell relation: η = τ × G∞would result in a viscosity significantly lower than 1012 Pa·s at Tg corresponding to a relaxation time τ of ~102 s. For example, in the case of supercooled ZnCl2 liquid, previous Brillouin scattering measurements [10] indicated G∞ ~ 109.5 Pa near Tg, which would result in a lower value of the high-temperature fragility m′ (~50) compared to that reported by Lucas et al. (m′ = 59). On the other hand, the lowtemperature fragility m can be calculated for this liquid using the em-
dφ d 2φ I ⎛ 2 ⎞ + J ⎛ ⎞ + D (φ) = 0 ⎝ dt ⎠ ⎝ dt ⎠ ⎜
⎟
(4)
where I is the moment of inertia of the empty cup and suspension, t is time, D is the force constant of the torsion wire, φ is the angular displacement of any segment of the liquid from its equilibrium position and J is a function of the density ρ and the viscosity η of the liquid, the internal radius of the cylindrical container R and the height of the liquid H. There are several mathematical treatments for the solution of Eq. (4) that yield different viscosity values from the same experimental parameters. The most popular approach for determining the viscosity, which has been used in practically all studies on metallic and chalcogenide glass-forming liquids, is the analysis by Roscoe which relates viscosity to the period of oscillation θ, its logarithmic decrement between consecutive swings λ, H, ρ and R. According to the Roscoe approach, viscosity is expressed as [12,14]:
Iδ ⎞2 1 η=⎛ 3 ⎝ πR HZ ⎠ πρθ where,
( )a − ( + ) + ( + ) ( ) ; and a = 1 + + ( ) .
Z= 1+ 3λ 4π
(5)
3 8
λ 2π
R 4H 2
0
3 2
4R πH
λ 4π
1 p
3 8
1 8
λ 2 2π
9R 4H
a2 ; 2p2
p=
πρ ηθ
R;
a0 = 1 − − 2 A major source of error in obtaining the viscosity from these experiments can be associated with the increasing experimental uncertainty in extrapolating H and R values at high temperatures [12–14]. Furthermore, previous studies have indicated that the wetting properties of the liquid are also important. For example, if the liquid does not wet the walls of the container then it may slip during the oscillation, which would result in increased damping. The wetting behavior may also be sensitively dependent on temperature and on any chemical reaction that may occur between the liquid and the container, especially at high temperatures [12–14]. Finally, it has been suggested that the effect of the liquid meniscus on H and the finite length of the cylindrical container have not been adequately weighted in the Roscoe treatment. Besides these inherent drawbacks of the technique, Cheng et al. [14] have recently noted that, unlike other viscosity measurement techniques, there are no specified standards that have been adopted for the oscillation viscometry technique. Thus, it is not surprising that, even as recently as in 1988, Iida and Guthrie [15] noted a spread of ~400% in
ΔCp (Tg )
pirical relation [11]: m = 40 ΔS , where ΔCp(Tg) is the jump in heat m capacity across glass transition and ΔSm is the entropy of melting of crystalline ZnCl2. Previously reported results [11] on ΔCp(Tg) and ΔSm of ZnCl2 indicate m ~ 40 near Tg, compared to the value of m ~ 30 reported by Lucas et al. [7]. These estimates bring the high- and lowtemperature fragilities quite close (50 vs. 40) and cast reasonable doubt on the existence of a fragile-to-strong crossover for ZnCl2. However, similar corrections for the fragility values are not significant enough to question the validity of such a transition in the case of metallic or some chalcogenide liquids where the F ratios are significantly higher than that reported for ZnCl2 [4,7–9]. It is important to note that compared to ZnCl2, these latter classes of liquids are characterized by significantly higher melting points and the reported high temperature viscosities are measured exclusively using an oscillating viscometer. Recent reviews of the viscosity measurement techniques that are commonly adopted at high temperature, especially for molten metals 103
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W. Zhu et al.
the reported viscosity of molten aluminum and of nearly ~100% for that of molten iron. In the present work we take issue with the accuracy of this technique at high temperatures and demonstrate that the viscosity of chalcogenide liquids of composition Ge15Se85 and Ge20Se80, as measured by this technique, is underestimated by nearly an order of magnitude compared to when measured by the more conventional parallel plate technique. The implications of these results on the existence of FST in chalcogenides and metallic glass-forming liquids are discussed.
Table 2 Viscosity data for Ge15Se85 liquid measured in the present study.
2. Experimental details
Table 3 Viscosity data for Ge20Se80 liquid measured in the present study.
The Se, Ge15Se85 and Ge20Se80 glasses were synthesized by the typical melt-quench method. Constituent elements (≥99.999% purity, metals basis) were loaded into vacuum sealed quartz ampoule and melted in a rocking furnace for 24 h before quenching in water. The viscosity of the supercooled Ge15Se85 liquid was measured at temperatures ranging between 563 K and 723 K, using a capillary rheometer (Rosand RH2000, Malvern). The glass was loaded into the barrel of the rheometer and heated to the desired temperature. After reaching thermal equilibrium, the melt was extruded with a piston at a desired speed, within the Newtonian regime, through a capillary die (diameter 0.75 mm) at the end of the barrel. The pressure at the entrance of the die and the piston speed were recorded to calculate the shear stress, shear rate and shear viscosity η. The details of these calculations and of the experimental setup for capillary rheometry (CR) can be found in a previous publication [16]. The same technique was used to measure the viscosity of the supercooled Ge20Se80 liquid at 638 K, while viscosity of this liquid at temperatures ranging between 673 K and 823 K was measured using parallel plate rheometry (PPR) with an ARES G2 (TA Instruments) rheometer. The same rheometer was also used to measure the viscosity of liquid selenium at temperatures ranging between 523 K and 673 K. In this rheometer two 8 mm diameter plates are utilized. The lower plate is connected to the motor to apply the strain while the upper plate measures the corresponding torque. A furnace that can go up to 873 K controls the temperature of the whole measuring system with a constant flow of N2. Before the measurement, the sample was first heated, pressed and trimmed at high temperature to establish a sandwich geometry with a thickness around 1.5 mm. Once thermal equilibrium was reached at the desired temperature, the Newtonian viscosity was measured under a steady shearing mode with a shear rate of 100 s−1, which allows a large enough torque signal while small enough to stay in the Newtonian range within the temperature range. Multiple measurements were performed at each temperature, for both capillary and parallel plate rheometry and the average value was taken as the Newtonian viscosity. These average viscosity values and the corresponding experimental uncertainties are listed in Tables 1–3.
Table 1 Viscosity data for liquid selenium measured in the present study. Error (Pa·s)
353 523 573 623 673
787,000 1.16 0.32 0.12 0.057
24,468 0.009 0.002 0.001 0.0005
Error (Pa·s)
290 342 350 375 450
5888 398 354 129 12
474.5 12 15 15 2.4
Temperature (K)
Viscosity (Pa·s)
Error (Pa·s)
638 673 723 773 823
6764 2670 393 82 22
392 22 2.1 0.34 0.06
shear viscosity in Pa·s using the density of these liquids as reported in the literature [24,25]. It is clear from Fig. 1 that the viscosity of selenium, measured over a wide range with a variety of techniques in different laboratories, does not display any signature of FST. The hightemperature viscosity of liquid selenium, measured in this study using the PPR technique, is found to be in good agreement with that measured in previous studies using the oscillation viscometry technique (Fig. 1). In contrast, for the supercooled Ge15Se85 and Ge20Se80 liquids, a comparison between the low- and high-temperature viscosity data reported in the literature shows indication of an apparently abrupt change in the fragility (Figs. 2, 3). These high-temperature viscosity data were collected exclusively using the oscillation viscometry technique [19,22]. Besides fitting these viscosity data with the VFT equation (Eq. (2)), we have also used the MYEGA equation (see Eq. (6) below) to fit the low- and high-temperature viscosity data separately to obtain the low- and high-temperature fragilities m and m′, respectively.
The temperature dependence of the viscosity of these three chalcogenide liquids is shown in Figs. 1–3, along with the corresponding data available in the literature for these compositions [17–23]. The high-temperature oscillation viscometry data in the literature are reported as the kinematic viscosity in Stokes. These data are converted to
Viscosity (Pa·s)
Viscosity (Pa·s)
Fig. 1. Viscosity of selenium in the supercooled and stable states. PP rheometry data obtained in the present study are shown as filled squares. Experimental uncertainties are within the size of the symbols. Other symbols represent data from previous reports in the literature (see legend for the sources). Solid and dashed lines represent fits to VFT and MYEGA equations, respectively. Corresponding fragility values are listed in the inset.
3. Results and discussion
Temperature (K)
Temperature (K)
104
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W. Zhu et al.
Fig. 2. Viscosity of Ge15Se85 liquid in supercooled and stable states. Capillary rheometry data obtained in the present study are shown as filled squares and are compared with the data reported by Nemilov [23]. Experimental uncertainties for the data obtained in the present study are within the size of the symbols. Solid and dashed blue lines represent fits to these datasets using VFT and MYEGA equations, respectively. Open triangles correspond to oscillation viscometry data for Ge15Se85 liquid, reported by Laugier et al. [22]. Solid and dashed red lines through these data points represent fits to VFT and MYEGA equations, respectively. Corresponding fragility values are listed in the inset. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Viscosity of Ge20Se80 liquid in supercooled and stable states. Capillary and PP rheometry data obtained in the present study are shown as filled triangle and squares, respectively, and are compared with the data for supercooled Ge20Se80 liquid (open circles), reported by Nemilov [23]. Experimental uncertainties for the data obtained in the present study are within the size of the symbols. Solid and dashed blue lines represent fits to these datasets using VFT and MYEGA equations, respectively. Open upright and inverted triangles correspond to oscillation viscometry data for this liquid, reported by Laugier et al. [22] and Glazov et al. [19], respectively. Solid (dashed) red and pink lines through these data points represent least squares fits of VFT (MYEGA) equation to data from Laugier et al. and Glazov et al., respectively. Corresponding fragility values are listed in the inset. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The MYEGA equation can be written as [26]:
log10 η (T ) = log10 η∞ + (log10 ηTg − log10 η∞ ) ⎤ ⎞ Tg − 1 ⎛ − 1⎞ ⎥ ⎟⎝ T ⎠ ⎥ ⎠ ⎦ ⎜
Tg T
⎡⎛ m exp ⎢ ⎜ log η − log η Tg 10 10 ∞ ⎢ ⎣⎝
regard, further insight can be obtained by comparing the abovementioned F values for Ge15Se85 and Ge20Se80 liquids to the corresponding value of F = 2.5, recently reported by Lucas et al. [7] for the GeSe2 liquid and F = 1 (i.e. no FST) for liquid Se. This comparison demonstrates that F, when determined from oscillation viscometry data, monotonically increases with increasing temperature of measurement of the high-temperature viscosity of these liquids. This trend is further validated in Fig. 4 where the F values of Ge-Se liquids are plotted as a function of the average temperature of the oscillation viscometry measurements. The clear positive correlation between F and temperature is a telltale sign of the problem with the accuracy of the oscillation technique which becomes worse with increasing temperature of measurement. This temperature effect is also apparent in Fig. 3 when one compares the viscosity data by Laugier et al. [22] with those of Glazov [19]; the latter data collected at higher temperature yield an even higher fragility, which again suggests that the oscillation viscometry results become less accurate at higher temperatures. As mentioned above, this trend is somewhat expected since the accuracy of this viscometry technique is critically dependent on: (i) the behavior of the liquid-container interface in terms of chemical reaction as well as changing surface tension at high temperatures, which may affect the meniscus and the stick vs. slip condition, as well as on (ii) the extrapolations of the dimensions of the container and the height of the liquid at high temperatures.
⎟
(6)
In Eq. (6) log10ηTg and log10η∞correspond to the logarithm of viscosity at Tg and at infinite temperature, respectively, and m is the melt fragility. The parameters log10η∞ and m are used as fitting variables in the present study. Typically, in the absence of the G∞data, ηTg in Eq. (6) is taken to be 1012 Pa·s. However, G∞ at Tg for Ge15Se85 and Ge20Se80 glasses has been reported in the literature to be ~5.3 GPa and 4.8 GPa [27,28], respectively, which consequently yields ηTg = 100 ∗ G∞ = 5.3 × 1011 and 4.81 × 1011 Pa·s, respectively. The resulting VFT fits yield m = 35 and m′ = 53 (F = 1.5) for the Ge15Se85 liquid and m = 29 and m′ = 70 (F = 2.4) for the Ge20Se80 liquid (Figs. 2, 3). Corresponding values from the MYEGA fits are: m = 31 and m′ = 44 (F = 1.4) for the Ge15Se85 liquid and m = 29 and m′ = 49 (F = 1.7) for the Ge20Se80 liquid (Figs. 2, 3). However, and more interestingly, it is clear from Figs. 2 and 3, that the viscosity data for the Ge15Se85 and Ge20Se80 liquids, obtained in the present study with the PPR and CR techniques, differ by nearly an order of magnitude from those collected using the oscillation viscometry technique at similar temperatures. Furthermore, a comparison of the high-temperature data collected in the present study with the lowtemperature data show no obvious sign of FST for these two compositions (Figs. 2, 3). In fact, for each composition, a single MYEGA/VFT equation can be used to fit the viscosity data over the entire temperature range (Fig. 3). Therefore, these results cast significant doubt on the validity of the existence of a FST in these chalcogenide liquids. In this
4. Summary Viscosity of elemental Se and binary Ge15Se85 and Ge20Se80 liquids 105
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Nature 410 (2001) 259. [2] C. Angell, Relaxation in liquids, polymers and plastic crystals—strong/fragile patterns and problems, J. Non-Cryst. Solids 131 (1991) 13–31. [3] L.-M. Martinez, C. Angell, A thermodynamic connection to the fragility of glassforming liquids, Nature 410 (2001) 663. [4] C. Zhang, L. Hu, Y. Yue, J.C. Mauro, Fragile-to-strong transition in metallic glassforming liquids, J. Chem. Phys. 133 (2010) 014508. [5] 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. [6] I. Saika-Voivod, P.H. Poole, F. Sciortino, Fragile-to-strong transition and polyamorphism in the energy landscape of liquid silica, Nature 412 (2001) 514. [7] P. Lucas, G.J. Coleman, M. Venkateswara Rao, A.N. Edwards, C. Devaadithya, S. Wei, A.Q. Alsayoud, B. Potter Jr., K. Muralidharan, P.A. Deymier, Structure of ZnCl2 melt. Part II: fragile-to-strong transition in a tetrahedral liquid, J. Phys. Chem. B 121 (2017) 11210–11218. [8] J. Orava, H. Weber, I. Kaban, A.L. Greer, Viscosity of liquid Ag–In–Sb–Te: evidence of a fragile-to-strong crossover, J. Chem. Phys. 144 (2016) 194503. [9] S. Stølen, T. Grande, H.-B. Johnsen, Fragility transition in GeSe 2–Se liquids, PCCP 4 (2002) 3396–3399. [10] H.G. Knape, Hypersonic relaxation studies of molten zinc chloride, J. Chem. Phys. 80 (1984) 4788–4793. [11] L.-M. Wang, C.A. Angell, R. Richert, Fragility and thermodynamics in nonpolymeric glass-forming liquids, J. Chem. Phys. 125 (2006) 074505. [12] M.J. Assael, K. Kakosimos, R.M. Banish, J. Brillo, I. Egry, R. Brooks, P.N. Quested, K.C. Mills, A. Nagashima, Y. Sato, Reference data for the density and viscosity of liquid aluminum and liquid iron, J. Phys. Chem. Ref. Data 35 (2006) 285–300. [13] J. Brillo, Thermophysical Properties of Multicomponent Liquid Alloys, Walter de Gruyter GmbH & Co KG, 2016. [14] J. Cheng, J. Gröbner, N. Hort, K. Kainer, R. Schmid-Fetzer, Measurement and calculation of the viscosity of metals—a review of the current status and developing trends, Meas. Sci. Technol. 25 (2014) 062001. [15] T. Iida, R.I. Guthrie, The Physical Properties of Liquid Metals, 1988 Clarendon Press, Walton Street, Oxford OX 2 6 DP, UK, 1988. [16] W. Zhu, M. Marple, B. Aitken, S. Sen, Universality in the non-Newtonian viscous flow behavior of As x Se 1 − x liquids: results from capillary rheometry, J. NonCryst. Solids 453 (2016) 42–45. [17] K. Bernatz, I. Echeverría, S. Simon, D. Plazek, Characterization of the molecular structure of amorphous selenium using recoverable creep compliance measurements, J. Non-Cryst. Solids 307 (2002) 790–801. [18] S. Dzhalilov, S. Orudzhevo, Effect of contamination with antimony on viscosity of amorphous selenium, Russ. J. Phys. Chem. 40 (1966). [19] V.M. Glazov, O.V. Situlina, Physico-chemical analysis of liquid binary A4-Se systems, Dokl. Akad. Nauk SSSR 187 (1969) 799 (Sov. Phys. Dokl.). [20] A. Kadoun, G. Chaussemy, J. Fornazero, J. Mackowski, Kinematic viscosity of AsxSe1 − x glass forming liquids, J. Non-Cryst. Solids 57 (1983) 101–108. [21] P. Koštál, J. Málek, Viscosity of selenium melt, J. Non-Cryst. Solids 356 (2010) 2803–2806. [22] A. Laugier, G. Chaussemy, J. Fornazero, Viscosity of the glass-forming Ge-Se liquid solutions, J. Non-Cryst. Solids 23 (1977) 419–429. [23] S. Nemilov, Viscosity and structure of glasses in the selenium-germanium system, Zh. Prikl. Khim. 37 (1964) 1020–1024. [24] G. Avetikyan, L. Baidakov, Temperature dependence of the density of seleniumenriched glasses of the system Ge-Se and their thermal expansion, Neorg. Mater 8 (1972) 1489–1490. [25] J. Ruska, H. Thurn, Change of short-range order with temperature and composition in liquid GexSe1 − x as shown by density measurements, J. Non-Cryst. Solids 22 (1976) 277–290. [26] J.C. Mauro, Y. Yue, A.J. Ellison, P.K. Gupta, D.C. Allan, Viscosity of glass-forming liquids, Proc. Natl. Acad. Sci. 106 (2009) 19780–19784. [27] Y. Gueguen, T. Rouxel, P. Gadaud, C. Bernard, V. Keryvin, J.-C. Sangleboeuf, Hightemperature elasticity and viscosity of Ge x Se 1 − x glasses in the transition range, Phys. Rev. B 84 (2011) 064201. [28] J.P. Guin, T. Rouxel, J.C. Sanglebœuf, I. Melscoët, J. Lucas, Hardness, toughness, and scratchability of germanium–selenium chalcogenide glasses, J. Am. Ceram. Soc. 85 (2002) 1545–1552.
Fig. 4. F values of Ge-Se liquids obtained from fragility values listed in Figs. 1–3; where m′ corresponds to the fragility obtained exclusively from oscillation viscometry data (red and pink curves in Figs. 2, 3) while m is the fragility obtained from viscosity data obtained using conventional techniques (blue curves in Figs. 2, 3). The variation of F is shown as a function of the average temperature where the oscillation viscometry measurements were carried out for different compositions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
is measured in their supercooled and stable liquid states using conventional capillary and parallel-plate rheometry. The viscosity data are in good agreement with low-temperature measurements reported in the literature, and when taken together, do not display any temperature dependent change in the fragility index. On the other hand, a comparison with the superliquidus viscosity data obtained in previous studies using oscillation viscometry indicate that the latter technique may systematically underestimate the viscosity, often by an order of magnitude, especially at high temperatures. This large discrepancy in viscosity results, taken in conjunction with the known drawbacks of the oscillation viscometry technique, not only questions the existence of FST in glass-forming liquids but also warrants careful consideration of the accuracy of majority of the oscillation viscometry data reported in the literature for molten metals and alloys. Acknowledgement This work was supported by a GOALI grant from the National Science Foundation (NSF-DMR 1505185). References [1] P.G. Debenedetti, F.H. Stillinger, Supercooled liquids and the glass transition,
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Letter to the editor
An experimental critique on the existence of fragile-to-strong transition in glass-forming liquids W. Zhua, M.A.T. Marplea, M.J. Lockhartb, B.G. Aitkenb, S. Sena, a b
T
⁎
Dept. of Materials Science & Engineering, University of California, Davis, CA 95616, USA Science & Technology Division, Corning Incorporated, Corning, NY 14831, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: Viscosity Fragile-to-strong transition Oscillation viscometry Chalcogenide Super-liquidus
A number of recent studies in the literature have reported the existence of unusual temperature dependence of the shear viscosity for a variety of glass-forming liquids that cannot be described with a single fragility index. Rather, these liquids display an apparent fragile-to-strong transition (FST) with lowering of temperature that was suggested to be indicative of a structural transition. In the present work we critique the accuracy of the hightemperature oscillation viscometry technique that was used almost exclusively for viscosity measurements of these liquids above their liquidus temperature. Viscosity measurements are carried out on binary Ge-Se chalcogenide liquids in their supercooled and stable states using conventional capillary and parallel-plate rheometry techniques and are compared with the oscillation viscometry data in the literature. Such comparisons conclusively demonstrate that the latter measurement technique often underestimates the viscosity by nearly an order of magnitude, especially at high temperatures. These results, when taken together, indicate that the observation of FST in glass-forming liquids is likely an artifact of the inaccuracy of the oscillation viscometry technique. Moreover, these results cast serious doubt on the reliability of the high-temperature oscillation viscometry data for metals and alloys that are prevalent in the literature.
1. Introduction Viscosity is a key transport property that is at the heart of the processing of a wide variety of liquids, including metals, alloys, glasses and polymers. The remarkable increase in viscosity of glass-forming liquids by more than twelve orders of magnitude with decreasing temperature in the supercooled regime holds the clue for an atomistic understanding of the phenomenon of glass transition [1]. The temperature dependence of the shear viscosity η of glass-forming liquids displays behaviors ranging from approximately Arrhenius, with an activation energy that is nearly independent of temperature, to strongly non-Arrhenius, where the activation energy increases rapidly with decreasing temperature. The degree of departure from an Arrhenius behavior has been used by Angell to classify glass-forming liquids into “strong” vs. “fragile” via the use of the fragility index m which is defined as [2,3]:
m=
dlog10 η d
( ) Tg T
T = Tg
(1)
where Tgand T are the glass transition temperature and the absolute ⁎
Corresponding author. E-mail address: [email protected] (W. Zhu).
https://doi.org/10.1016/j.jnoncrysol.2018.05.009 Received 10 March 2018; Received in revised form 4 May 2018; Accepted 5 May 2018 0022-3093/ © 2018 Elsevier B.V. All rights reserved.
temperature, respectively. In most cases, the temperature dependence of the viscosity of a glass-forming liquid can be described reasonably well, over the entire range of supercooling, using the Vogel–Fulcher–Tammann (VFT) equation [1]:
log10 η = A +
B T − T0
(2)
where A, B and T0 are material-dependent constants. In this case, the fragility index can be expressed in terms of the VFT parameters as:
() m= ⎡1 − ( ) ⎤⎦ ⎣ B Tg
T0 Tg
2
(3)
Since, at T = Tg the viscosity of any glass-forming liquid typically T reaches a value of η ≈ 1012Pa · s, the log10η vs. g behavior of a glassT forming liquid over the entire range of supercooling is expected to be characterized by a single value of m. In a recent study Zhang et al. reported experimental viscosity data on a variety of metallic glassforming liquids that displayed a violation of this behavior [4]. As these liquids are prone to crystallization at intermediate temperatures, the
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and alloys, can be found in the literature [12–14]. Here we focus on the most commonly adopted technique of oscillation viscometry, since all reports in the literature on the fragile-to-strong transition in glassforming chalcogenide and metallic liquids have exclusively used the high-temperature viscosity data generated by this technique. In the application of the oscillation technique the liquid under consideration is typically contained in a cylindrical vessel which is vertically suspended using a torsion wire and is set into oscillatory motion about the vertical axis. This motion is damped by the presence of the liquid due to its absorption and dissipation of the frictional energy and the viscosity of the liquid is determined from its relation to the time period of the oscillatory motion and its decrement which can be measured with relative ease. Relatively small amounts of liquid can be used with this technique, which allows for the attainment of stable thermal equilibrium and extremely low viscosities (1 mPa·s or less) can be measured [12–14]. In spite of the ease with which the technique can be used at very high temperatures under controlled atmosphere conditions and the parameters of the oscillatory motion can be measured, this method suffers from some serious disadvantages [12–14]. Most important among these is the fact that the viscosity is derived from the measured parameters through the solution of a second order differential equation of oscillatory motion of the system which also involves the approximate (linearized) solutions of the Navier-Stokes equations for the motion of the liquid in the container. In the case of a liquid in a cylindrical container the equation of motion for its damped oscillatory motion is [12,14]:
viscosity data were obtained near Tg using either calorimetric or threepoint beam bending techniques and near the liquidus temperature with an oscillating viscometer. It was shown that the temperature dependence of the shear viscosity of these liquids cannot be explained with a single fragility index [4]. Rather, these liquids are characterized by a low value of m i.e. behave as strong liquids at low temperatures near Tg, while their viscosity near liquidus is characteristic of that of a fragile liquid with high m. This apparent fragile-to-strong transition (FST) with lowering of temperature was ascribed to a polyamorphic transformation of the structure of these liquids with temperature. It was hypothesized that such a transformation is accompanied by the destruction of the medium-range and even short-range structural order in these liquids with increasing temperature [4]. It is to be noted here that such transitions were predicted by molecular dynamics simulations to be prevalent in tetrahedral glasses-forming liquids such as H2O and SiO2 [5,6]. A recent study by Lucas et al. reported fragility measurements on two other tetrahedral glass-forming liquids namely, ZnCl2 and GeSe2 using differential scanning calorimetry [7]. It was shown that an FST scenario, similar to that mentioned above, needs to be invoked when near Tg fragility data for both systems are compared to the high-temperature viscosity data available in the literature [7]. Conjectures regarding the existence of such a transition have also been made previously in the literature for other Ge-Se liquids as well as phase change telluride liquids (e.g. Ag-In-Sb-Te, Ge-Te), based on similar comparisons between the DSC-based fragility and the high-temperature viscosity data [8,9]. The ratio F of the high- and low-temperature fragilities has been used as an indicator of the strength of this transition which typically ranges between 2 ≤ F ≤ 8, with metallic liquids being characterized by some of the highest F values [4]. It is clear from Eq. (1) that fragility m is defined at T = Tg. However, for many of the glass-forming liquids that display the FST, viscosity measurements are often not feasible near Tg due to their instability against crystallization or their extreme hygroscopic nature. Under these circumstances, any comparison between the fragility derived from fitting high-temperature viscosity data over a limited viscosity range and that obtained from calorimetric methods near Tg, has its caveats. In the case of ZnCl2, the high-temperature viscosity was fitted to an empirical equation under the inherent assumption of a viscosity of 1012 Pa·s at T = Tg. While this assumption is quite accurate for oxide liquids, in the case of chalcogenides and halides the high-frequency shear modulus G∞ is lower than 1010 Pa, which, on the basis of the Maxwell relation: η = τ × G∞would result in a viscosity significantly lower than 1012 Pa·s at Tg corresponding to a relaxation time τ of ~102 s. For example, in the case of supercooled ZnCl2 liquid, previous Brillouin scattering measurements [10] indicated G∞ ~ 109.5 Pa near Tg, which would result in a lower value of the high-temperature fragility m′ (~50) compared to that reported by Lucas et al. (m′ = 59). On the other hand, the lowtemperature fragility m can be calculated for this liquid using the em-
dφ d 2φ I ⎛ 2 ⎞ + J ⎛ ⎞ + D (φ) = 0 ⎝ dt ⎠ ⎝ dt ⎠ ⎜
⎟
(4)
where I is the moment of inertia of the empty cup and suspension, t is time, D is the force constant of the torsion wire, φ is the angular displacement of any segment of the liquid from its equilibrium position and J is a function of the density ρ and the viscosity η of the liquid, the internal radius of the cylindrical container R and the height of the liquid H. There are several mathematical treatments for the solution of Eq. (4) that yield different viscosity values from the same experimental parameters. The most popular approach for determining the viscosity, which has been used in practically all studies on metallic and chalcogenide glass-forming liquids, is the analysis by Roscoe which relates viscosity to the period of oscillation θ, its logarithmic decrement between consecutive swings λ, H, ρ and R. According to the Roscoe approach, viscosity is expressed as [12,14]:
Iδ ⎞2 1 η=⎛ 3 ⎝ πR HZ ⎠ πρθ where,
( )a − ( + ) + ( + ) ( ) ; and a = 1 + + ( ) .
Z= 1+ 3λ 4π
(5)
3 8
λ 2π
R 4H 2
0
3 2
4R πH
λ 4π
1 p
3 8
1 8
λ 2 2π
9R 4H
a2 ; 2p2
p=
πρ ηθ
R;
a0 = 1 − − 2 A major source of error in obtaining the viscosity from these experiments can be associated with the increasing experimental uncertainty in extrapolating H and R values at high temperatures [12–14]. Furthermore, previous studies have indicated that the wetting properties of the liquid are also important. For example, if the liquid does not wet the walls of the container then it may slip during the oscillation, which would result in increased damping. The wetting behavior may also be sensitively dependent on temperature and on any chemical reaction that may occur between the liquid and the container, especially at high temperatures [12–14]. Finally, it has been suggested that the effect of the liquid meniscus on H and the finite length of the cylindrical container have not been adequately weighted in the Roscoe treatment. Besides these inherent drawbacks of the technique, Cheng et al. [14] have recently noted that, unlike other viscosity measurement techniques, there are no specified standards that have been adopted for the oscillation viscometry technique. Thus, it is not surprising that, even as recently as in 1988, Iida and Guthrie [15] noted a spread of ~400% in
ΔCp (Tg )
pirical relation [11]: m = 40 ΔS , where ΔCp(Tg) is the jump in heat m capacity across glass transition and ΔSm is the entropy of melting of crystalline ZnCl2. Previously reported results [11] on ΔCp(Tg) and ΔSm of ZnCl2 indicate m ~ 40 near Tg, compared to the value of m ~ 30 reported by Lucas et al. [7]. These estimates bring the high- and lowtemperature fragilities quite close (50 vs. 40) and cast reasonable doubt on the existence of a fragile-to-strong crossover for ZnCl2. However, similar corrections for the fragility values are not significant enough to question the validity of such a transition in the case of metallic or some chalcogenide liquids where the F ratios are significantly higher than that reported for ZnCl2 [4,7–9]. It is important to note that compared to ZnCl2, these latter classes of liquids are characterized by significantly higher melting points and the reported high temperature viscosities are measured exclusively using an oscillating viscometer. Recent reviews of the viscosity measurement techniques that are commonly adopted at high temperature, especially for molten metals 103
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the reported viscosity of molten aluminum and of nearly ~100% for that of molten iron. In the present work we take issue with the accuracy of this technique at high temperatures and demonstrate that the viscosity of chalcogenide liquids of composition Ge15Se85 and Ge20Se80, as measured by this technique, is underestimated by nearly an order of magnitude compared to when measured by the more conventional parallel plate technique. The implications of these results on the existence of FST in chalcogenides and metallic glass-forming liquids are discussed.
Table 2 Viscosity data for Ge15Se85 liquid measured in the present study.
2. Experimental details
Table 3 Viscosity data for Ge20Se80 liquid measured in the present study.
The Se, Ge15Se85 and Ge20Se80 glasses were synthesized by the typical melt-quench method. Constituent elements (≥99.999% purity, metals basis) were loaded into vacuum sealed quartz ampoule and melted in a rocking furnace for 24 h before quenching in water. The viscosity of the supercooled Ge15Se85 liquid was measured at temperatures ranging between 563 K and 723 K, using a capillary rheometer (Rosand RH2000, Malvern). The glass was loaded into the barrel of the rheometer and heated to the desired temperature. After reaching thermal equilibrium, the melt was extruded with a piston at a desired speed, within the Newtonian regime, through a capillary die (diameter 0.75 mm) at the end of the barrel. The pressure at the entrance of the die and the piston speed were recorded to calculate the shear stress, shear rate and shear viscosity η. The details of these calculations and of the experimental setup for capillary rheometry (CR) can be found in a previous publication [16]. The same technique was used to measure the viscosity of the supercooled Ge20Se80 liquid at 638 K, while viscosity of this liquid at temperatures ranging between 673 K and 823 K was measured using parallel plate rheometry (PPR) with an ARES G2 (TA Instruments) rheometer. The same rheometer was also used to measure the viscosity of liquid selenium at temperatures ranging between 523 K and 673 K. In this rheometer two 8 mm diameter plates are utilized. The lower plate is connected to the motor to apply the strain while the upper plate measures the corresponding torque. A furnace that can go up to 873 K controls the temperature of the whole measuring system with a constant flow of N2. Before the measurement, the sample was first heated, pressed and trimmed at high temperature to establish a sandwich geometry with a thickness around 1.5 mm. Once thermal equilibrium was reached at the desired temperature, the Newtonian viscosity was measured under a steady shearing mode with a shear rate of 100 s−1, which allows a large enough torque signal while small enough to stay in the Newtonian range within the temperature range. Multiple measurements were performed at each temperature, for both capillary and parallel plate rheometry and the average value was taken as the Newtonian viscosity. These average viscosity values and the corresponding experimental uncertainties are listed in Tables 1–3.
Table 1 Viscosity data for liquid selenium measured in the present study. Error (Pa·s)
353 523 573 623 673
787,000 1.16 0.32 0.12 0.057
24,468 0.009 0.002 0.001 0.0005
Error (Pa·s)
290 342 350 375 450
5888 398 354 129 12
474.5 12 15 15 2.4
Temperature (K)
Viscosity (Pa·s)
Error (Pa·s)
638 673 723 773 823
6764 2670 393 82 22
392 22 2.1 0.34 0.06
shear viscosity in Pa·s using the density of these liquids as reported in the literature [24,25]. It is clear from Fig. 1 that the viscosity of selenium, measured over a wide range with a variety of techniques in different laboratories, does not display any signature of FST. The hightemperature viscosity of liquid selenium, measured in this study using the PPR technique, is found to be in good agreement with that measured in previous studies using the oscillation viscometry technique (Fig. 1). In contrast, for the supercooled Ge15Se85 and Ge20Se80 liquids, a comparison between the low- and high-temperature viscosity data reported in the literature shows indication of an apparently abrupt change in the fragility (Figs. 2, 3). These high-temperature viscosity data were collected exclusively using the oscillation viscometry technique [19,22]. Besides fitting these viscosity data with the VFT equation (Eq. (2)), we have also used the MYEGA equation (see Eq. (6) below) to fit the low- and high-temperature viscosity data separately to obtain the low- and high-temperature fragilities m and m′, respectively.
The temperature dependence of the viscosity of these three chalcogenide liquids is shown in Figs. 1–3, along with the corresponding data available in the literature for these compositions [17–23]. The high-temperature oscillation viscometry data in the literature are reported as the kinematic viscosity in Stokes. These data are converted to
Viscosity (Pa·s)
Viscosity (Pa·s)
Fig. 1. Viscosity of selenium in the supercooled and stable states. PP rheometry data obtained in the present study are shown as filled squares. Experimental uncertainties are within the size of the symbols. Other symbols represent data from previous reports in the literature (see legend for the sources). Solid and dashed lines represent fits to VFT and MYEGA equations, respectively. Corresponding fragility values are listed in the inset.
3. Results and discussion
Temperature (K)
Temperature (K)
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Fig. 2. Viscosity of Ge15Se85 liquid in supercooled and stable states. Capillary rheometry data obtained in the present study are shown as filled squares and are compared with the data reported by Nemilov [23]. Experimental uncertainties for the data obtained in the present study are within the size of the symbols. Solid and dashed blue lines represent fits to these datasets using VFT and MYEGA equations, respectively. Open triangles correspond to oscillation viscometry data for Ge15Se85 liquid, reported by Laugier et al. [22]. Solid and dashed red lines through these data points represent fits to VFT and MYEGA equations, respectively. Corresponding fragility values are listed in the inset. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Viscosity of Ge20Se80 liquid in supercooled and stable states. Capillary and PP rheometry data obtained in the present study are shown as filled triangle and squares, respectively, and are compared with the data for supercooled Ge20Se80 liquid (open circles), reported by Nemilov [23]. Experimental uncertainties for the data obtained in the present study are within the size of the symbols. Solid and dashed blue lines represent fits to these datasets using VFT and MYEGA equations, respectively. Open upright and inverted triangles correspond to oscillation viscometry data for this liquid, reported by Laugier et al. [22] and Glazov et al. [19], respectively. Solid (dashed) red and pink lines through these data points represent least squares fits of VFT (MYEGA) equation to data from Laugier et al. and Glazov et al., respectively. Corresponding fragility values are listed in the inset. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The MYEGA equation can be written as [26]:
log10 η (T ) = log10 η∞ + (log10 ηTg − log10 η∞ ) ⎤ ⎞ Tg − 1 ⎛ − 1⎞ ⎥ ⎟⎝ T ⎠ ⎥ ⎠ ⎦ ⎜
Tg T
⎡⎛ m exp ⎢ ⎜ log η − log η Tg 10 10 ∞ ⎢ ⎣⎝
regard, further insight can be obtained by comparing the abovementioned F values for Ge15Se85 and Ge20Se80 liquids to the corresponding value of F = 2.5, recently reported by Lucas et al. [7] for the GeSe2 liquid and F = 1 (i.e. no FST) for liquid Se. This comparison demonstrates that F, when determined from oscillation viscometry data, monotonically increases with increasing temperature of measurement of the high-temperature viscosity of these liquids. This trend is further validated in Fig. 4 where the F values of Ge-Se liquids are plotted as a function of the average temperature of the oscillation viscometry measurements. The clear positive correlation between F and temperature is a telltale sign of the problem with the accuracy of the oscillation technique which becomes worse with increasing temperature of measurement. This temperature effect is also apparent in Fig. 3 when one compares the viscosity data by Laugier et al. [22] with those of Glazov [19]; the latter data collected at higher temperature yield an even higher fragility, which again suggests that the oscillation viscometry results become less accurate at higher temperatures. As mentioned above, this trend is somewhat expected since the accuracy of this viscometry technique is critically dependent on: (i) the behavior of the liquid-container interface in terms of chemical reaction as well as changing surface tension at high temperatures, which may affect the meniscus and the stick vs. slip condition, as well as on (ii) the extrapolations of the dimensions of the container and the height of the liquid at high temperatures.
⎟
(6)
In Eq. (6) log10ηTg and log10η∞correspond to the logarithm of viscosity at Tg and at infinite temperature, respectively, and m is the melt fragility. The parameters log10η∞ and m are used as fitting variables in the present study. Typically, in the absence of the G∞data, ηTg in Eq. (6) is taken to be 1012 Pa·s. However, G∞ at Tg for Ge15Se85 and Ge20Se80 glasses has been reported in the literature to be ~5.3 GPa and 4.8 GPa [27,28], respectively, which consequently yields ηTg = 100 ∗ G∞ = 5.3 × 1011 and 4.81 × 1011 Pa·s, respectively. The resulting VFT fits yield m = 35 and m′ = 53 (F = 1.5) for the Ge15Se85 liquid and m = 29 and m′ = 70 (F = 2.4) for the Ge20Se80 liquid (Figs. 2, 3). Corresponding values from the MYEGA fits are: m = 31 and m′ = 44 (F = 1.4) for the Ge15Se85 liquid and m = 29 and m′ = 49 (F = 1.7) for the Ge20Se80 liquid (Figs. 2, 3). However, and more interestingly, it is clear from Figs. 2 and 3, that the viscosity data for the Ge15Se85 and Ge20Se80 liquids, obtained in the present study with the PPR and CR techniques, differ by nearly an order of magnitude from those collected using the oscillation viscometry technique at similar temperatures. Furthermore, a comparison of the high-temperature data collected in the present study with the lowtemperature data show no obvious sign of FST for these two compositions (Figs. 2, 3). In fact, for each composition, a single MYEGA/VFT equation can be used to fit the viscosity data over the entire temperature range (Fig. 3). Therefore, these results cast significant doubt on the validity of the existence of a FST in these chalcogenide liquids. In this
4. Summary Viscosity of elemental Se and binary Ge15Se85 and Ge20Se80 liquids 105
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Fig. 4. F values of Ge-Se liquids obtained from fragility values listed in Figs. 1–3; where m′ corresponds to the fragility obtained exclusively from oscillation viscometry data (red and pink curves in Figs. 2, 3) while m is the fragility obtained from viscosity data obtained using conventional techniques (blue curves in Figs. 2, 3). The variation of F is shown as a function of the average temperature where the oscillation viscometry measurements were carried out for different compositions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
is measured in their supercooled and stable liquid states using conventional capillary and parallel-plate rheometry. The viscosity data are in good agreement with low-temperature measurements reported in the literature, and when taken together, do not display any temperature dependent change in the fragility index. On the other hand, a comparison with the superliquidus viscosity data obtained in previous studies using oscillation viscometry indicate that the latter technique may systematically underestimate the viscosity, often by an order of magnitude, especially at high temperatures. This large discrepancy in viscosity results, taken in conjunction with the known drawbacks of the oscillation viscometry technique, not only questions the existence of FST in glass-forming liquids but also warrants careful consideration of the accuracy of majority of the oscillation viscometry data reported in the literature for molten metals and alloys. Acknowledgement This work was supported by a GOALI grant from the National Science Foundation (NSF-DMR 1505185). References [1] P.G. Debenedetti, F.H. Stillinger, Supercooled liquids and the glass transition,
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