Linking Equilibrium and Nonequilibrium Dynamics in Glass-Forming

Article pubs.acs.org/JPCB

Linking Equilibrium and Nonequilibrium Dynamics in Glass-Forming Systems Xiaoju Guo,† Morten M. Smedskjaer,*,†,‡ and John C. Mauro*,† †

Science and Technology Division, Corning Incorporated, Corning, New York 14831, United States Department of Chemistry and Bioscience, Aalborg University, 9220 Aalborg, Denmark

‡

ABSTRACT: Understanding nonequilibrium glassy dynamics is of great scientific and technological importance. However, prediction of the temperature, thermal history, and composition dependence of nonequilibrium viscosity is challenging due to the noncrystalline and nonergodic nature of the glassy state. Here, we show that the nonequilibrium glassy dynamics are intimately connected with the equilibrium liquid dynamics. This is accomplished by deriving a new functional form for the thermal history dependence of nonequilibrium viscosity, which is validated against experimental measurements of industrial silicate glasses and computed viscosities for selenium over a wide range of conditions. Since the temperature and composition dependence of liquid viscosity can be predicted using temperature-dependent constraint theory, our work also opens the possibility to improve understanding of the physics of nonequilibrium viscosity.

I. INTRODUCTION The fundamental physics governing the liquid-to-glass transition and the spontaneous relaxation of glass toward the liquid state are topics that continue to attract widespread attention in the condensed matter physics community.1−11 Although substantial progress has been made in understanding these phenomena, many questions remain. In particular, for several high-tech applications of glass, it is important to understand the dependence of glass relaxation behavior on composition, temperature, and thermal history. At sufficiently low temperatures the relaxation process is essentially frozen owing to the slow dynamics of the system on an experimentally accessible time scale. The relaxation process is accelerated at elevated temperatures, such as those required for processing of liquid crystal display substrates12 and ion-exchangeable cover glasses for personal electronic devices.13,14 A fundamental understanding of the thermal relaxation behavior is thus of vital scientific and technological interest. Both thermodynamics and kinetics govern glass relaxation behavior. As a nonequilibrium material, a glass continuously relaxes toward the equilibrium liquid state.15 The microscopic relaxation typically involves continuous changes in volume and other macroscopic properties as the glass approaches the liquid state. While the presence of the thermodynamic driving force is a necessity for glass relaxation to occur, glass relaxation is often limited by the kinetics, which are determined by glass composition, temperature, and thermal history. Glass relaxation is primarily governed by the nonequilibrium viscosity of glass,16 as the kinetics controlled by the nonequilibrium viscosity can easily vary over many orders of magnitude, whereas the thermodynamic driving force exhibits comparatively small variations. As an aside, we note that nonequilibrium viscosity is different from non-steady-state viscosity. The latter can refer © 2016 American Chemical Society

to any time-dependent viscosity measurement, whereas the former is due to the departure from equilibrium as the glassforming liquid is cooled through the glass transition range. Nonequilibrium viscosity is also distinct from isostructural viscosity,17,18 which refers to a situation with no relaxation and thus no change in structure. Hence, isostructural viscosity is a special case of nonequilibrium viscosity, which is, in turn, a special case of non-steady-state viscosity. To understand glass relaxation behavior, it thus becomes essential to be able to experimentally determine the nonequilibrium viscosity of glass. Recently, a custom designed beam-bending viscometer was introduced by Mauro et al.,16 which allows viscosities up to 1016 Pa s to be directly measured. Combined with the development of the Mauro−Allan−Potuzak (MAP) model, nonequilibrium viscosities can be determined as a function of temperature and thermal history as quantified by the fictive temperature, Tf = Tf[T(t)], which is a function of the thermal history, T(t), of the system.19 While the model provides good agreement with the experimental data, it has two drawbacks. First, the nonequilibrium viscosity model is phenomenological and is thus not derived from fundamental physics. Second, as a consequence of the first drawback, some of the parameters of the model lack a direct physical interpretation, which complicates the prediction of composition effects on nonequilibrium viscosity. Prediction of nonequilibrium glass viscosity as a function of composition is important, as beam-bending viscometry measurements are intrinsically time-consuming due to the high nonequilibrium viscosity. On the other hand, numerous studies have investigated the Received: January 6, 2016 Revised: March 8, 2016 Published: March 9, 2016 3226

DOI: 10.1021/acs.jpcb.6b00141 J. Phys. Chem. B 2016, 120, 3226−3231

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The Journal of Physical Chemistry B composition dependence of equilibrium liquid viscosity.20−22 Hence, it would be desirable if the knowledge of equilibrium viscosity scaling with composition could be used to predict the composition dependence of nonequilibrium viscosity. In this paper, we show that it is indeed possible to connect the equilibrium liquid dynamics to the nonequilibrium glassy dynamics. We do so by deriving a new form for the nonequilibrium viscosity based on enthalpy landscape and temperature-dependent constraint theories. This new model is validated against experimental measurements of Corning EAGLE XG and Jade glasses and computed viscosities for selenium over a wide range of thermal histories.

transition between basins i and j is denoted Kij. Equation 6 can be rewritten as35 −1 ⎛ Ω Ω ⎛ ΔHij ⎞⎞ ⎟ ⎜ τ[T (t )] = ⎜ν ∑ pi [T (t )] ∑ gj exp⎜ − ⎟ ⎝ kT (t ) ⎠⎟⎠ ⎝ i=1 j≠1

where ν is the attempt frequency, k is Boltzmann’s constant, gj is the degeneracy of basin j, and ΔHij is the activation barrier for transitioning from basin i to j. Since the degeneracy depends on the localized region of the landscape, it also depends on the fictive temperature. Moreover, we assume a dominant activation enthalpy for isostructural flow, ΔH = ΔHij,9 and then rewrite eq 7 as

II. THEORETICAL ANALYSIS As presented in detail in ref 16, the MAP model of nonequilibrium viscosity is given by

τ(T , Tf ) =

log η(T , Tf ) = x log ηeq (Tf ) + (1 − x) log ηne(T , Tf )

⎛ ΔH ⎞ 1 ⎟ exp⎜ ⎝ kT ⎠ νg (Tf )

(8)

or (1)

where ηeq is equilibrium contribution to viscosity, ηne is the nonequilibrium contribution to viscosity, and x is an ergodicity parameter defined by ⎛ min(T , T f) ⎞ ⎟⎟ x = ⎜⎜ ⎝ max(T , T f) ⎠

log τ(T , Tf ) = −log ν − log g (Tf ) +

log ηne(T , Tf ) = A − BTf +

(2)

Tg Tf

(3)

log ηeq (Tf ) = log η∞ +

Here, Tg is the glass transition temperature, defined by (4)

m is the liquid fragility index, defined by25 m≡

d log ηeq d(T /Tg)

T = Tg

(5)

and η∞ = 10−2.9 Pa·s is the infinite temperature limit of liquid viscosity, which is a universal composition independent constant for silicate liquids. This universal limit of η∞ is based on the MYEGA model of viscosity and is discussed in detail in ref 26. Following the energy landscape view of glassy dynamics,27−34 nonequilibrium glass viscosity is governed by basin hopping in an enthalpy landscape.9 Specifically, ηne is proportional to the average relaxation time τ from enthalpy landscape theory:35 Ω

i=1

Sc(Tf ) =

j≠1

(10)

C TSc(Tf )

(11)

⎡ T⎛ ⎞⎤ S∞ m g exp⎢ − ⎜⎜ − 1⎟⎟⎥ ln 10 ⎢⎣ Tf ⎝ 12 − log η∞ ⎠⎥⎦

(12)

where S∞ is the configurational entropy in the infinite temperature limit. Next, we consider Boltzmann’s equation for entropy Sc(Tf ) = k log g (Tf )

(13)

which allows the scaling of degeneracy with fictive temperature to be solved:

Ω

τ[T (t )] = (∑ pi [T (t )] ∑ K ij[T (t )])−1

ΔH kT ln 10

where C is a constant. While not derived rigorously from fundamental physics, the Adam−Gibbs equation has met with remarkable success in fitting experimental data for various systems.6,37−39 By equating the MYEGA model with the Adam−Gibbs model, the configurational entropy of the liquid is given by

12

ηeq (Tg) = 10 Pa·s

(9)

which is the form for ηne used in the original MAP model.16 This form provides an excellent fit to measured nonequilibrium viscosity values, but it is not particularly amenable for understanding and predicting the composition dependence of ηne (due to the empirical parameter B). It also does not demonstrate any link to the equilibrium viscosity. We revisit the problem of deriving a form for ηne(T,Tf) starting with the Adam−Gibbs equation,36 which relates equilibrium viscosity to the configurational entropy of the liquid, Sc(Tf):

Here, p is a constant governing the sharpness of the glass transition, which scales linearly with the liquid fragility, as discussed in detail in ref 16. For the equilibrium viscosity, we employ the Mauro−Yue−Ellison−Gupta−Allan (MYEGA) equation23 derived from topological constraint theory:24

⎡⎛ ⎞⎤ ⎞⎛ Tg m × exp⎢⎜⎜ − 1⎟⎟⎜⎜ − 1⎟⎟⎥ ⎢⎣⎝ 12 − log η∞ ⎠⎝ Tf ⎠⎥⎦

ΔH kT ln 10

In ref 16 we note that log g(Tf) scales approximately linearly with fictive temperature, leading to

p

log ηeq (Tf ) = log η∞ + (12 − log η∞)

(7)

log g (Tf ) =

(6)

where pi is the probability of occupying basin i upon experiencing the thermal history given by T(t). The total number of nondegenerate basins is denoted Ω and the rate of

⎡ T⎛ ⎞⎤ S∞ m g − 1⎟⎟⎥ exp⎢ − ⎜⎜ k ln 10 ⎢⎣ Tf ⎝ 12 − log η∞ ⎠⎥⎦

(14)

By inserting this expression into eq 9, we obtain the new functional form for nonequilibrium viscosity: 3227

DOI: 10.1021/acs.jpcb.6b00141 J. Phys. Chem. B 2016, 120, 3226−3231

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The Journal of Physical Chemistry B S∞ ΔH − kT ln 10 k ln 10 ⎡ T⎛ ⎤ ⎞ m g × exp⎢ − ⎜⎜ − 1⎟⎟⎥ ⎢⎣ Tf ⎝ 12 − log η∞ ⎠⎥⎦

log ηne(T , Tf ) = A +

(15)

The nonequilibrium viscosity of the glass is therefore a function of both the thermal temperature, T, and the fictive temperature, Tf, of the glass. If Tf ≠ T, the fictive temperature will evolve as a function of time, Tf = Tf(t), such that the nonequilibrium viscosity given in eq 15 also becomes a function of time. However, in the limit of low temperature, the dynamics of the system are too slow for structural relaxation to occur on a typical laboratory time scale. This corresponds to a glass with a fixed Tf. In this limit, the nonequilibrium viscosity of eq 15 shows an Arrhenius dependence on temperature, with the slope governed by the enthalpy barrier ΔH seen by the glass in the localized region of the enthalpy landscape where it has become essentially frozen. This functional form has the advantage of being physically derived and more amenable for understanding the composition dependence of nonequilibrium viscosity. It also shows the direct link between the equilibrium liquid dynamics (as expressed by Tg and m) and the nonequilibrium glassy dynamics. This linkage between equilibrium and nonequilibrium dynamics is completely absent in the previous model of eq 10. Thus, the new eq 15 shows that the equilibrium viscosity has a direct and profound impact on the glassy dynamics.

Figure 1. Fictive temperature (Tf) dependence of selenium glass viscosity (log η) at Tg/T = 1.5 (T = 212 K). The viscosities are computed using an energy landscape model of selenium.9 The phenomenological (eq 10) and physically derived (eq 15) models of nonequilibrium viscosity are fit to the computed values.

that the two models predict essentially the same thermal history dependence of ηne. However, for the highest and lowest fictive temperatures, the two models deviate, with eq 15 having the best agreement with the computed values (Figure 1). Hence, the new form of the nonequilibrium viscosity model appears to offer a slightly improved fit to the energy landscape calculated viscosities. Next, we validate the new form against experimental measurements of Corning EAGLE XG and Corning Jade glasses.16,50 The measurements were performed using a custom designed beam-bending viscometer capable of accurate nonequilibrium viscosity measurements up to 1016 Pa·s, as described in detail elsewhere.16 Figure 2 shows the measured viscosities as a function of time during isothermal holds at three different temperatures. The new model (eq 15) is fit to the data using the equilibrium values of Tg, m, and η∞ as input parameters with the values of p, A, ΔH, and S∞ listed in Table 1 for both EAGLE XG and Jade glasses. The fictive temperatures for as-prepared glasses as well as those subjected to isothermal holds during viscosity measurement are calculated by the same set of parameters combined with a stretched exponent of β = 3/ 7, which is derived from the Phillips diffusion-trap model of stretched exponential relaxation.11,51−53 The initial thermal history of these commercial glasses is set by the fusion downdraw process54 through which they were manufactured without annealing. We find that the model gives good fits to the viscosity data at all three temperatures for both EAGLE XG (Figure 2a) and Jade (Figure 2b) glasses. Hence, the model accurately represents the temperature and thermal history dependence of nonequilibrium viscosity and demonstrates that the nonequilibrium glassy dynamics are directly linked with equilibrium liquid dynamics. This is the first time such a direct connection between equilibrium and nonequilibrium viscosity has been demonstrated. We have previously shown that the phenomenological model (eq 10) also provides an excellent fit to the experimental data.16,50 Figure 3 shows a direct comparison of eqs 10 and 15 in fitting the nonequilibrium dynamics of EAGLE XG glass. Here, the comparison is done by plotting viscosity as a function of fictive temperature. The two models have approximately the same fitting quality. This suggests that it is a good approximation to assume that the nonequilibrium viscosity scales linearly with fictive temperature (with slope B). This is shown in Figure 4, in which the exponential term in eq 15 is

III. RESULTS AND DISCUSSION The previously used phenomenological model (eq 10) for nonequilibrium viscosity provides excellent agreement with experimental measurements of nonequilibrium viscosity. Hence, despite the inherent advantages of the new model (eq 15), it should also be able to accurately capture the temperature and thermal history dependence of nonequilibrium viscosity. We first address this problem using the enthalpy landscape model of Mauro and Loucks for selenium,9 an elemental glass former. This model is derived from fundamental physics, combining ab initio-derived interatomic potentials40 with the enthalpy landscape approach41−43 and nonequilibrium statistical mechanics techniques.44−46 First, the continuous enthalpy landscape is mapped to a discrete set of inherent structures and transition points.43 Then the inherent structure density of states is calculated,47 and finally the dynamics of the selenium glass-forming system are computed using a master equation solver accounting for a continuous breakdown of ergodicity at the glass transition.35 This model enables the calculation of nonequilibrium glass viscosity over a wide range of thermal histories without any empirical fitting parameters.16 The model has previously been validated against the experimental data of Varshneya and co-workers.48,49 Here, we apply the selenium landscape model to compute ηne with cooling rates from 10−12 to 1012 K/s, as shown in Figure 1 for a fixed temperature of 212 K (Tg/T = 1.5). The nonequilibrium viscosity is lower for the faster cooling rates (higher fictive temperature), since the glass is trapped in a metabasin with more available transition paths. This corresponds to a lower effective free energy barrier for structural transitions; i.e., the glass viscosity decreases with increasing fictive temperature. Figure 1 shows that both the phenomenological model (eq 10) and the theoretically derived model (eq 15) account for this qualitative dependence. Indeed, we find 3228

DOI: 10.1021/acs.jpcb.6b00141 J. Phys. Chem. B 2016, 120, 3226−3231

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The Journal of Physical Chemistry B

Figure 2. Experimental and model viscosities as a function of time (t) during isothermal holds at three different temperatures for (a) EAGLE XG and (b) Jade glasses. The model fits are done using eq 15. The errors are smaller than the size of the lines.

Table 1. Parameters Obtained When Fitting the New Model (Eq 15) to the High-Temperature (Equilibrium) and LowTemperature (Nonequilibrium) Viscosity Data for Both EAGLE XG and Jade Glassesa EAGLE XG JADE

log (η∞/Pa·s)

Tg (°C)

m

p

A

ΔH/(k ln10)

S∞/(k ln10)

−2.9 −2.9

735 794

35.3 36.8

10.88 11.34

46.19 46.19

4136.73 4136.73

135.09 149.40

η∞ is the infinite temperature limit of liquid viscosity, Tg the glass transition temperature, m the liquid fragility index, p a constant governing the sharpness of the glass transition, A the glass viscosity in the limit of Tf → 0 and T → ∞, ΔH the activation enthalpy for isostructural flow, and S∞ the infinite temperature configurational entropy. a

Figure 3. Experimental and model viscosities as a function of fictive temperature (Tf) at three different heat treatment temperatures for EAGLE XG glass. The model fits are done using both the phenomenological model (eq 10) and the theoretically derived model (eq 15).

Figure 4. Exponential term of viscosity in eq 15 plotted as a function of fictive temperature (Tf) at three different heat treatment temperatures for EAGLE XG glass. The solid lines represent linear fits representative of the original MAP model for nonequilibrium viscosity.16

plotted against fictive temperature. The linearity only breaks down at sufficiently high fictive temperatures. The new model of eq 15 can thus be used to describe accurately the nonequilibrium viscosity over a wide range of temperatures and fictive temperature. It also opens up a way for understanding and predicting the composition (x) dependence of nonequilibrium viscosity, with important implications for advanced glass applications. This is because the composition dependence of Tg and m in eq 15 can already be calculated as a function of composition using topological constraint theory.55 As originally developed by Phillips and Thorpe,56−59 constraint theory considers each atom in the glass as having translational degrees of freedom that are removed by the imposition of radial and angular bond constraints. Topological constraint theory was originally derived for zero temperature conditions but has recently been extended by Gupta and Mauro60,61 to include an explicit temperature dependence of the bond constraints. Rigid bond constraints set in as the system is cooled, where the strongest bonds are the first ones to become rigid.62−65 Each

time a new rigid constraint is imposed, there is a loss of one floppy mode of deformation in the glass network. Naumis showed that the configurational entropy of the system is largely proportional to the number of these floppy modes.66,67 When the floppy modes are removed through the onset of rigid bond constraints, there is a concurrent loss of configurational entropy. The glass transition temperature has been shown to scale inversely with the available topological degrees of freedom.61,62 A greater number of floppy modes lead to a lower glass transition temperature, since more thermal energy must be taken from the system before it becomes completely rigid. Liquid fragility scales proportionally with the temperature derivative of configurational entropy.8,25,68−70 Hence, the composition dependence of both Tg and m can be predicted. In order to predict the composition dependence of nonequilibrium viscosity, we still need to account for the composition dependence of four parameters: p, A, ΔH, and S∞. However, all of these parameters are governed by the equilibrium viscosity parameters, viz., the fragility and the 3229

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(9) Mauro, J. C.; Loucks, R. J. Selenium Glass Transition: A Model Based on the Enthalpy Landscape Approach and Nonequilibrium Statistical Mechanics. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 76, 174202. (10) Smedskjaer, M. M.; Huang, L.; Scannell, G.; Mauro, J. C. Elastic Interpretation of the Glass Transition in Aluminosilicate Liquids. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 85, 144203. (11) Welch, R. C.; Smith, J. R.; Potuzak, M.; Guo, X. J.; Bowden, B. F.; Kiczenski, T. J.; Allan, D. C.; King, E. A.; Ellison, A. J.; Mauro, J. C. Dynamics of Glass Relaxation at Room Temperature. Phys. Rev. Lett. 2013, 110, 265901. (12) Ellison, A. J.; Cornejo, I. A. Glass Substrates for Liquid Crystal Displays. Int. J. Appl. Glass Sci. 2010, 1, 87−103. (13) Varshneya, A. K. Chemical Strengthening of Glass: Lessons Learned and Yet To Be Learned. Int. J. Appl. Glass Sci. 2010, 1, 131− 142. (14) Tandia, A.; Vargheese, K. D.; Mauro, J. C.; Varshneya, A. K. Atomistic Understanding of the Network Dilation Anomaly in IonExchanged Glass. J. Non-Cryst. Solids 2012, 358, 316−320. (15) Varshneya, A. K.; Mauro, J. C. Comment on Misconceived ASTM Definition of “Glass”. Glass Technol.: Eur. J. Glass Sci. Technol. A 2010, 51, 28−30. (16) Mauro, J. C.; Allan, D. C.; Potuzak, M. Nonequilibrium Viscosity of Glass. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 80, 094204. (17) Yue, Y. Z. The Iso-Structural Viscosity, Configurational Entropy and Fragility of Oxide Liquids. J. Non-Cryst. Solids 2009, 355, 737− 744. (18) Gupta, P. K.; Heuer, A. Physics of the Iso-Structural Viscosity. J. Non-Cryst. Solids 2012, 358, 3551−3558. (19) Mauro, J. C.; Loucks, R. J.; Gupta, P. K. Fictive Temperature and the Glassy State. J. Am. Ceram. Soc. 2009, 92, 75−86. (20) Chodhury, P.; Pal, S. K.; Ray, H. S. On the Prediction of Viscosity of Glasses from Optical Basicity. J. Appl. Phys. 2006, 100, 113502. (21) Avramov, I.; Russel, C.; Keding, R. Effect of Chemical Composition on Viscosity of Oxide Glasses. J. Non-Cryst. Solids 2003, 324, 29−35. (22) Leko, V. K.; Mazurin, O. V. Analysis of Regularities in Composition Dependence of the Viscosity for Glass-Forming Oxide Melts: I. Viscosity of Melts in the Na2O-RO-SiO2 and Na2O-R2OSiO2 Systems. Glass Phys. Chem. 2000, 26, 612−614. (23) Mauro, J. C.; Yue, Y. Z.; Ellison, A. J.; Gupta, P. K.; Allan, D. C. Viscosity of Glass-Forming Liquids. Proc. Natl. Acad. Sci. U. S. A. 2009, 106, 19780−19784. (24) Mauro, J. C.; Ellison, A. J.; Allan, D. C.; Smedskjaer, M. M. Topological Model for the Viscosity of Multicomponent GlassForming Liquids. Int. J. Appl. Glass Sci. 2013, 4, 408−413. (25) Angell, C. A. Formation of Glasses from Liquids and Biopolymers. Science 1995, 267, 1924−1935. (26) Zheng, Q. J.; Mauro, J. C.; Ellison, A. J.; Potuzak, M.; Yue, Y. Z. Universality of the High-Temperature Viscosity Limit of Silicate Liquids. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 212202. (27) Stillinger, F. H.; Weber, T. A. Hidden Structure in Liquids. Phys. Rev. A: At., Mol., Opt. Phys. 1982, 25, 978−989. (28) Stillinger, F. H.; Weber, T. A. Dynamics of Structural Transitions in Liquids. Phys. Rev. A: At., Mol., Opt. Phys. 1983, 28, 2408−2416. (29) Sastry, S.; Debenedetti, P. G.; Stillinger, F. H. Signatures of Distinct Dynamical Regimes in the Energy Landscape of a GlassForming Liquid. Nature 1998, 393, 554−557. (30) Stillinger, F. H. Supercooled Liquids, Glass Transitions, and the Kauzmann Paradox. J. Chem. Phys. 1988, 88, 7818−7825. (31) Debenedetti, P. G.; Stillinger, F. H.; Truskett, T. M.; Roberts, C. J. The Equation of State of an Energy Landscape. J. Phys. Chem. B 1999, 103, 7390−7397. (32) Debenedetti, P. G.; Stillinger, F. H. Supercooled Liquids and the Glass Transition. Nature 2001, 410, 259−267.

glass transition temperature, for the following reasons. First, p is the ergodicity exponent, which governs the sharpness of the continuous glass transition. Mauro et al.16 have shown that p scales linearly with m. Second, A is the glass viscosity in the limit of Tf → 0 and T → ∞, which has also been shown to decrease linearly with increasing fragility.16 Third, ΔH is the activation enthalpy for isostructural flow. While the thermal history dependence of ΔH is generally small,16 it increases with increasing glass transition temperature and fragility.71 The scaling of ΔH with m is linear with Tg as the scaling parameter.16 Fourth, S∞ is the infinite temperature configurational entropy, which relates to the high-temperature limit of liquid viscosity, which is generally taken as a constant.26 Thus, the composition dependence of nonequilibrium viscosity is directly related to that of the equilibrium liquid dynamics (through Tg(x) and m(x)). This is a remarkable result in that it shows how nonequilibrium viscosity is related to equilibrium viscosity parameters and directly to the underlying topological constraints of the glass network.

IV. CONCLUSIONS We have introduced a new functional form for the nonequilibrium glass viscosity. It has been derived based on enthalpy landscape and temperature-dependent constraint theories and has been validated against experimental measurements of Corning EAGLE XG and Jade glasses and computed viscosities for selenium over a wide range of thermal histories. We find that the model gives excellent fits to the viscosity data. This implies that the nonequilibrium glassy dynamics are directly governed by equilibrium liquid dynamics, as quantified by liquid viscosity parameters such as glass transition temperature and fragility index.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected]; Tel +45 5142 7627 (M.M.S.). *E-mail [email protected]; Tel +1 607 974 2185 (J.C.M.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful for valuable discussions with D. C. Allan, T. J. Kiczenski, and A. J. Ellison of Corning Incorporated. The beam-bending viscosity measurements were conducted by M. Potuzak.

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REFERENCES

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DOI: 10.1021/acs.jpcb.6b00141 J. Phys. Chem. B 2016, 120, 3226−3231

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DOI: 10.1021/acs.jpcb.6b00141 J. Phys. Chem. B 2016, 120, 3226−3231