Nonequilibrium viscosity of glass

Since glass is a nonequilibrium material, its properties depend on both composition and thermal history. While most prior studies have focused on equilibrium liquid viscosity, an accurate description of nonequilibrium viscosity is essential for understanding the low temperature dynamics of glass. Departure from equilibrium occurs as a glass-forming system is cooled through the glass transition range. The glass transition involves a continuous breakdown of ergodicity as the system gradually becomes trapped in a subset of the available configurational phase space. At very low temperatures a glass is perfectly nonergodic (or “isostructural”), and the viscosity is described well by an Arrhenius form. However, the behavior of viscosity during the glass transition range itself is not yet understood. In this paper, we address the problem of glass viscosity using the enthalpy landscape model of Mauro and Loucks [Phys. Rev. B 76, 174202 (2007)] for selenium, an elemental glass former. To study a wide range of thermal histories, we compute nonequilibrium viscosity with cooling rates from ${10}^{-12}$ to ${10}^{12}\text{K}/\text{s}$. Based on these detailed landscape calculations, we propose a simplified phenomenological model capturing the essential physics of glass viscosity. The phenomenological model incorporates an ergodicity parameter that accounts for the continuous breakdown of ergodicity at the glass transition. We show a direct relationship between the nonequilibrium viscosity parameters and the fragility of the supercooled liquid. The nonequilibrium viscosity model is validated against experimental measurements of Corning EAGLE XG™ glass. The measurements are performed using a specially designed beam-bending apparatus capable of accurate nonequilibrium viscosity measurements up to ${10}^{16}\text{Pa}\text{s}$. Using a common set of parameters, the phenomenological model provides an accurate description of EAGLE XG™ viscosity over the full range of measured temperatures and fictive temperatures.

Figure 1

Average inherent structure enthalpy and density of states as a function of molar volume. Only noncrystalline inherent structures are considered.

Figure 2

(Color online) (a) Volume-temperature diagrams for five selenium glasses cooled from the melting temperature $\left(T_m=490\text{K}\right)$ to 200 K at rates of ${10}^{12}$, ${10}^6$, 1, ${10}^{-6}$, and ${10}^{-12}\text{K}/\text{s}$. The equilibrium supercooled liquid line is also shown. (b) Molar volume at $T=212\text{K}$ for cooling rates ranging from ${10}^{-12}$ to ${10}^{12}\text{K}/\text{s}$. This temperature corresponds to $T_g/T=1.5$ for the 1 K/s cooled glass (which has $T_g=318\text{K}$).

Figure 3

(Color online) (a) Plot of ${\text{log}}_{10}\eta$ as a function of $T_g/T$ for selenium glass prepared using cooling rates of ${10}^{-12}$, ${10}^{-6}$, 1, ${10}^6$, and ${10}^{12}\text{K}/\text{s}$. For plotting purposes we consider a reference glass transition temperature of $T_g=318\text{K}$, which corresponds to the 1 K/s cooled glass. The equilibrium supercooled liquid line is also shown. (b) Plot of ${\text{log}}_{10}\eta$ at $T_g/T=1.5$ $\left(T=212\text{K}\right)$ for selenium with linear cooling rates ranging from ${10}^{-12}$ to ${10}^{12}\text{K}/\text{s}$.

Figure 4

Relaxation of ${\text{log}}_{10}\eta$ of selenium glass toward the equilibrium supercooled liquid value at $T=212\text{K}$ $\left(T_g/T=1.5\right)$. The selenium glass was prepared using a cooling rate of 1 K/s and relaxation is shown on the time scale of (a) minutes and (b) years. Full equilibration occurs on a time scale of billions of years.

Figure 5

(a) Fictive temperature, $T_f$, for selenium after cooling from the melting temperature (490 K) to 200 K with linear cooling rates ranging from ${10}^{-12}$ to ${10}^{12}\text{K}/\text{s}$. (b) Molar volume as a function of fictive temperature (equals supercooled liquid volume as a function of temperature). (c) ${\text{log}}_{10}\eta$ at $T_g/T=1.5$ $\left(T=212\text{K}\right)$ as a function of fictive temperature of the glass. In all plots, the solid line shows a linear fit in the high fictive temperature regime.

Figure 6

(Color online) Computed ${\text{log}}_{10}\eta$ of selenium using the enthalpy landscape approach and fit with the Narayanaswamy, Adam-Gibbs, Mazurin, and Avramov expressions of Eqs. (15, 19, 20, 21), respectively. (a) Viscosity as a function of normalized inverse temperature for selenium glass cooled at a rate of 1 K/s. (b) Viscosity as a function of fictive temperature for selenium glass at $T_g/T=1.5$ $\left(T=212\text{K}\right)$. Fictive temperature is computed using Eq. (14).

Figure 7

(Color online) Computed ${\text{log}}_{10}\eta$ of selenium using the enthalpy landscape approach and fit with the modified Mazurin expression of Eq. (22) and our current nonequilibrium viscosity model of Eq. (23). (a) Viscosity as a function of normalized inverse temperature for selenium glass cooled at a rate of 1 K/s. (b) Viscosity as a function of fictive temperature for selenium glass at $T_g/T=1.5$ $\left(T=212\text{K}\right)$. Fictive temperature is computed using Eq. (14).

Figure 8

Experimental results of deflection at the center of the EAGLE XG™ flat beam as a function of time. Experiments were conducted isothermally at (a) $600°\text{C}$ using an applied uniaxial stress of 61.8 MPa, (b) $650°\text{C}$ using a stress of 61.8 MPa, and $675°\text{C}$ using a stress of 21.6 MPa.

Figure 9

(Color online) EAGLE XG™ deflection vs time measured and various model fits described in the text: (a) at $600°\text{C}$; (b) at $650°\text{C}$; and (c) at $675°\text{C}$.

Figure 10

(Color online) Empirical and model ${\text{log}}_{10}\eta$ vs fictive temperature $T_f$ for EAGLE XG™. The empirical (gray curves) and current model viscosities (red curves) are close together and have higher slope vs $T_f$ than those of the other models, especially at $600°\text{C}$.

Figure 11

(Color online) Empirical and model viscosities vs time for EAGLE XG™. The empirical (gray curves) and current model viscosities (red curves) are close together at each temperature and have a greater change in value over the time interval shown.

Figure 12

Variation in fragility with enthalpy barrier.

Figure 13

(Color online) Computed (a) volume-temperature and (b) ${\text{log}}_{10}\eta$ vs temperature curves for three glass-forming systems with different values of fragility. The cooling rate for all systems is 1 K/s. Increasing the fragility of the supercooled liquid results in a sharper glass transition and a higher nonequilibrium viscosity.

Figure 14

(Color online) Variation in (a) fictive temperature with cooling rate and (b) nonequilibrium viscosity with fictive temperature for three glass-forming systems with different values of fragility.

Figure 15

Scaling of nonequilibrium viscosity parameters with fragility: (a) $A$ from Eq. (27); (b) $B$ from Eq. (27); and (c) $p$ from Eq. (24). The $\Delta H$ parameter from Eq. (27) scales as in Fig. 12.

Figure 16

(Color online) Evolution of the ergodicity parameter $x$ from Eq. (24) for three glass-forming systems with different values of fragility. The cooling rate for all systems is 1 K/s. Increasing the fragility of the supercooled liquid results in a sharper loss of ergodicity (i.e., a sharper glass transition).