# Fast contribution to the activation energy of a glass-forming liquid

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Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved July 18, 2019 (received for review March 20, 2019)

## Significance

One of the main questions in glass science is what causes the spectacular temperature dependence of the dynamics of a supercooled liquid. Answering this would establish the fundamental physics of glass-forming liquids and, in particular, settle the long-standing controversy about whether or not the relaxation time diverges at a finite temperature. This paper reports experiments on a glass-forming silicone oil, monitoring how the system equilibrates after temperature jumps carried out just below the glass-transition temperature. We find that the high-frequency shear modulus

## Abstract

This paper presents physical-aging data for the silicone oil tetramethyl-tetraphenyl trisiloxane. The density and the high-frequency plateau shear modulus

Physical aging is the term used for changes of material properties over time caused by adjustments of the positions of a system’s atoms or molecules (1⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–16). Although aging and degradation of materials are often due to chemical reactions, physical aging is important for amorphous solids like polymers and oxide glasses during production as well as in subsequent use. The experimental study of physical aging requires measurements of high accuracy and considerable patience; simulations of aging are likewise demanding in terms of computational power requirements (17⇓–19). Aging has been studied by monitoring density (1), enthalpy (2, 3), dc and ac electrical response (20⇓–22), etc. Interestingly, these and other quantities all age in a very similar fashion.

Property changes due to physical aging rarely follow a simple exponential function in time, and even temperature changes as small as 1% usually lead to a response that is far from linear. Physical aging can only be observed just below the system’s glass transition temperature because at lower temperatures, aging takes place on geological time scales. Many studies of physical aging monitor a quantity during and after the system’s glass transition (2⇓–4, 7, 14). A conceptually simpler case involves first equilibrating the system at one temperature by long-time annealing, after which the temperature is changed rapidly to a new, constant value where the full approach to equilibrium is monitored as a function of time. This is referred to as an ideal temperature-jump experiment if no relaxation takes place before the temperature is constant throughout the sample and if the system is monitored until equilibrium has been reached (21, 23). We report below close-to-ideal temperature-jump aging data for a silicone oil.

## The Material Time and Single-Parameter Aging

A breakthrough in the description of physical aging was made in 1971 by Ford Motor Company engineer O. S. Narayanaswamy (2). He showed that physical aging can be described by a standard linear-response-theory–type convolution integral if the integration variable is changed from time to a so-called “material time,” the rate of change of which itself evolves as the structure ages (2). This finding is still not fully understood theoretically (24). Yet, what became known as the Tool–Narayanaswamy (TN) aging formalism has been used in industry for decades. An excellent introduction to the TN formalism can be found in Scherer’s 1986 textbook (7); we use this theoretical framework below to interpret data for the physical aging of the silicone oil DC704 (tetramethyl-tetraphenyl trisiloxane) in temperature-jump experiments.

Consider a temperature jump that starts from a state of equilibrium at the temperature *B*.

According to Narayanaswamy, a glass-forming liquid has what may be thought of as an “internal” clock that quantifies how fast molecular rearrangements take place (2, 7, 28). In equilibrium, the clock rate is constant and equal to the inverse of the structural (alpha) relaxation time. In an aging system, the clock rate changes with time. If the material time is denoted by ξ, at any given time t the aging (clock) rate

The simplest case is that of single-parameter aging (SPA) (2, 7, 32). To introduce this concept, we write for any 2 physical quantities **4** must apply also for out-of-equilibrium conditions.

Since relaxation phenomena are generally thermally activated, we write the aging rate in terms of a time-dependent activation energy as follows:

Fig. 1*A* shows data for the aging of DC704 monitored by measuring the real part of the high-frequency dielectric constant, *B* shows the 3 normalized relaxation functions plotted as a function of the logarithm of the time since the jump. Comparing the up and down jumps to 208.5 K, blue and red, respectively, demonstrates the above-mentioned nonlinear nature of physical aging even for small temperature jumps.

Fig. 1*C* shows the Kovacs–McKenna (KM) aging rate defined (1, 28) by

Eqs. **2**, **3**, and **5** imply for the Kovacs–McKenna aging rate **3** is the same function of R for all jumps, one has by inversion *C*, *Inset*, which shows that the difference of the logarithms of the 2 KM aging rates is a linear function of R.

The 2 solid collapsing curves in the main plot in Fig. 1*C* show the density KM aging rates compensated for the exponential R dependence. This gives the KM aging rate predicted for an infinitesimal temperature jump,

The density data in Fig. 1 conform to the SPA prediction of a linear relation between the activation energy and the property measured; for DC704 this has also been established for the high-frequency shear-mechanical plateau modulus **4** is obeyed because the same prediction applies for a more general version of SPA (Eq. **6**). Recently, glycerol was also shown to conform to the SPA prediction (35), which is interesting because hydrogen-bonding systems are generally more complex than van der Waals bonded systems.

A major challenge in glass science is what determines the activation energy of the dynamics of supercooled liquids. Is it the configurational entropy as in the Adam–Gibbs model from 1965 (36, 37) and later more sophisticated theories (38, 39)? Is it the density as in free-volume models (40, 41)? Is it the high-frequency plateau shear modulus

## Comparing Density and G ∞ Aging

Fig. 2*B* compares the normalized relaxation functions of ρ and *C*; the *A*). Fig. 2*B* shows that ρ and *B*, *Inset* shows that after rescaling the 2 relaxation curves superpose.

The different magnitudes of the fast changes right after **4**) because *C* and ref. 32 relate only to the relaxing part of the quantity X monitored, however, and these tests apply also if **4**) and “general SPA” if the more general relation **6** applies.

## Determining the Aging-Rate Activation Energy

Fig. 2*B* showed that ρ and **4**, i.e., at most one of them obeys rigorous SPA. To investigate this more closely, one needs to find the aging-rate activation energy, which is done by arguing as follows (21). Fig. 3*A* shows the equilibrium dielectric loss of DC704 at 2 temperatures above the glass transition. The loss-peak frequency is identified with the equilibrium aging rate. If temperature is lowered, the equilibrium loss curve moves to lower frequencies. Fig. 3*A* illustrates the so-called shift-factor method of determining the aging rate *B* validates this method of determining the aging rate by showing that it predicts equilibrium aging rates consistent with extrapolations of higher-temperature equilibrium loss-peak data (black circles).

Due to the time it takes to establish a spatially constant temperature profile in our measuring cell after a jump, *A* shows the aging-rate activation energy *C* compares the normalized relaxation function of the activation energy to that of the *B*). The 2 curves superpose to a good approximation.

Is the activation energy proportional to the plateau modulus **4** in which case *A* plots the change of the activation energy from its equilibrium value at the starting temperature divided by the same for *B* the same ratio where instead the long-time limits were subtracted. Finally, Fig. 5*C* demonstrates directly that

## Thermodynamic Description of General Single-Parameter Aging

A single-parameter description of linear scalar thermoviscoelasticity was proposed some time ago, based on the following reasoning (53, 54). If temperature and pressure are externally controlled, each of their complementary variables entropy and volume is a linear combination of the temperature and pressure deviations from equilibrium plus a relaxing variable, i.e., a quantity that cannot change abruptly and which contains all memory of the thermal prehistory. A single-parameter description applies if the 2 relaxing variables are proportional in their time variation (53).

According to the Narayanaswamy recipe, linear scalar thermoviscoelasticity generalizes to the nonlinear case by replacing time by the material time. In the spirit of SPA it is reasonable to expect that the aging-rate activation energy is also a linear combination of temperature, pressure, and the relaxing variable. By expressing the latter in terms of volume, temperature, and pressure, a description of general SPA is arrived at which involves the system’s temperature T, pressure p, and density ρ. Considering only small variations as in the above experiments, a thermodynamic description of general SPA is thus (5, 6, 55)

In ambient-pressure experiments **7** reduces to

Recently Niss (22) presented high-precision data for the aging rate as a function of volume for polyisobutylene 625 subjected to a number of different temperature jumps at ambient pressure (lower part of figure 2a in ref. 22). These data conform to the **7**). Struik’s (5) old polymer shift-factor aging data also obey Eq. **7** to a good approximation (figure 85 in ref. 5). For R-simple systems, i.e., those obeying hidden scale invariance believed to include most van der Waals liquids and metals (56), because of the strong correlations between virial and potential energy, Eq. **7** implies that besides temperature and density, the potential energy determines the aging-rate activation energy. This is the main assumption of the potential-energy-clock model of Adolf et al. (57, 58), which describes well several different experiments.

## Concluding Remarks

We have introduced 2 versions of SPA for a quantity X monitored during aging, rigorous SPA defined by **4**) and general SPA for which **6**), which includes a term that follows the temperature instantaneously on the aging time scale. The density of the silicone oil DC704 obeys general SPA (Fig. 1) (32). The activation energy *C*). This means that

Our findings for the out-of-equilibrium situation of physical aging are consistent with the shoving model for the equilibrium non-Arrhenius temperature dependence of the alpha relaxation time. According to this model, which describes the temperature dependence of the relaxation time of some but not all glass-forming liquids (59), the activation energy may be identified with the elastic work done locally on the surroundings to, for a brief moment, create the space needed for the molecules to rearrange (43, 60, 61).

Which function fits the aging data? The linear *SI Appendix*). This function is the green dashed line in Fig. 2*B*. If

A long-standing discussion in the field is whether or not the relaxation time of a metastable supercooled liquid diverges at a finite temperature (14, 36, 45, 62, 63). Prominent models like the Adam–Gibbs entropy model (36) and the more sophisticated random first-order theory (RFOT) of Wolynes and Lubchenko (64) predict such a divergence, while a recent related approach predicts a zero-temperature divergence of the relaxation time (65). Experimental evidence for a finite-temperature divergence has been reported (66), but other experiments question this conclusion (45, 48). The findings of this paper suggest that, at least for the silicone oil in question, the relaxation time does not diverge at a finite temperature because it is difficult to imagine that

Our results imply that

## Materials and Methods

The setup used is described in *SI Appendix*.

## Acknowledgments

We are indebted to Kristine Niss for insightful discussions on aging over the years as well as her comments on the manuscript. Lisa Roed, Birthe Riechers, and Tage Christensen are also thanked for their comments on the manuscript. This work was supported by the VILLUM Foundation’s grant Matter (16515).

## Footnotes

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^{1}To whom correspondence may be addressed. Email: tihe{at}ruc.dk or dyre{at}ruc.dk.

Author contributions: N.B.O. designed research; T.H., N.B.O., and J.C.D. performed research; T.H. analyzed data; and T.H. and J.C.D. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1904809116/-/DCSupplemental.

Published under the PNAS license.

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