Predicting nonlinear physical aging of glasses from equilibrium relaxation via the material time

Nonlinear susceptibilities and higher-order responses related to physical aging: Wiener-Volterra approach and extended Tool-Narayanaswamy-Moynihan models

Large-amplitude thermal excursions imposed on deeply supercooled liquids modulate the nonlinear time evolution of their structural rearrangements. The consequent aftereffects are treated within a Wiener-Volterra expansion in laboratory time that allows one to calculate the associated physical-aging and thermal response functions. These responses and the corresponding higher-harmonic susceptibilities are illustrated using calculations based on the Tool-Narayanaswamy-Moynihan (TNM) model. The conversion from laboratory to material time is thoroughly discussed. Similarities and differences to field-induced higher-harmonic susceptibilities are illustrated using Lissajous and Cole-Cole plots and discussed in terms of aging nonlinearity parameters. For the Lissajous plots, banana-type shapes emerge, while the Cole-Cole plots display cardioidic and other visually appealing patterns. For application beyond the regime in which conventional single-parameter aging concepts work, the Wiener-Volterra material-time-series is introduced as the central tool. Calculations and analyses within this general framework in conjunction with suitable choices of higher-order memory kernels and employing correspondingly extended TNM models yield at least qualitative agreement with recent large-perturbation physical aging experiments. Implications for differential scanning calorimetry and related methods are discussed. The introduced concepts and analyses provide a solid foundation for a generalized description of nonlinear thermal out-of-equilibrium dynamics of glass forming materials, differing from the nonlinear responses known from rheology and dielectric spectroscopy.

Structural Relaxation Time of a Polymer Glass during Deformation

In order to determine the structural relaxation time of a polymer glass during deformation, a strain rate switching experiment is performed in the steady-state plastic flow regime. A lightly cross-linked poly(methylmethacrylate) glass was utilized and, simultaneously, the segmental motion in the glass was quantified using an optical probe reorientation method. After the strain rate switch, a nonmonotonic stress response is observed, consistent with previous work. The correlation time for segmental motion, in contrast, monotonically evolves toward a new steady state, providing an unambiguous measurement of the structural relaxation time during deformation, which is found to be approximately equal to the segmental correlation time. The Chen-Schweizer model qualitatively predicts the changes in the segmental correlation time and the observed nonmonotonic stress response. In addition, our experiments are reasonably consistent with the material time assumption used in polymer deformation modeling; in this approach, the response of a polymer glass to a large deformation is described by combining a linear-response model with a time-dependent segmental correlation time.

Crossing the Frontier of Validity of the Material Time Approach in the Aging of a Molecular Glass

We studied the physical aging of glycerol in response to upward temperature steps of amplitude ranging from 0.3 to 18 K. This was done using a specially designed experimental setup allowing quick heating of a liquid film while measuring the evolution of its dielectric properties. Despite the nonlinear evolution of these observables for large steps, a fictive temperature could be obtained. In the case of moderate step amplitudes, we checked that the material time approach in its simplest form, the single parameter aging (SPA), applies well. The memory kernel extracted from the quasi-linear regime was used to test its frontiers of validity for significant step amplitudes. We showed that the observations deviate from the prediction of the material time framework and of SPA simultaneously. As these approaches link aging to equilibrium dynamics, our results help set the bounds beyond which new theoretical arguments are needed.

Length Scale of Molecular Motions Governing Glass Equilibration in Hyperquenched and Slow-Cooled Polystyrene

Polymer glasses attain thermodynamic equilibrium owing to structural relaxation at various length scales. Herein, calorimetry experiments were conducted to trace the macroscopic relaxation of slow-cooled (SC) and hyperquenched (HQ) polystyrene (PS) glasses and based on detailed comparisons with molecular dynamics probed by dye reorientation, we discussed the possible molecular process governing the equilibration of PS glasses near the glass transition temperatures (Tg). Both SC and HQ glasses equilibrate owing to the cooperative segment motion above a characteristic temperature (Tc) slightly lower than the Tg. In contrast, below the Tc, the localized backbone motion with an apparent activation energy of 290 ± 20 kJ/mol, involving approximately six repeating units, assists equilibrium recovery of PS glasses on the experimentally accessible time scales. The results possibly indicate the presence of an alternative mechanism other than the α-cooperative process controlling physical aging of materials in their deep glassy states.

On the origin of time-aging-time superposition

Time-aging-time superposition and the concept of single-parameter aging refer to the experimentally verified scenario in which the relaxation profile is shifted as a whole along the logarithmic time or frequency scale during physical aging, i.e., without changing the shape of the susceptibility spectrum or decay function. This homogeneous aspect of aging and structural recovery appears to contrast the heterogeneous nature of structural relaxation in equilibrium. A picture is proposed in which both structural recovery and relaxation are heterogeneous, but lacking a local correlation of time constants. This scenario is consistent with time-aging-time superposition and single-parameter aging, as well as with recovery and relaxation processes being subject to practically the same time constant dispersion.

Fast vs slow physical aging of a glass forming liquid

Using electric fields to initiate the process of physical aging has facilitated measurements of structural recovery dynamics on the time scale of milliseconds. This, however, complicates the interesting comparison with aging processes due to a temperature jump, as these are significantly slower. This study takes a step toward comparing the results of field and temperature perturbations by providing data on field-induced structural recovery of vinyl ethylene carbonate at two different time scales: 1.0 ms at 181 K and 33 s at 169 K, i.e., 4.5 decades apart. It is found that structural recovery is a factor of two slower than structural relaxation in equilibrium, with the latter determined via dielectric relaxation in the limit of linear response. The relation between recovery and relaxation dynamics remains temperature invariant across the present experimental range.

Pressure Scanning Volumetry of Physically Aged Polymer Glasses, Fictive Pressure, and Memory Effect

Physical aging of a glass decreases its volume, V, entropy, and enthalpy, H, toward the equilibrium state values. For glasses usually formed by cooling a melt, the effect is modeled in terms of non-exponential, nonlinear structural relaxation by using a plot of the heat capacity, Cp = (dH/dT)p, against T obtained from differential scanning calorimetry (DSC) cooling and heating scans. A melt becomes glass also on isothermal pressurizing and the glass formed becomes liquid on depressurizing, showing a hysteresis of the sigmoid-shape plot of -(dV/dp)T against p, which resembles the thermal hysteresis observed in the Cp against T plots. By analogy with DSC, it was named pressure scanning volumetry (PSV). Here, we use the known values of non-exponential and nonlinearity parameters β and x and volume of activation for structural relaxation time, ΔV*, of atactic poly(propylene) to investigate the effect of aging pressure, page, of aging time, tage, and of the pressurizing rate on aging features in PSV scans. The scans show a post-p g → l feature on depressurizing before the -(dV/dp)T overshoot peak appears. We provide quantitative plots (i) of the monotonic decrease of V and increase of fictive pressure, pf, with tage and (ii) of the memory (Kovacs) effect in V and pf of the polymer and (iii) provide generic plots of -(dV/dp)T against p for different combinations of β, x, and ΔV*. The study is of academic significance because PSV scans show a change in the density fluctuation response. It is of technological significance in polymer-extrusion processing and it may stimulate the commercial development of computer-controlled, high-pressure equipment.

Size-dependent vitrification in metallic glasses

Reducing the sample size can profoundly impact properties of bulk metallic glasses. Here, we systematically reduce the length scale of Au and Pt-based metallic glasses and study their vitrification behavior and atomic mobility. For this purpose, we exploit fast scanning calorimetry (FSC) allowing to study glassy dynamics in an exceptionally wide range of cooling rates and frequencies. We show that the main α relaxation process remains size independent and bulk-like. In contrast, we observe pronounced size dependent vitrification kinetics in micrometer-sized glasses, which is more evident for the smallest samples and at low cooling rates, resulting in more than 40 K decrease in fictive temperature, Tf, with respect to the bulk. We discuss the deep implications on how this outcome can be used to convey glasses to low energy states.

Dynamic Heterogeneity and Kovacs' Memory Effects in Supercooled Water

Understanding the properties of supercooled water is important for developing a comprehensive theory for liquid water and amorphous ices. Because of rapid crystallization for deeply supercooled water, experiments on it are typically carried out under conditions in which the temperature and/or pressure are rapidly changing. As a result, information on the structural relaxation kinetics of supercooled water as it approaches (metastable) equilibrium is useful for interpreting results obtained in this experimentally challenging region of phase space. We used infrared spectroscopy and the fast time resolution obtained by transiently heating nanoscale water films to investigate relaxation kinetics (aging) in supercooled water. When the structural relaxation of the water films was followed using a temperature jump protocol analogous to the classic experiments of Kovacs, similar memory effects were observed. In particular, after suitable aging at one temperature, water's structure displayed an extremum versus the number of heat pulses upon changing to a second temperature before eventually relaxing to a steady-state structure characteristic of that temperature. A random double well model based on the idea of dynamic heterogeneity in supercooled water accounts for the observations.

Two-step devitrification of ultrastable glasses

The discovery of ultrastable glasses raises novel challenges about glassy systems. Recent experiments studied the macroscopic devitrification of ultrastable glasses into liquids upon heating but lacked microscopic resolution. We use molecular dynamics simulations to analyze the kinetics of this transformation. In the most stable systems, devitrification occurs after a very large time, but the liquid emerges in two steps. At short times, we observe the rare nucleation and slow growth of isolated droplets containing a liquid maintained under pressure by the rigidity of the surrounding glass. At large times, pressure is released after the droplets coalesce into large domains, which accelerates devitrification. This two-step process produces pronounced deviations from the classical Avrami kinetics and explains the emergence of a giant lengthscale characterizing the devitrification of bulk ultrastable glasses. Our study elucidates the nonequilibrium kinetics of glasses following a large temperature jump, which differs from both equilibrium relaxation and aging dynamics, and will guide future experimental studies.

Waiting time dependence of aging

Aging phenomena have been observed in many non-equilibrium systems such as polymers and glasses, where physical properties depend on the waiting time between the starting time of observation and the time when the temperature is changed. The aging is classified into two types on the basis of the waiting time dependence of an instantaneous relaxation time: When the relaxation time is always an increasing function of the waiting time, the aging is called Type I and when it depends on the protocol of the temperature change, the aging is called Type II. Aging of a random walk in three dimensions is investigated when the free energy landscape controlling the jump rate responds to temperature change with a delay. It is shown that the intermediate scattering function of the random walk model exhibits Type II aging. It is also shown that the relaxation time of the free energy landscape can be deduced from the waiting time dependence of the instantaneous relaxation time.

To age or not to age: Anatomy of a supercooled liquid's response to a high alternating electric field

Physical aging and structural recovery are the processes with which the structure of a system approaches equilibrium after some perturbation. Various methods exist, that initiate structural recovery, such as changing the temperature or applying a strong, external static field. This work is concerned with high alternating electric fields and their suitability to study structural recovery and aging. The present work demonstrates that rationalizing the nonlinear dielectric response of a supercooled liquid to high-amplitude ac-fields requires multiple fictive temperatures. This feature is in stark contrast to structural recovery after a temperature down-jump or a considerable increase in the static electric field, for which a single parameter, the fictive temperature or material time, describes the structural change. In other words, the structural recovery from a high ac-field does not adhere to time aging-time superposition, which is so characteristic of genuine aging processes.

Nongeneric structural-relaxation shape of supercooled liquids: Insights from linear and nonlinear experiments on propylene glycol

Currently, there is a debate whether the structural relaxation of polar liquids is more faithfully reflected (i) by the generically shaped response detected by dynamic light scattering or rather (ii) by the slower, more stretched, system-dependent susceptibility response recorded by dielectric spectroscopy. In this work, nonlinearly induced transients probing structural relaxation reveal that near the glass transition, alternative (ii) is appropriate for propylene glycol. Results from shear rheology and from calorimetry corroborate this finding, underscoring the previously advanced notion (Moch et al., Phys. Rev. Lett. 128, 228001, 2022) that the reorientationally probed structural susceptibility of viscous liquids displays a nongeneric spectral shape.

One experiment makes a direct comparison of structural recovery with equilibrium relaxation

For a molecular glass-former, propylene glycol, we directly compare the equilibrium fluctuations, measured as "structural" relaxation in the regime of linear response, with structural recovery, i.e., field induced physical aging in the limit of a small perturbation. The two distinct correlation functions are derived from a single experiment. Because the relaxation time changes only 2% during structural recovery, no aging model is needed to analyze the results. Although being conceptually different processes, dielectric relaxation and recovery dynamics are observed to be identical for propylene glycol, whereas single-particle dynamics as seen by photon correlation spectroscopy are significantly faster. This confirms the notion that structural recovery and aging are governed by all modes observed by dielectric spectroscopy, i.e., including cross correlations, not only by single-particle dynamics. A comparison with analogous results for other materials suggests that the relation between relaxation and recovery time scales may be material specific rather than universal.

From Subaging to Hyperaging in Structural Glasses

We demonstrate nonequilibrium scaling laws for the aging and equilibration dynamics in glass formers that emerge from combining a relaxation equation for the static structure with the equilibrium scaling laws of glassy dynamics. Different scaling regimes are predicted for the evolution of the structural relaxation time τ with age (waiting time t_{w}), depending on the depth of the quench from the liquid into the glass: "simple" aging (τ∼t_{w}) applies for quenches close to the critical point of mode-coupling theory (MCT) and implies "subaging" (τ≈t_{w}^{δ} with δ<1) as a broad equilibration crossover for quenches to nearly arrested equilibrium states; "hyperaging" (or superaging, τ∼t_{w}^{δ^{'}} with δ^{'}>1) emerges for quenches deep into the glass. The latter is cut off by non-mean-field fluctuations that we account for within a recent extension of MCT, the stochastic β-relaxation theory (SBR). We exemplify the scaling laws with a schematic model that quantitatively fits simulation data.

Distance-as-time in physical aging

Although it has been known for half a century that the physical aging of glasses in experiments is described well by a linear thermal-history convolution integral over the so-called material time, the microscopic definition and interpretation of the material time remains a mystery. We propose that the material-time increase over a given time interval reflects the distance traveled by the system's particles. Different possible distance measures are discussed, starting from the standard mean-square displacement and its inherent-state version that excludes the vibrational contribution. The viewpoint adopted, which is inspired by and closely related to pioneering works of Cugliandolo and Kurchan from the 1990s, implies a "geometric reversibility" and a "unique-triangle property" characterizing the system's path in configuration space during aging. Both of these properties are inherited from equilibrium, and they are here confirmed by computer simulations of an aging binary Lennard-Jones system. Our simulations moreover show that the slow particles control the material time. This motivates a "dynamic-rigidity-percolation" picture of physical aging. The numerical data show that the material time is dominated by the slowest particles' inherent mean-square displacement, which is conveniently quantified by the inherent harmonic mean-square displacement. This distance measure collapses data for potential-energy aging well in the sense that the normalized relaxation functions following different temperature jumps are almost the same function of the material time. Finally, the standard Tool-Narayanaswamy linear material-time convolution-integral description of physical aging is derived from the assumption that when time is replaced by distance in the above sense, an aging system is described by the same expression as that of linear-response theory.

Relaxation Dynamics of Biomass-Derived Copolymers With Promising Gas-Barrier Properties

This article presents an experimental study on the relaxation dynamics of a series of random copolymers based on bio-friendly comonomers with interesting gas barrier properties. We analyze the relaxation response in the glassy and ultraviscous regime of poly (trimethylene furanoate/sebacate) random copolymers via dielectric spectroscopy. We report lower values of dynamic fragility [a dimensionless index introduced in 1985 (Angell, Relaxations in Complex Systems, 1985)] in comparison to popular polyesters widely used in industry, such as poly (ethylene terephthalate), suggesting that the amorphous phase of these furanoate-based polyesters adopt an efficient chain packing. This is consistent with their low permeability to gases. We also discuss on different equations (phenomenological and theory-based approaches) for fitting the temperature-evolution of the alpha relaxation time.

A density scaling conjecture for aging glasses

The aging rate of glasses has traditionally been modeled as a function of temperature, T , andfictive temperature, while density, ρ, is not explicitly included as a parameter. However, this de-scription does not naturally connect to the modern understanding of what governs the relaxationrate in equilibrium. In equilibrium it is well known that the relaxation rate, γeq , depends on tem-perature and density. In addition a large class of systems obey density scaling which means therate specifically depends on the scaling parameter, Γ = e(ρ)/T , where e(ρ) is a system specificfunction. This paper present a generalization of the fictive temperature concept in terms of a fic-tive scaling paramter, Γfic , and a density scaling conjecture for aging glasses in which the agingrate depends on Γ and Γfic .

"Inner clocks" of glass-forming liquids

Providing a physically sound explanation of aging phenomena in non-equilibrium amorphous materialsis a challenging problem in modern statistical thermodynamics. The slow evolution of physical propertiesafter quenches of control parameters is empirically well interpreted via the concept of material time (orinternal clock), based on the Tool-Narayanaswamy-Moynihan (TNM) model. Yet, the fundamental reasonsof its striking success remain unclear. We propose a microscopic rationale behind the material time onthe basis of the linear laws of irreversible thermodynamics and its extension that treats the correspondingkinetic coefficients as state functions of a slowly evolving material state. Our interpretation is based onthe recognition that the same mathematical structure governs both the Tool model and the recently devel-oped non-equilibrium extension of the self-consistent generalized Langevin equation theory (NE-SCGLE),guided by the universal principles of Onsager's theory of irreversible processes. This identification opensthe way for a generalization of the material-time concept to aging systems where several relaxation modeswith very different equilibration processes must be considered, and partially frozen glasses manifest theappearance of partial ergodicity breaking, and hence materials with multiple very distinct inner clocks.