Communication: Direct tests of single-parameter aging

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.

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.

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.

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 .

Temperature dependence of aging dynamics in highly non-equilibrium model polymer glasses

A central feature of the non-equilibrium glassy “state” is its tendency to age toward equilibrium, obeying signatures identified by Kovacs over 50 years ago. The origin of these signatures, their fate far from equilibrium and at high temperatures, and the underlying nature of the glassy “state” far from equilibrium remain unsettled. Here, we simulate physical aging of polymeric glasses, driven much farther from equilibrium and at much higher temperatures than possible in experimental melt-quenched glasses. While these glasses exhibit Kovacs’ signatures of glassy aging at sufficiently low temperatures, these signatures disappear above the onset TA of non-Arrhenius equilibrium dynamics, suggesting that TA demarcates an upper bound to genuinely glassy states. Aging times in glasses after temperature up-jumps are found to obey an Arrhenius law interpolating between equilibrium dynamics at TA and at the start of the temperature up-jump, providing a zero-parameter rule predicting their aging behavior and identifying another unrecognized centrality of TA to aging behavior. This differs qualitatively from behavior of our glasses produced by temperature down-jumps, which exhibit a fractional power law decoupling relation with equilibrium dynamics. While the Tool–Narayanaswamy–Moynihan model can predict the qualitative single-temperature behavior of these systems, we find that it fails to predict the disappearance of Kovacs signatures above TA and the temperature dependence of aging after large temperature up-jumps. These findings highlight a need for new theoretical insights into the aging behavior of glasses at ultra-high fictive temperatures and far from equilibrium.

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

The noncrystalline glassy state of matter plays a role in virtually all fields of materials science and offers complementary properties to those of the crystalline counterpart. The caveat of the glassy state is that it is out of equilibrium and therefore exhibits physical aging, i.e., material properties change over time. For half a century, the physical aging of glasses has been known to be described well by the material-time concept, although the existence of a material time has never been directly validated. We do this here by successfully predicting the aging of the molecular glass 4-vinyl-1,3-dioxolan-2-one from its linear relaxation behavior. This establishes the defining property of the material time. Via the fluctuation-dissipation theorem, our results imply that physical aging can be predicted from thermal-equilibrium fluctuation data, which is confirmed by computer simulations of a binary liquid mixture.

Single-parameter aging in a binary Lennard-Jones system

This paper studies physical aging by computer simulations of a 2:1 Kob-Andersen binary Lennard-Jones mixture, a system that is less prone to crystallization than the standard 4:1 composition. Starting from thermal-equilibrium states, the time evolution of the following four quantities is monitored by following up and down jumps in temperature: potential energy, virial, average squared force, and the Laplacian of the potential energy. Despite the fact that significantly larger temperature jumps are studied here than in typical similar experiments, to a good approximation, all four quantities conform to the single-parameter-aging scenario derived and validated for small jumps in experiments [T. Hecksher, N. B. Olsen, and J. C. Dyre, J. Chem. Phys. 142, 241103 (2015)]. As a further confirmation of single-parameter aging with a common material time for the four different quantities monitored, their relaxing parts are found to be almost identical for all temperature jumps.

Spatial Heterogeneities in Structural Temperature Cause Kovacs' Expansion Gap Paradox in Aging of Glasses

Volume and enthalpy relaxation of glasses after a sudden temperature change has been extensively studied since Kovacs' seminal work. One observes an asymmetric approach to equilibrium upon cooling versus heating and, more counterintuitively, the expansion gap paradox, i.e., a dependence on the initial temperature of the effective relaxation time even close to equilibrium when heating. Here, we show that a distinguishable-particle lattice model can capture both the asymmetry and the paradox. We quantitatively characterize the energetic states of the particle configurations using a physical realization of the fictive temperature called the structural temperature, which, in the heating case, displays a strong spatial heterogeneity. The system relaxes by nucleation and expansion of warmer mobile domains having attained the final temperature, against cooler immobile domains maintained at the initial temperature. A small population of these cooler regions persists close to equilibrium, thus explaining the paradox.

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

This paper presents physical-aging data for the silicone oil tetramethyl-tetraphenyl trisiloxane. The density and the high-frequency plateau shear modulus [Formula: see text] were monitored following temperature jumps starting from fully equilibrated conditions. Both quantities exhibit a fast change immediately after a temperature jump. Adopting the material-time formalism of Narayanaswamy, we determine from the dielectric loss at 0.178 Hz the time evolution of the aging-rate activation energy. The relative magnitude of the fast change of the activation energy differs from that of the density, but is identical to that of [Formula: see text] In fact, the activation energy is proportional to [Formula: see text] throughout the aging process, with minor deviations at the shortest times. This shows that for the silicone oil in question the dynamics are determined by [Formula: see text] in-as well as out of-equilibrium.

Isomorph theory of physical aging

This paper derives and discusses the configuration-space Langevin equation describing a physically aging R-simple system and the corresponding Smoluchowski equation. Externally controlled thermodynamic variables like temperature, density, and pressure enter the description via the single parameter T_s/T, in which T is the bath temperature and T_s is the "systemic" temperature defined at any time t as the thermodynamic equilibrium temperature of the state point with density ρ(t) and potential energy U(t). In equilibrium, T_s ≅ T with fluctuations that vanish in the thermodynamic limit. In contrast to Tool's fictive temperature and other effective temperatures in glass science, the systemic temperature is defined for any configuration with a well-defined density, even if it is not close to equilibrium. Density and systemic temperature define an aging phase diagram, in which the aging system traces out a curve. Predictions are discussed for aging following various density-temperature and pressure-temperature jumps from one equilibrium state to another, as well as for a few other scenarios. The proposed theory implies that R-simple glass-forming liquids are characterized by the dynamic Prigogine-Defay ratio being equal to unity.

Mapping Isobaric Aging onto the Equilibrium Phase Diagram

The linear volume relaxation and the nonlinear volume aging of a glass-forming liquid are measured, directly compared, and used to extract the out-of-equilibrium relaxation time. This opens a window to investigate how the relaxation time depends on temperature, structure, and volume in parts of phase space that are not accessed by the equilibrium liquid. It is found that the temperature dependence of relaxation time is non-Arrhenius even in the isostructural case-challenging the Adam-Gibbs entropy model. Based on the presented data and the idea that aging happens through quasiequilibrium states, we suggest a mapping of the out-of-equilibrium states during isobaric aging to the equilibrium phase diagram. This mapping implies the existence of isostructural lines in the equilibrium phase diagram. The relaxation time is found to depend on the bath temperature, density, and a just single structural parameter, referred to as an effective temperature.

Model for the alpha and beta shear-mechanical properties of supercooled liquids and its comparison to squalane data

This paper presents data for supercooled squalane's frequency-dependent shear modulus covering frequencies from 10 mHz to 30 kHz and temperatures from 168 K to 190 K; measurements are also reported for the glass phase down to 146 K. The data reveal a strong mechanical beta process. A model is proposed for the shear response of the metastable equilibrium liquid phase of supercooled liquids. The model is an electrical equivalent-circuit characterized by additivity of the dynamic shear compliances of the alpha and beta processes. The nontrivial parts of the alpha and beta processes are each represented by a "Cole-Cole retardation element" defined as a series connection of a capacitor and a constant-phase element, resulting in the Cole-Cole compliance function well-known from dielectrics. The model, which assumes that the high-frequency decay of the alpha shear compliance loss varies with the angular frequency as ω^-1/2, has seven parameters. Assuming time-temperature superposition for the alpha and beta processes separately, the number of parameters varying with temperature is reduced to four. The model provides a better fit to the data than an equally parametrized Havriliak-Negami type model. From the temperature dependence of the best-fit model parameters, the following conclusions are drawn: (1) the alpha relaxation time conforms to the shoving model; (2) the beta relaxation loss-peak frequency is almost temperature independent; (3) the alpha compliance magnitude, which in the model equals the inverse of the instantaneous shear modulus, is only weakly temperature dependent; (4) the beta compliance magnitude decreases by a factor of three upon cooling in the temperature range studied. The final part of the paper briefly presents measurements of the dynamic adiabatic bulk modulus covering frequencies from 10 mHz to 10 kHz in the temperature range from 172 K to 200 K. The data are qualitatively similar to the shear modulus data by having a significant beta process. A single-order-parameter framework is suggested to rationalize these similarities.