Temperature oscillations provide access to high-order physical aging harmonics of a glass forming melt Free

Subjected to external perturbations, such as those provided by large mechanical loadings or thermal stresses, many materials modify their properties in a way that widely differs from those observed under small-amplitude excitations. Founded on linear-response theory, which describes the latter regime, one of the challenges in this area is to probe and predict the material response under aperiodic, step-like, or oscillatory large-scale excitations of any amplitude. A field of basic and practical relevance in which this approach is witnessing tremendous activity and consequently undergoing rapid progress is rheology. Here, Fourier transform (FT)^1,2 and related techniques³ allowed one to characterize the dynamics of glass forming materials, such as polymers, in great detail. By virtue, for example, of the coupling of various relaxational modes within these materials, apart from the fundamental (1ω), also the third (3ω) and higher harmonics (nω with n > 300) could occasionally be generated.¹ Often, the periodically imposed stresses or strains generate nonthermal states that eventually lead to wear, fatigue, and even complete material failure, a fascinating topic of its own.⁴ 

Cubic and higher-order susceptibility responses provide insights that can hardly be accessed in the linear-response regime, for instance, regarding the dynamically heterogeneous nature and the cooperativity length of the structural relaxation of supercooled liquids,⁵ regarding the effective ionic jump lengths and the emergence of anomalous Wien effects in ionic conductors,⁶ and regarding morphological and dynamical rheological fingerprints that control the functionality of non-associating and associating polymers.^7,8 One of the challenges encountered not only in rheological but also in nonlinear dielectric experiments⁹ is that the linear-response component is typically overwhelmingly large. Hence, clever schemes for its suppression have been devised.⁵ An approach circumventing this problem is to exploit the so-called cross-experiments, where, for example, in rheodielectric spectroscopy,¹⁰ the excitation and the detection channels are well separated.

The field of thermal responses is of similar fundamental and technological relevance. Here, modulated calorimetry that implements both excitation and detection on the thermal channel is usually carried out in the quasi-linear¹¹ regime. We are aware of only a few modulated cross-experiments that employ dielectric detection of the time dependent structural relaxation (in the quasi-linear regime) or of its thermally induced response.^12,13 The corresponding changes, called physical aging, are otherwise traditionally probed using temperature up- or down-jumps.¹⁴ With the asymmetry¹⁵ of the jump response being questioned,¹⁶ and avoiding cumbersome experimental implementations of a temperature “step,” in the present work, we devise a cross-experiment featuring thermal modulation and, in our case, rheological detection that taps the full FT potential for investigating nonequilibrium phenomena.

Analogous to the nonlinearity-probing FT rheology,¹ we are introducing “FT physical aging” spectroscopy with the goal of increasing the resolution framework in which temperature-related nonequilibrium phenomena can be investigated. In FT physical aging, oscillations of temperature are employed on the input channel and physical aging is monitored in terms of a suitable output observable. As detailed below, in this work the observable is the shear rheological loss tangent that is detected at a fixed shear frequency. Although this approach can be used for any complex system, in the present work, we exploit it to monitor the periodic structural recovery for glycerol, a paradigmatic glass former. For such materials, FT physical aging not only provides access to high-order material individualities (similar to FT rheology) but can also be used to test and potentially to discriminate various descriptions of their structural recovery.^12,17–20

For the development of the method, we chose glycerol,²¹ mainly for its low tendency to crystallize. This glass former (from Sigma-Aldrich, stated purity >99%) was investigated as received in an MCR 502 rheometer from Anton-Paar. While oscillating the temperature, the complex shear response was probed in a 4 mm geometry using small strain amplitudes (typ. 0.03%) at a frequency ν_s. The latter was chosen to be much larger than (1) the inverse molecular relaxation time 1/(2πτ) and (2) the sub-mHz frequency ν_T of the externally imposed sinusoidal temperature modulation T(t) = T_b + ΔT sin(2πν_Tt) with amplitude ΔT.²² Base temperatures T_b above as well as below glycerol’s calorimetric glass transition temperature T_g = 189 K²³ were chosen. As shown later in this paper, the thermally well insulated rheometer oven ensured a highly accurate oscillatory temperature input.

The shear response will be discussed in terms of the loss tangent tan δ, where δ refers to the phase lag between the applied strain and the detected stress. Since both parts of the complex shear modulus G* = G′ + iG″ are proportional to the instantaneous shear modulus, G_∞(T), the loss tangent, tan δ = G″/G′, is insensitive to temperature variations of this quantity. The inset of Fig. 1 illustrates the frequency dependent quantity log(tan δ) that we isothermally recorded for glycerol above and slightly below T_g. In the T_b range from 193 to 187 K, the tan δ spectra follow power laws. Hence, at a fixed frequency ν_s, the (vertical) log(tan δ) variation directly gauges the (horizontal) change in the instantaneous structural relaxation time τ. The proportionality between log(tan δ) and log τ is given by the slope s of the lines in the inset of Fig. 1. For this simple relationship to hold, the oscillation amplitude ΔT must not be too large. As the inset of Fig. 1 reveals, s may change when employing ΔT larger than currently used.

As the sine curves in Fig. 1 schematically illustrate, in the end, it is important how the T oscillation modulates log τ, irrespective of the particular choice for the quantity that is probed, in our case log(tan δ), or even of the experimental method, in our case shear rheology. Similar to traditional step-like experiments,²⁴ the vehicle used to detect aging should not matter since it is not expected to alter the final outcome of the structural recovery.

Figure 2(a) presents the rheological log(tan δ(t)), probed at a shear frequency of ν_s = 20 Hz, induced by oscillating the temperature with frequency ν_T = 0.5 mHz and amplitude ΔT = 3 K about a base temperature T_b = 193 K. In this experiment, the temperature does not cross glycerol’s calorimetric T_g and the appearance of the log(tan δ(t)) response is essentially symmetric with respect to the heating and cooling lobes. This is different for the tests carried out at 190 ± 3 K (for ν_T = 0.5 mHz and ν_s = 12 Hz) and at 187 ± 4 K (for ν_T = 0.33 mHz and ν_s = 0.3 Hz), with the corresponding results shown in Figs. 2(b) and 2(c), respectively. The cycling about T_b = 187 K mostly explores the range below T_g, i.e., the glassy state. Here, physical aging clearly affects the temperature modulated output signal. Considering the asymmetry of the latter, not only odd but also even²⁵ harmonics can be expected to occur in the Fourier analysis of the log(tan δ(t)) response. In other words, since aging dynamics depend on the absolute value of the temperature, which is constantly altered during the oscillation cycle, and on the direction of temperature change,²⁶ a symmetry forbidding the appearance of even harmonics does not exist. In fact, even harmonics are expected to be present in temperature oscillation aging experiments.

At T_b = 193 K, that is when T(t) never crosses T_g for the chosen ΔT and ν_T, the appearance of the weak second and third harmonics in Fig. 3(a) reveals that log τ obeys Arrhenius or VFT rather than linear temperature dependences [the latter implies similar normalized Fourier coefficients for the log τ(t) output and T(t) input]. Considering that deviations from log τ ∝ T are fairly indiscernible from Fig. 1, the presence of “equilibrium” 2ω and 3ω components attests to the high sensitivity of the present experimental approach.

While for T_b = 193 K, the harmonics with n ≥ 4 are all buried in the noise, this changes for T_b = 190 K, where the intensity I₄ is already nonzero, and even more so for T_b = 187 K, where the sixth harmonic is still significant.

To assess the “trivial” nth-order contributions to the output intensities, Fig. 2 includes log(tan δ(t)) estimates with T_f in Eq. (3) set equal to the thermodynamic temperature T. This condition suppresses any dependence on the nonlinearity parameter x, as Eq. (2) reduces to an Arrhenius law that approximates the equilibrium temperature dependence of the relaxation time. Figure 3 shows that for T_b = 193 K, the resulting “trivial” intensities corresponding to the Arrhenius and the VFT approximations match the experimental I_n/I₁ results well.³² However, for T_b = 190 and 187 K, the experimentally determined intensities are significantly larger than the calculated background nonlinearities, demonstrating that these cannot be attributed to equilibrium structural relaxation.

Exploiting the proportionality between log τ and log(tan δ) (see the inset of Fig. 1), we performed TNM calculations also for the nontrivial case. At this point, we emphasize that this proportionality as well as the temperature insensitivity noted above just simplifies the modeling. Therefore, choosing an observable other than log(tan δ), as long as it couples to the structural relaxation, could easily be implemented in the description.

For the nontrivial TNM calculations, we drop the condition for T_f and then solve Eq. (3) self-consistently for this quantity. Figure 2 shows that the resulting predictions for log(tan δ(t)) closely trace the experimental data, thereby providing additional confidence in the parameters used for the TNM analysis. The higher-order intensities displayed in Figs. 3(b) and 3(c) allow for a more sensitive check of the oscillatory aging approach: The calculated intensities furnish a more or less quantitative description of all of the measured intensities. Thus, our results demonstrate that the present method can be used to access feeble nonlinear fingerprints of relaxation processes, similar to what is achieved using FT rheology.^1,2 Although the main goal of this work is to introduce a new method to detect high-order nonlinearities related to physical aging, we note that the lack of “excellent” agreement between the experiment and TNM estimations implies that either the literature parameters are not the most reliable ones, or that overall, this approach is not the most suitable one to describe the obtained results.

We note that systematically increasing the oscillation amplitudes ΔT from mK to several K will open the exciting possibility to experimentally relate the structural fluctuation regime to that governed by structural recovery. Furthermore, complementing the concept of fictive field³³ (instead of fictive temperature), the introduced approach facilitates a generalized description of nonlinear dielectrically, rheologically, and thermally induced responses of glass forming materials.

To conclude, this Communication adopts the Fourier transform approach used in frequency domain high-field dissipation studies (such as nonlinear dielectric and mechanical experiments) to access high-order nonlinearities associated with physical aging of glass forming materials. From an experimental standpoint, the employed temperature oscillation protocol precludes some pitfalls of traditional isothermal aging tests, such as finite rate and overshoots/undershoots of intended step-like temperature changes and partial aging occurring during thermal equilibration. As demonstrated here for glycerol by means of shear rheological detection, temperature oscillation aging experiments provide access to high-order coefficients of time-dependent relaxation times, unraveling up to the sixth harmonic.

These high-resolution aging results could be described well using the TNM approach, and they can be further used to test or to develop alternative theoretical aging descriptions. In future work, we plan to explore the systematic variation in the oscillation frequency for a given base temperature in order to additionally address aging processes in terms of the corresponding susceptibilities. To conclude, the present work supplies a powerful tool for the study of nonequilibrium phenomena in condensed matter and opens new venues for nonlinear probes in a multidimensional time-temperature space.

The authors gratefully acknowledge the financial support provided by the Deutsche Forschungsgemeinschaft under Grant No. 461147152.

The authors have no conflicts to disclose.

Kevin Moch: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal). Roland Böhmer: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal). Catalin Gainaru: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.