A high-resolution, temperature oscillation-based probe of physical aging in complex systems is introduced. The Fourier analysis of the measured responses allows one to extract high-order, aging-related nonlinearities that are not accessible via traditional temperature-jump and temperature-ramp procedures. To demonstrate the potential of this oscillatory approach, we analyze the periodic time evolution of glycerol’s structural relaxation using shear rheology as a vehicle. Thereby, we access up to the sixth harmonic and detect aging fingerprints within a resolution range of three orders of magnitude for temperature amplitudes of up to 4 K. The even harmonics are present since aging is not symmetrical with respect to the direction of temperature change. The high-order aging coefficients obtained for glycerol are described reasonably well within the Tool–Narayanaswamy–Moynihan formalism.

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 techniques^{3} 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 (1*ω*), also the third (3*ω*) 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 experiments^{9} 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-linear^{11} 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 asymmetry^{15} 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/(2π*τ*) and (2) the sub-mHz frequency *ν*_{T} of the externally imposed sinusoidal temperature modulation *T*(*t*) = *T*_{b} + Δ*T* sin(2π*ν*_{T}*t*) with amplitude Δ*T*.^{22} Base temperatures *T*_{b} above as well as below glycerol’s calorimetric glass transition temperature *T*_{g} = 189 K^{23} 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*′ + i*G*″ 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 Δ*T* must not be too large. As the inset of Fig. 1 reveals, *s* may change when employing Δ*T* 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 Δ*T* = 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 even^{25} 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.

*δ*(

*t*)), is set by deviations from the (ideally) sinusoidal temperature input. Using the present setup, the green squares in Fig. 3 show that for their relative intensity, we find

*θ*

_{n}/

*θ*

_{1}≤ 10

^{−3}. Here, the

*n*th-order thermal intensity

*θ*

_{n}= $|Xn*|$ is defined via

*X*(

*t*) =

*T*(

*t*). The length of the detection period is designated

*t*

_{p}. As we will demonstrate below, this very low input distortion level enables the reliable detection of up to the sixth harmonic in log(tan

*δ*(

*t*)) output.

^{27}We emphasize that the experimental achievement of a temperature oscillation profile close to a “perfect” sinusoidal is highly non-trivial. Based on the shear responses presented in Fig. 2, Fig. 3 summarizes the

*n*th-order Fourier intensities

*I*

_{n}, defined in analogy to

*θ*

_{n}, but now with

*X*(

*t*) = log(tan

*δ*(

*t*)). This choice for

*X*(

*t*) reflects the fact that to an excellent approximation, log(tan

*δ*), and not for instance tan

*δ*, is proportional to log

*τ*.

At *T*_{b} = 193 K, that is when *T*(*t*) never crosses *T*_{g} for the chosen Δ*T* 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” 2*ω* and 3*ω* 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 *I*_{4} is already nonzero, and even more so for *T*_{b} = 187 K, where the sixth harmonic is still significant.

^{12}but others showed that more elaborate models are necessary to fully understand physical aging far from equilibrium.

^{28}To start with, in this work, we will test the single-parameter approach named after Tool, Narayanaswamy, and Moynihan (TNM). Within the premises of this formalism, the time evolution of

*τ*is given by

*R*is the ideal gas constant,

*A*and

*h*are the Arrhenius parameters describing the equilibrium (

*x*= 0) time evolution of

*τ*in the temperature range of interest. The sinusoidal input is represented by

*T*(

*t*″),

*x*is the so-called nonlinearity parameter, and

*T*

_{f}is the fictive temperature, which, by definition, is the “temperature at which the corresponding liquid structure and properties are frozen in upon cooling.”

^{29}The time variation of

*T*

_{f}can be deduced from

*T*

_{0}=

*T*

_{b}denotes the initial temperature at time

*t*→

*t*

_{0}, the exponent

*β*represents a stretch exponent, and

*q*(

*t*′) =

*dT*(

*t*′)/

*dt*′ represents a rate. For the present analyses, we use the effective equilibrium parameters

*A*= 1.7 × 10

^{−54}s and

*h*= 200 kJ/mol

^{30}and the reported TNM parameters

*x*= 0.29 and

*β*= 0.51.

^{31}

To assess the “trivial” *n*th-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}/*I*_{1} 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 Δ*T* 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 field^{33} (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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**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).

## DATA AVAILABILITY

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

## REFERENCES

*Nonlinear Polymer Rheology: Macroscopic Phenomenology and Molecular Foundation*

^{3}

*Q*

_{0}(ω) in MAOS

*cis*-polyisoprene under uniform steady shear and LAOS

_{3})

_{2}]

_{0.4}(KNO

_{3})

_{0.6}

*α*-relaxation dynamics determined by broadband dielectric spectroscopy

Based on its thermal mass, the high-frequency limit *ν*_{T,max} of our experimental assembly is estimated to be 0.1 Hz at most and in any case much larger than the presently used cycling frequencies. The low-frequency limit *ν*_{T,min} essentially reflects the patience of the experimenter and the long-time stability of the setup.

We performed period-by-period analyses of the data. For each experiment, we found that at least the last two periods were stationary in the sense that their Fourier coefficients deviated by less than 1%, at least for the first seven harmonics. Figure 3 shows these stationary values.

*α*relaxation

The given Arrhenius parameters are obtained by fitting the dielectric data from Ref. 21 in the 193–187 K range.

Note that the log *τ* ∝ *T* approximation results in normalized intensities identical to those characterizing the *T*(*t*) input, which are, therefore, not shown in Fig. 3.