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 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 ( 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 νs. The latter was chosen to be much larger than (1) the inverse molecular relaxation time 1/(2πτ) and (2) the sub-mHz frequency νT of the externally imposed sinusoidal temperature modulation T(t) = Tb + ΔT sin(2πνTt) with amplitude ΔT.22 Base temperatures Tb above as well as below glycerol’s calorimetric glass transition temperature Tg = 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 Tg. In the Tb range from 193 to 187 K, the tan δ spectra follow power laws. Hence, at a fixed frequency νs, 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.

FIG. 1.

In the covered temperature range, glycerol’s dielectric relaxation times,21 i.e., log τ (blue solid circles), depend approximately (i) linearly on T (green dashed line). With the goal to describe the same data points, the red dotted-dashed line represents (ii) an Arrhenius law, log τ ∝ 1/T, and the blue dotted line represents (iii) a Vogel–Fulcher–Tammann (VFT) law, log τ ∝ 1/(TTK) with a fit variable TK.21 The sinusoidal lines schematically visualize how a temperature oscillation leads to a modulated log τ. Note that the amplitudes do not correspond to those employed in the actual experiments. The inset presents glycerol’s frequency dependent shear loss tangent, log(tan δ). This quantity is probed in the stress–strain linear-response regime and shown here for two of the employed base temperatures. The red solid lines demonstrate the approximate linear relation between log(tan δ) and log ν. The vertical arrows indicate the frequencies, νs, used for the shear measurements during the temperature oscillations. Note that the temperature oscillation frequency is νT ≪ 10−3 νs. The crossed arrows schematically illustrate that a temperature-induced (horizontal) change in log ν leads to a (vertical) tan δ variation.

FIG. 1.

In the covered temperature range, glycerol’s dielectric relaxation times,21 i.e., log τ (blue solid circles), depend approximately (i) linearly on T (green dashed line). With the goal to describe the same data points, the red dotted-dashed line represents (ii) an Arrhenius law, log τ ∝ 1/T, and the blue dotted line represents (iii) a Vogel–Fulcher–Tammann (VFT) law, log τ ∝ 1/(TTK) with a fit variable TK.21 The sinusoidal lines schematically visualize how a temperature oscillation leads to a modulated log τ. Note that the amplitudes do not correspond to those employed in the actual experiments. The inset presents glycerol’s frequency dependent shear loss tangent, log(tan δ). This quantity is probed in the stress–strain linear-response regime and shown here for two of the employed base temperatures. The red solid lines demonstrate the approximate linear relation between log(tan δ) and log ν. The vertical arrows indicate the frequencies, νs, used for the shear measurements during the temperature oscillations. Note that the temperature oscillation frequency is νT ≪ 10−3 νs. The crossed arrows schematically illustrate that a temperature-induced (horizontal) change in log ν leads to a (vertical) tan δ variation.

Close modal

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 νs = 20 Hz, induced by oscillating the temperature with frequency νT = 0.5 mHz and amplitude ΔT = 3 K about a base temperature Tb = 193 K. In this experiment, the temperature does not cross glycerol’s calorimetric Tg 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 νT = 0.5 mHz and νs = 12 Hz) and at 187 ± 4 K (for νT = 0.33 mHz and νs = 0.3 Hz), with the corresponding results shown in Figs. 2(b) and 2(c), respectively. The cycling about Tb = 187 K mostly explores the range below Tg, 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.

FIG. 2.

The symbols represent the shear loss tangent probed for glycerol at the indicated shear frequencies νs during temperature oscillations with frequencies νT and amplitudes ΔT about the base temperatures Tb. The black solid lines are calculations based on the Tool–Narayanaswamy–Moynihan model; cf. Eqs. (2) and (3). The other lines reflect “trivial” equilibrium expectations defined by assuming that fictive and thermodynamic temperatures agree. The corresponding lines refer to the linear, Arrhenius, and VFT dependences of log τ(T); see Fig. 1.

FIG. 2.

The symbols represent the shear loss tangent probed for glycerol at the indicated shear frequencies νs during temperature oscillations with frequencies νT and amplitudes ΔT about the base temperatures Tb. The black solid lines are calculations based on the Tool–Narayanaswamy–Moynihan model; cf. Eqs. (2) and (3). The other lines reflect “trivial” equilibrium expectations defined by assuming that fictive and thermodynamic temperatures agree. The corresponding lines refer to the linear, Arrhenius, and VFT dependences of log τ(T); see Fig. 1.

Close modal
The detection limit of the harmonics in the output channel, log(tan 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 θn/θ1 ≤ 10−3. Here, the nth-order thermal intensity θn = |Xn*| is defined via
Xn*=Xn+iXn=2tp0tpX(t)e2πiνTntdt
(1)
with |Xn*|=(Xn)2+(Xn)2 and X(t) = T(t). The length of the detection period is designated tp. 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 nth-order Fourier intensities In, defined in analogy to θn, 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.
FIG. 3.

Normalized (with respect to the intensity of the fundamental mode) nth-order Fourier coefficients obtained for temperature oscillation signals generated at (a) 193 K with ΔT = 3 K, (b) 190 K with ΔT = 3 K, and (c) 187 K with ΔT = 4 K: Shown are the intensities In/I1 of the measured output, log(tan δ) ∝ log τ (black filled circles), of the TNM model (plus symbols), and of the “trivial” log(tan δ) calculations32 (for visual clarity represented as dotted-dashed and dotted lines). Clearly, the higher-order output signals (black filled circles) grow much above the “trivial” signals as the base temperature is lowered. The normalized intensity θn/θ1 of the temperature input (green open squares) sets the resolution limit for the experimentally determined output coefficients as also highlighted by the green dashed horizontal lines.

FIG. 3.

Normalized (with respect to the intensity of the fundamental mode) nth-order Fourier coefficients obtained for temperature oscillation signals generated at (a) 193 K with ΔT = 3 K, (b) 190 K with ΔT = 3 K, and (c) 187 K with ΔT = 4 K: Shown are the intensities In/I1 of the measured output, log(tan δ) ∝ log τ (black filled circles), of the TNM model (plus symbols), and of the “trivial” log(tan δ) calculations32 (for visual clarity represented as dotted-dashed and dotted lines). Clearly, the higher-order output signals (black filled circles) grow much above the “trivial” signals as the base temperature is lowered. The normalized intensity θn/θ1 of the temperature input (green open squares) sets the resolution limit for the experimentally determined output coefficients as also highlighted by the green dashed horizontal lines.

Close modal

At Tb = 193 K, that is when T(t) never crosses Tg for the chosen ΔT and νT, 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 τT 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 Tb = 193 K, the harmonics with n ≥ 4 are all buried in the noise, this changes for Tb = 190 K, where the intensity I4 is already nonzero, and even more so for Tb = 187 K, where the sixth harmonic is still significant.

Most studies are compatible with a single-parameter aging description,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
τ(t)=AexpxhRT(t)+(1x)hRTf(t),
(2)
where 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 Tf 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 Tf can be deduced from
Tf(t)=T0+t0t1expttdtτ(t)βq(t)dt,
(3)
where T0 = Tb denotes the initial temperature at time tt0, 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/mol30 and the reported TNM parameters x = 0.29 and β = 0.51.31 

To assess the “trivial” nth-order contributions to the output intensities, Fig. 2 includes log(tan δ(t)) estimates with Tf 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 Tb = 193 K, the resulting “trivial” intensities corresponding to the Arrhenius and the VFT approximations match the experimental In/I1 results well.32 However, for Tb = 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 Tf 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.

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32.

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

33.
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