Fast temperature up steps as a test of the Tool-Narayanaswamy formalism

arXiv:2607.15988 · cond-mat.soft · Submitted 2026-07-17 · Read on arXiv

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Introduction to the show: ident: Condensed Matter. The day in condensed matter: materials, phases and the measurements under them.

Kai: We're looking at the paper: "Fast temperature up steps as a test of the Tool-Narayanaswamy formalism".

Mira: Here's the gist of the paper. Fast temperature up steps as a test of the Tool-Narayanaswamy formalism investigates aging dynamics in triethyl-2-acetylcitrate (TEAC) following fast temperature up steps with amplitudes ranging from 0.3 to 13.6 K, testing the predictive power of the Tool-Narayanaswamy (TN) formalism for nonlinear relaxation dynamics in glasses Experimental Setup and Protocol The study investigated the aging dynamics of TEAC following fast temperature up steps with amplitudes ranging from 0.3 to 13.6 K, starting from either an equilibrium or an out-of-equilibrium state The initial states were either at equilibrium or prepared at increasing levels of out-of-equilibrium through a prior down step experiment Measurements were performed on a triethyl-2-acetylcitrate sample (Sigma-Aldrich, 99%, Tg = 199 K) through an experimental setup previously described in ref.6 and which we briefly explain here Up-steps from T0 to T1 were performed by applying a tup = 40 ms continuous current pulse through the resistive electrode, which rapidly heats the entire sample due to its thinness (see Fig.

Kai: 1a) Both types of step were characterized (see Fig.

Mira: 1b) by operating at a temperature T0 = 185 K, small enough compared to Tg so that no relaxation occurs on the measurement time window The dielectric permittivity of the liquid ϵ⋆ can thus be used as a thermometer TN Parameter Determination The TN formalism is assumed to link the non-linear re-equilibration of a liquid to its linear response to a small perturbation The authors determined the TN parameters for steps with small to moderate amplitude (≤ 3.3 K) and crucially took advantage of down step aging experiments below the glass transition temperature to constrain the determination of the equilibrium relaxation time For these smaller steps, they found very good agreement for differences in fictive temperature characterizing the distance from equilibrium as high as 10 K The memory kernel M was fitted as a stretched exponential M(ξ) = e−(ξ/τ0)β with parameters τ0 = 0.98 and β = 0.58 Testing the Formalism Limits For larger steps, the prediction progressively fails to capture the aging dynamics, likely indicating that the re-equilibration mechanism is no longer related to the equilibrium dynamics The authors identified a limit for the validity of the TN for full re-equilibration dynamics based on whether ttr, 50% is well predicted or not This limit corresponds to a contrast between the initial and final relaxation time τ (0+)/τ1 of ≈ 102 and a transformation time ttr, 50%/τ1 ≈ 10 for steps with ∆Tf > 10 K Low-Temperature Extrapolation A further difficulty addressed was the extrapolation of the VFT law for τα(T) below Tg, which was constrained using aging experiments following temperature down steps They parameterized τα(T) for T < Tlim by assuming an Arrhenius dependence with an activation energy Ea = 235 kJ·mol−1 and Tlim = 205.3 K This constraint led to a good agreement, especially for T0 = 196 K and 194.6 K The TN formalism proved to be equivalent to the SPA ansatz in the regime explored, by determining its parameters for moderate amplitude up steps (∆T ≤ 3.3 K) and by estimating the equilibrium relaxation time below Tg in a self-consistent manner The discrepancy observed for large steps corresponds to a contrast between the initial and final relaxation time τ (0+)/τ1 of ≈ 102 and a transformation time ttr, 50%/τ1 ≈ 10 This threshold for the limit of validity of the TN appears consistent with previous work from the literature on others systems Comparison to Single Parameter Ansatz The single parameter aging (SPA) ansatz was shown to give an excellent agreement with aging experiments on molecular liquids, and in this work, it was found that the SPA ansatz provides aging predictions of comparable quality to those of the TN approach The two approaches yield similar results because for TEAC, ln τα(T) is almost linear with T This suggests that the simplest expression, Eq.

Title and authors: Kai: So this paper is titled "Fast temperature up steps as a test of the Tool-Narayanaswamy formalism". It’s testing how well the Tool-Narayanaswamy formalism works when you have fast temperature up steps on triethyl-two-acetylcitrate, or TEAC <ref:2607.15988#pg1>.

Mira: I see. So they’re looking at aging dynamics in this glass following these fast temperature up steps with amplitudes ranging from zero point three K all the way to thirteen point six K, and they're checking if the TN formalism can predict that non-linear relaxation behavior in glasses correctly <ref:2607.15988#pg1,fast temperature up steps with amplitudes ranging from 0.3>.

Lev: It sounds like they're pushing the limits of what we think this formalism can handle by using these big temperature changes as a test case for its predictive power.

Kai: Exactly. The authors are using TEAC, which has a glass transition temperature of one hundred ninety-nine K, and they start from either an equilibrium state or an already out-of-equilibrium state prepared by some prior down step experiment <ref:2607.15988#pg2>.

Mira: They're setting up this test by performing the up steps at a starting temperature T0 of one hundred eighty-five K, which is small enough compared to the glass transition temperature so that they don't see any relaxation during the measurement time window <ref:2607.15988#pg2>.

Lev: And they’re using the dielectric permittivity of the liquid as a thermometer to track everything happening inside.

Kai: Right. They determined some TN parameters for steps with small to moderate amplitudes, specifically those less than or equal to three point three K, and they used down step aging experiments below the glass transition temperature to constrain how they figure out the equilibrium relaxation time <ref:2607.15988#pg1,down step aging experiments below the glass transition temperature to constrain>.

Mira: For those smaller steps, they found very good agreement for the difference in fictive temperature that characterizes how far you are from equilibrium as high as ten K <ref:2607.15988#pg1,from equilibrium as high as 10 K>.

Lev: That’s a pretty solid result for small steps, but what happens when we push those step amplitudes higher?

Kai: That’s what they test next. They found that for larger steps, the prediction progressively fails to capture the aging dynamics, which suggests that the re-equilibration mechanism isn't related to equilibrium dynamics anymore.

Mira: So they identified a limit for when this TN formalism is valid based on whether a specific relaxation time, ttr, fifty percent, is well predicted or not.

Lev: That distinction between the initial and final relaxation time, tau (zero plus)/tau1 of approximately one hundred two versus a transformation time ttr, fifty percent/tau1 of approximately ten for steps with delta Tf greater than ten K is a key finding here.

Kai: It seems they found that the TN formalism is equivalent to the SPA ansatz in the regime they explored, provided you determine its parameters for moderate amplitude up steps and estimate the equilibrium relaxation time below Tg in a self-consistent way.

Mira: And they found that for TEAC, since ln tau(T) is almost linear with T, this means that even if you use the simpler single parameter aging ansatz, it gives predictions of comparable quality to the TN approach.

Lev: So they’ve shown that for this system, the two approaches yield similar results because of the way TEAC relaxes.

Kai: It seems like a strong validation for using these formalisms in this type of experimental setup when you keep those step amplitudes low enough.

Mira: But the failure at larger steps tells us there’s a fundamental contrast between how fast things relax initially and how they relax after the perturbation.

The paper's summary: Kai: So to summarize what this paper is doing, they are investigating the aging dynamics of TEAC after fast temperature up steps with amplitudes between zero point three K and thirteen point six K, specifically to test the Tool-Narayanaswamy formalism for nonlinear relaxation in glasses <ref:2607.15988#pg1,fast temperature up steps with amplitudes>.

Mira: They set up an experiment starting from either an equilibrium state or a state made out-of-equilibrium by a prior down step experiment, measuring the sample's response through its dielectric permittivity.

Lev: The core idea is to link the non-linear re-equilibration of a liquid to its linear response to a small perturbation, which is what the TN formalism assumes.

Kai: They then determined parameters for steps under three point three K and used down step aging experiments below Tg to constrain the equilibrium relaxation time determination for those smaller steps <ref:2607.15988#pg1,down step aging experiments below>.

Mira: They confirmed good agreement for differences in fictive temperature up to ten K, which is a measure of the distance from equilibrium in temperature units, characterizing how far they are from that starting point <ref:2607.15988#pg1,good agreement for differences in fictive temperature>.

Lev: But they also found limits on the formalism; specifically, when the prediction for ttr, fifty percent divided by tau1 is not well predicted for steps with delta Tf greater than ten K.

Kai: This suggests that the TN formalism breaks down when you move into larger temperature changes because the underlying re-equilibration mechanism changes fundamentally.

Mira: They also addressed another hurdle: extrapolating the Vogel-Fulcher-Tammann law for relaxation times below Tg, which they constrained using aging experiments following temperature down steps.

Lev: They parameterized the relaxation time below Tlim by assuming an Arrhenius dependence with an activation energy Ea of two hundred thirty-five kJ·mol−one and a limit temperature Tlim of two hundred five point three K, which gave them good agreement for certain starting temperatures like T0 = one hundred ninety-six K and one hundred ninety-four point six K.

Kai: It’s interesting how they handled the low-temperature extrapolation by tying it to down step constraints rather than just a simple fit of the Vogel-Fulcher-Tammann law.

Mira: The conclusion is that the TN formalism is equivalent to the SPA ansatz in this explored regime because they could determine its parameters for moderate amplitude up steps and estimate the equilibrium relaxation time below Tg in a self-consistent manner.

Lev: So, basically, they show that for TEAC, you can get good results using these formalisms as long as you respect those specific constraints on the step size and temperature.

Kai: The main thing to remember is that this formalism has a clear limit based on the ratio of relaxation times, tau (zero plus)/tau1 of approximately one hundred two versus ttr, fifty percent/tau1 of approximately ten for steps with delta Tf greater than ten K.

The paper's improvements: Kai: The authors suggest a few ways to improve the study and the formalism itself. First, they propose using a material time approach that accounts for out-of-equilibrium states by using an equation like Tf i+one = Ti+one minus sum j<=i M(xi+one − xj) dT/dxi j Delta xi j <ref:2607.15988#pg2>.

Mira: That material time approach is interesting because it allows the system to evolve based on its entire thermal history instead of just looking at a single temperature step, which seems like a significant improvement for modeling complex dynamics.

Lev: From an engineering standpoint, incorporating the full thermal history into that equation would mean the error term lands in a much more complex way when you try to run it on real hardware.

Kai: They also suggest developing a general criterion for the limit of validity of the TN formalism by comparing it to other systems from the literature, specifically noting that large steps beyond delta Tf = ten K show noticeable discrepancies in curve shape and dynamics <ref:2607.15988#pg1>.

Mira: I agree with that, identifying a systematic failure threshold based on temperature difference seems like a much more useful way to understand the physics than just observing where it breaks down empirically.

Lev: That kind of criterion would be very helpful if we were trying to build error correction schemes because we need to know exactly when our model is going to fail.

Kai: They also suggest using the initial contrast tau (zero plus)/tau1 as a driving force for characterizing aging experiments instead of relying solely on step amplitude, since that metric correlates better with the limit of validity for the TN formalism <ref:2607.15988#pg1>.

Mira: Using that contrast as a measure seems much more physically grounded than just using the step amplitude, especially if it relates to how far away you are from equilibrium.

Lev: If we can link those two metrics together, it could help us understand why they both correlate with that limit more robustly for real hardware testing.

Kai: And finally, they suggest refining the model for low-temperature relaxation times below Tg by combining an Arrhenius law with constraints derived from temperature down steps to ensure continuity of the derivative to get a more robust extrapolation than simple Vogel-Fulcher-Tammann fits.

Mira: That sounds like a way to make the low-temperature behavior more reliable without just relying on fitting functions that might not capture the underlying physics perfectly.

Lev: For building actual hardware, having a model that respects the physical constraints from down steps is definitely better than just using an Arrhenius law blindly if we need accuracy.

Kai: So they’re basically saying we should move toward a more history-dependent model and establish clearer rules for when the TN formalism stops being reliable based on those contrasts in relaxation times.

Conclusion: Mira: So to wrap up, the paper "Fast temperature up steps as a test of the Tool-Narayanaswamy formalism" found that while it works well for moderate amplitude steps and small amplitudes, it fails when you go beyond that threshold.

Kai: And they showed that for TEAC, the TN formalism is equivalent to the SPA ansatz in this regime because they could determine its parameters self-consistently.

Lev: From an engineering perspective, it means we need to build systems where we know exactly when our model will break down so we can design things that don't rely on assumptions being true for every temperature range.

Kai: They established a clear limit based on the contrast between tau (zero plus)/tau1 of approximately one hundred two and ttr, fifty percent/tau1 of approximately ten for steps with delta Tf greater than ten K.

Mira: So, if you’re looking at your own experiments, that contrast is what tells you when the TN formalism is no longer giving you reliable answers for large steps.

Lev: For me, that means we need to keep building models that account for those history effects because if we ignore them, the overhead of error correction will be huge later on.

Kai: We’re moving toward a model where the prediction depends on how much history the system has accumulated, not just one quick step.

Mira: And this work with TEAC provides a concrete example of how these formalisms can be used to probe complex aging dynamics in real materials.

Lev: It’s good to see that we have a clearer picture of the limitations for what we're trying to do for real hardware testing when you’re dealing with non-equilibrium states.

SPEC · CEA · CNRS · University Paris Saclay

cond-mat.soft

Submitted: 2026-07-17

Updated: 2026-10-29

Comments: 10 pages, 9 figures

Journal ref: J. Chem. Phys. 165, 124514 (2026)

DOI: 10.1063/5.0352723

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 69/100

The gist: Fast temperature up steps as a test of the Tool-Narayanaswamy formalism investigates aging dynamics in triethyl-2-acetylcitrate (TEAC) following fast temperature up steps with amplitudes ranging from

Terminology

Summary

Fast temperature up steps as a test of the Tool-Narayanaswamy formalism investigates aging dynamics in triethyl-2-acetylcitrate (TEAC) following fast temperature up steps with amplitudes ranging from 0.3 to 13.6 K, testing the predictive power of the Tool-Narayanaswamy (TN) formalism for nonlinear relaxation dynamics in glasses

Experimental Setup and Protocol

The study investigated the aging dynamics of TEAC following fast temperature up steps with amplitudes ranging from 0.3 to 13.6 K, starting from either an equilibrium or an out-of-equilibrium state The initial states were either at equilibrium or prepared at increasing levels of out-of-equilibrium through a prior down step experiment Measurements were performed on a triethyl-2-acetylcitrate sample (Sigma-Aldrich, 99%, Tg = 199 K) through an experimental setup previously described in ref.6 and which we briefly explain here Up-steps from T0 to T1 were performed by applying a tup = 40 ms continuous current pulse through the resistive electrode, which rapidly heats the entire sample due to its thinness (see Fig. 1a) Both types of step were characterized (see Fig. 1b) by operating at a temperature T0 = 185 K, small enough compared to Tg so that no relaxation occurs on the measurement time window The dielectric permittivity of the liquid ϵ⋆ can thus be used as a thermometer

TN Parameter Determination

The TN formalism is assumed to link the non-linear re-equilibration of a liquid to its linear response to a small perturbation The authors determined the TN parameters for steps with small to moderate amplitude (≤ 3.3 K) and crucially took advantage of down step aging experiments below the glass transition temperature to constrain the determination of the equilibrium relaxation time For these smaller steps, they found very good agreement for differences in fictive temperature characterizing the distance from equilibrium as high as 10 K The memory kernel M was fitted as a stretched exponential M(ξ) = e−(ξ/τ0)β with parameters τ0 = 0.98 and β = 0.58

Testing the Formalism Limits

For larger steps, the prediction progressively fails to capture the aging dynamics, likely indicating that the re-equilibration mechanism is no longer related to the equilibrium dynamics The authors identified a limit for the validity of the TN for full re-equilibration dynamics based on whether ttr, 50% is well predicted or not This limit corresponds to a contrast between the initial and final relaxation time τ (0+)/τ1 of ≈ 102 and a transformation time ttr, 50%/τ1 ≈ 10 for steps with ∆Tf > 10 K

Low-Temperature Extrapolation

A further difficulty addressed was the extrapolation of the VFT law for τα(T) below Tg, which was constrained using aging experiments following temperature down steps They parameterized τα(T) for T < Tlim by assuming an Arrhenius dependence with an activation energy Ea = 235 kJ·mol−1 and Tlim = 205.3 K This constraint led to a good agreement, especially for T0 = 196 K and 194.6 K

Conclusion on Validity

The TN formalism proved to be equivalent to the SPA ansatz in the regime explored, by determining its parameters for moderate amplitude up steps (∆T ≤ 3.3 K) and by estimating the equilibrium relaxation time below Tg in a self-consistent manner The discrepancy observed for large steps corresponds to a contrast between the initial and final relaxation time τ (0+)/τ1 of ≈ 102 and a transformation time ttr, 50%/τ1 ≈ 10 This threshold for the limit of validity of the TN appears consistent with previous work from the literature on others systems

Comparison to Single Parameter Ansatz

The single parameter aging (SPA) ansatz was shown to give an excellent agreement with aging experiments on molecular liquids, and in this work, it was found that the SPA ansatz provides aging predictions of comparable quality to those of the TN approach The two approaches yield similar results because for TEAC, ln τα(T) is almost linear with T This suggests that the simplest expression, Eq.

Improvements for AI systems

  1. Improve prediction accuracy for glass aging dynamics by using a material time approach that accounts for out-of-equilibrium states, as described by Eq. 5: Tf i+1 = Ti+1 - Sum j<=i M(xi+1 − xj) dT/dxi j Delta xi j. This allows the system to evolve based on its entire thermal history rather than just a single temperature step.

  2. Develop a general criterion for the limit of validity of the Tool-Narayanaswamy (TN) formalism, comparing it to other systems from the literature, specifically identifying that large steps beyond ∆Tf = 10 K show noticeable discrepancies in curve shape and dynamics.

  3. Enhance system characterization by using initial contrast τ (0+)/τ1 as a driving force for characterizing aging experiments instead of relying solely on step amplitude, as this metric correlates better with the limit of validity for the TN formalism.

  4. Create a refined model for low-temperature relaxation times below Tg by combining an Arrhenius law with constraints derived from temperature down steps, ensuring continuity of the derivative to obtain a more robust extrapolation than simple Vogel-Fulcher-Tammann fits.

  5. Improve prediction robustness for out-of-equilibrium initial states by incorporating the memory effects where the less time they are given to equilibrate at T0, the more they remember their previous state through memory effects, and thus the faster they return to it. This allows accurate prediction of relaxation following double step experiments.

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